8. Application of Trigonometric Ratios
Sign In
An error ocurred, try again later!
Chapter 2
8.
Application of Trigonometric Ratios
Trigonometry is not merely a collection of abstract concepts; it permeates our daily experiences. The lesson sheds light on the pivotal role of trigonometry, especially as it relates to right triangles, in real-world contexts. Whether determining distances in navigation, analyzing architectural designs, or decoding the mechanics of objects in motion, the applications of trigonometry are vast and varied. Furthermore, by focusing on right triangles, one gains a deeper appreciation for the foundational relationships that guide various phenomena in our environment. Embracing these principles can lead to a more informed perspective on both the natural and man-made world around us.
Show less Show more expand_more Lesson Settings & Tools
| | 10 Theory slides |
| | 9 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
The trigonometry of a right triangle, along with the Pythagorean Theorem, can be used to solve a wide range of real-world problems. In this lesson, some cases will be presented.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Try your knowledge on these topics.
a Choose the correct values of sine, cosine, and tangent of θ.
{"type":"pair","form":{"alts":[[{"id":0,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.69444em;vertical-align:0em;\"><\/span><span class=\"mop\">sin<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right:0.02778em;\">\u03b8<\/span><\/span><\/span><\/span>"},{"id":1,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.69444em;vertical-align:0em;\"><\/span><span class=\"mop\">cos<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right:0.02778em;\">\u03b8<\/span><\/span><\/span><\/span>"},{"id":2,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.69444em;vertical-align:0em;\"><\/span><span class=\"mop\">tan<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord mathdefault\" style=\"margin-right:0.02778em;\">\u03b8<\/span><\/span><\/span><\/span>"}],[{"id":0,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:2.00744em;vertical-align:-0.686em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">7<\/span><span class=\"mord\">.<\/span><span class=\"mord\">2<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">4<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span>"},{"id":1,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:2.00744em;vertical-align:-0.686em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">7<\/span><span class=\"mord\">.<\/span><span class=\"mord\">2<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">6<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span>"},{"id":2,"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:2.00744em;vertical-align:-0.686em;\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.32144em;\"><span style=\"top:-2.314em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">6<\/span><\/span><\/span><span style=\"top:-3.23em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"frac-line\" style=\"border-bottom-width:0.04em;\"><\/span><\/span><span style=\"top:-3.677em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord\">4<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.686em;\"><span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><\/span><\/span><\/span>"}]],"lockLeft":false,"lockRight":false},"formTextBefore":"","formTextAfter":"","answer":[[0,1,2],[0,1,2]]}
b Find the measure of acute angle ∠1. Round your answer to the closest degree.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.69224em;vertical-align:0em;\"><\/span><span class=\"mord mathdefault\">m<\/span><span class=\"mord\">\u2220<\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.674115em;vertical-align:0em;\"><\/span><span class=\"mord\"><span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.674115em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","answer":{"text":["51"]}}
c Write the given expressions in terms of sine or cosine. Write your answer without the degree symbol.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.674115em;vertical-align:0em;\"><\/span><span class=\"mop\">sin<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\">3<\/span><span class=\"mord\"><span class=\"mord\">4<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.674115em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["\\cos(56)"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.674115em;vertical-align:0em;\"><\/span><span class=\"mop\">cos<\/span><span class=\"mspace\" style=\"margin-right:0.16666666666666666em;\"><\/span><span class=\"mord\">5<\/span><span class=\"mord\"><span class=\"mord\">1<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.674115em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["\\sin(39)"]}}
d Determine the value of x that makes the equation true.
cos28∘=sinx∘
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.43056em;vertical-align:0em;\"><\/span><span class=\"mord mathdefault\">x<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["62"]}}
Challenge
Measuring Depths Using Trigonometry
A submarine sailed underwater near a whale that was swimming 9 feet deeper than the submarine. A sonar on that submarine measured the angles of depression to the whale's mouth and tail to be 42∘ and 14∘, respectively.
How can the sailors use this information to measure the length of the whale?
Example
Measuring Heights Using Trigonometry
On the way to work, a man started wondering how tall his office building is. Suppose he is standing 10 meters from the building, looking up toward the rooftop at an approximate angle of 70∘. What is the height of the building?
External credits: @freepik
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9580078125em;vertical-align:-0.2080078125em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Regular\">Height<\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":"m","answer":{"text":["27.5"]}}
Hint
Draw an angle at which the man is looking at the top of the building. Which trigonometric ratio can be used to find the building's height?
Solution
Start by drawing an angle at which the man is looking at the top of the building. Then identify a right triangle.
External credits: @freepik
tanθ tan70∘=AdjacentOpposite⇓= 10h
By solving this equation, the value of h can be found.
tan70∘=10h
MultEqn
LHS⋅10=RHS⋅10
10tan70∘=h
RearrangeEqn
Rearrange equation
h=10tan70∘
UseCalc
Use a calculator
h=27.474774…
RoundDec
Round to 1 decimal place(s)
h≈27.5
Example
Measuring Angles Using Trigonometry
A plane arriving at O'Hare International Airport is 34 meters above the ground and 280 meters from the expected touchdown point on a runway. At what angle is the plane supposed to descend to land successfully? 
Round the answer to the closest degree. 
Since the lengths of the opposite side and the hypotenuse are given, the angle of descent can be calculated by using the sine ratio.
Therefore, in order to land successfully, the plane should descend at an angle of 7∘.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.96826171875em;vertical-align:-0.2080078125em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Regular\">Angle<\/span><span class=\"mord\">\u00a0<\/span><span class=\"mord Roboto-Regular\">of<\/span><span class=\"mord\">\u00a0<\/span><span class=\"mord Roboto-Regular\">Descent<\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.674115em;vertical-align:0em;\"><\/span><span class=\"mord\"><span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.674115em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","answer":{"text":["7"]}}
Hint
Draw a right triangle so that the hypotenuse shows the expected path of the plane's descent. Analyze which side length is given to determine which trigonometric ratio should be used.
Solution
First, draw a right triangle, whose hypotenuse shows the expected path of descent of the plane and label the given distances on the diagram.
sinθ sinθ=HypotenuseOpposite⇓= 28034
To solve this equation for θ, the inverse of sine can be used.
sinθ=28034
ReduceFrac
ba=b/2a/2
sinθ=14017
sin-1(LHS)=sin-1(RHS)
θ=sin-114017
UseCalc
Use a calculator
θ=6.974556…
RoundDec
Round to 1 decimal place(s)
θ≈7∘
Discussion
Investigating Angles of Elevations and Depressions
In the previous two examples, the angles mentioned can be called the angle of elevation and angle of depression, respectively. To be able to refer to these angles, they should first be properly defined.
Concept
Angle of Elevation
An angle of elevation is defined as an angle between the horizontal plane and oblique line from the observer's eye to some object above. Suppose someone is standing on the ground looking up toward the top of a tree.
Concept
Angle of Depression
An angle of depression is defined as an angle between the horizontal plane and oblique line from the observer's eye to some object below. Suppose someone is standing on a cliff looking down toward a ship in the ocean below.
Example
Using Angles of Depressions to Solve Problems
A snowboarder is sliding down a ski hill that is 870 feet long. Its angle of depression is 12∘. What is the height of the ski hill?
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9580078125em;vertical-align:-0.2080078125em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Regular\">Height<\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.77001953125em;vertical-align:-0.009765625em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Regular\">ft<\/span><\/span><\/span><\/span><\/span>","answer":{"text":["181"]}}
Hint
Which trigonometric ratio describes the ratio between the height and the length of the ski hill in relation to the angle of depression?
Solution
It is given that the angle of depression of the ski hill is 12∘. That angle can be identified on the diagram as ∠θ. To determine the height of the hill, a line parallel to ℓ can be drawn at the bottom of the hill. By the Alternate Interior Angles Theorem, ∠θ and ∠1 are congruent angles.
External credits: @macrovector
It is also known that the length of the ski hill is 870 feet. Let h represent the height of the hill.
External credits: @macrovector
sinm∠1=870h
By substituting m∠1 with 12∘, the value of h can be calculated.
sinm∠1=870h
Substitute
m∠1=12∘
sin12∘=870h
▼
Solve for h
MultEqn
LHS⋅870=RHS⋅870
870sin12∘=h
RearrangeEqn
Rearrange equation
h=870sin12∘
UseCalc
Use a calculator
h=180.883171…
RoundInt
Round to nearest integer
h≈181
Example
Using Angles of Elevations to Solve Problems
Mark, who plays fútbol for South High School, takes a shot into a 10-foot tall goal post. The ball travels at a 30-degree angle of elevation toward the center of the goal. 
Find the farthest Mark's position from the goal post at which he can still score. Round the answer to the first decimal place. Use the Pythagorean Theorem to calculate the distance that the ball travels. Round the answer to the closest integer. 
Using the tangent of θ, the distance d can be calculated.
Next, using the Pythagorean Theorem, the distance that the ball travels can be determined. Note that the lengths of the triangle's legs are known, so they can be substituted into the formula.
It was derived that if Mark takes a shot 17.3 feet away from the goal post, the ball will travel approximately 20 feet.
External credits: @macrovector
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.77001953125em;vertical-align:-0.009765625em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Regular\">ft<\/span><\/span><\/span><\/span><\/span>","answer":{"text":["17.3"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.77001953125em;vertical-align:-0.009765625em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Regular\">ft<\/span><\/span><\/span><\/span><\/span>","answer":{"text":["20"]}}
Hint
Draw a right triangle formed by the ball and the goal post. Analyze the given side length and angle measures to determine which trigonometric ratio should be used.
Solution
Draw a right triangle based on the position of the ball and the goal post. In order to score, the ball should not rise higher than 10 feet. Therefore, to find the farthest position of Mark from the goal post, let one leg of the triangle be 10 feet long.
External credits: @macrovector
tanθ=d10
Substitute θ with 30∘ and solve the equation for d.
Therefore, standing 17.3 feet away from the goal post is the farthest position from which Mark can score a goal. External credits: @macrovector
a2+b2=c2⇓102+17.32=c2
Now, the equation needs to be solved for c.
102+17.32=c2
▼
Solve for c
CalcPow
Calculate power
100+299.29=c2
AddTerms
Add terms
399.29=c2
RearrangeEqn
Rearrange equation
c2=399.29
SqrtEqn
LHS=RHS
c=19.982242…
RoundInt
Round to nearest integer
c≈20
Example
Real Life Applications of Trigonometry
A family has a 7-foot tall sliding-glass door leading to the backyard. They want to buy an awning for the door that will be long enough to keep the Sun out when the Sun is at its highest point with an angle of elevation of 75∘.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9580078125em;vertical-align:-0.2080078125em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Regular\">Length<\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":"ft","answer":{"text":["1.9"]}}
Hint
Identify parallel lines and use the corresponding theorem to find the measure of an interior angle of a right triangle. Which trigonometric ratio can be used to find the length of the awning?
Solution
First, note that the concrete entrance way is parallel to the awning, so by the Alternate Interior Angles Theorem, ∠1 and the 75-degree angle are congruent angles.
cotm∠1 cot75∘=OppositeAdjacent⇓= 7ℓ
By solving the above equation, the value of ℓ can be found.
cot75∘=7ℓ
MultEqn
LHS⋅7=RHS⋅7
7cot75∘=ℓ
RearrangeEqn
Rearrange equation
ℓ=7cot75∘
UseCalc
Use a calculator
ℓ=1.875644…
RoundDec
Round to 1 decimal place(s)
ℓ≈1.9
Example
Using Trigonometry to Measure Distances on Earth
A mountaineer is planning to climb the highest mountain in the US, Denali, located in Alaska. When she reaches the peak, she wonders if she would be able to see the most eastern point of Russia, about 660 miles away from Denali.
{"type":"choice","form":{"alts":["The curvature does block her line of sight.","The curvature does not block her line of sight."],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.96826171875em;vertical-align:-0.2080078125em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Regular\">Angle<\/span><span class=\"mord\">\u00a0<\/span><span class=\"mord Roboto-Regular\">of<\/span><span class=\"mord\">\u00a0<\/span><span class=\"mord Roboto-Regular\">Depression<\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.674115em;vertical-align:0em;\"><\/span><span class=\"mord\"><span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.674115em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","answer":{"text":["87"]}}
Hint
Use the Pythagorean Theorem to find the distance from the top of the mountain to the horizon. The angle of depression can be found using one of the trigonometric ratios.
Solution
The exercise will be solved in two steps. First, the distance to the horizon will be calculated to find the answer to the given question. Then, the angle of depression will be determined.
Finding the Distance to the Horizon
Begin by representing the problem with a diagram. The horizon, or the farthest point the person could see, is the point where the line of sight of the mountaineer is tangent to the circle of the Earth. Let point C represent the center of the globe.
r2+d2=(r+h)2
First, it should be solved for d.
r2+d2=(r+h)2
▼
Solve for d
SubEqn
LHS−r2=RHS−r2
d2=(r+h)2−r2
ExpandPosPerfectSquare
(a+b)2=a2+2ab+b2
d2=r2+2rh+h2−r2
SubTerm
Subtract term
d2=2rh+h2
SqrtEqn
LHS=RHS
d=2rh+h2
Finding the Angle of Depression
Next, to find the angle of depression from the top of the mountain to the horizon, any of the trigonometric ratios can be used.
tanθ=dr
By substituting r with 3963 and d with 173.6, the measure of ∠θ can be determined.
The angle of depression measures approximately 87∘.
Closure
Measuring the Length of the Whale
Throughout the lesson, different real-life problems have been solved using trigonometric ratios. Through the use of the learned methods, the challenge presented at the beginning can now be solved.
A submarine sailed underwater near a whale that was swimming 9 feet deeper than the submarine. A sonar on that submarine measured the angles of depression to the whale's mouth and tail to be 42∘ and 14∘, respectively.
How can the sailors use this information to measure the length of the whale?
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9580078125em;vertical-align:-0.2080078125em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Regular\">Length<\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":"ft","answer":{"text":["26"]}}
Hint
Use one of the trigonometric ratios to determine the horizontal distance from the submarine to the whale's nose and tale.
Solution
It is given that the angle of depression from the submarine to the whale's front is 42∘. Mark this information on the diagram and form a corresponding right triangle.
Using the tangent ratio, the horizontal distance from the submarine to the whale labeled as d1 can be found.
tan42∘=d19
d1≈10
tan14∘=d29
d2≈36
Length=36−10=26 ft
Therefore, the whale is approximately 26 feet long.
The wooden box has an inside base area of 40×50 square centimeters and a height of 30 centimeters.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"cm","answer":{"text":["70.7"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.674115em;vertical-align:0em;\"><\/span><span class=\"mord\"><span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.674115em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","answer":{"text":["25.1"]}}
security
report
code
security
report
code
The longest pole we can fit inside the box would fit either between the upper left and lower right corner or the box's upper right and lower-left corner. Each combination of corners the pole touches would be on opposite sides of the box.
Notice that the pole forms the hypotenuse of a right triangle. In this case, the shorter leg is the box's height and the longer leg is the diagonal of the base area of the box.
To calculate the diagonal of the box's base area, we can use the Pythagorean Theorem.
The diagonal of the base area of the box is 10sqrt(41) centimeters.
Now we can calculate the length of the longest pole by using the Pythagorean Theorem once more.
The longest pole that fits inside the box is about 70.7 centimeters.
Let's mark the angle of elevation between the pole and the floor of the box.
Let's calculate θ by using the tangent ratio.
The angle of elevation is about 25.1 ^(∘).
security
report
code
A company uses drones to deliver packages from their warehouse to various locations in a big city. The streets are in a uniform grid with 0.1 miles between each street. A package needs to be delivered to a location 7 blocks north and 15 blocks west of the warehouse.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.674115em;vertical-align:0em;\"><\/span><span class=\"mord\"><span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.674115em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","answer":{"text":["65"]}}
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"miles","answer":{"text":["1.7"]}}
security
report
code
security
report
code
Let's make a graph of the situation. We will let the warehouse W be at the origin of a coordinate plane. This means the delivery address D, will be at the coordinates (15,7). We will also label the angle with which we will send the drone as θ.
To determine θ, we will use the tangent ratio. tan θ = Opposite/Adjacent From the diagram, we see that the opposite side is 15 and the adjacent side is 7. With this information, we can determine θ.
The drone should be sent off at an angle of about 65^(∘) to the north.
From the exercise, we know that the distance from one street to the next is 0.1 miles. This means the warehouse is 0.7 miles north and 1.5 miles east of the warehouse. Let's add this to the diagram.
Since the route of the drone is the hypotenuse of a right triangle, we can determine the distance the drone must fly by using the Pythagorean Theorem.
The delivery address is about 1.7 miles away.
security
report
code
A semicircular tunnel is measured to be 30 feet high at its highest point. One evening, Jordan is hoping to drive through the tunnel while driving her mobile home. Will she make it through if the vehicle measures to be 24 feet wide and 28 feet high?
{"type":"choice","form":{"alts":["Yes","No"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}
security
report
code
security
report
code
Let's add the dimensions of the tunnel and of the mobile home to the diagram. We will assume that Jordan is able to drive perfectly in the middle of the road.
If the vehicle can make it through, the distance from the ground to the ceiling must be greater than 28 feet at the top left and top right corners of the mobile home.
Since Jordan is driving in the middle of the tunnel, the distance from the ground to the ceiling at the critical point of the vehicle's corner, can be viewed as the leg of a right triangle with a hypotenuse equal to the tunnel's radius. The second leg will be half the width of the vehicle.
Using the Pythagorean Theorem, we can calculate a. Once again, if it is less than 28 feet, Jordan will not make it through.
The height of the tunnel where the sides of the mobile house would pass through, is less than 28 feet. Therefore, Jordan can not drive her mobile home through the tunnel. Good thing she did the math before trying!
security
report
code
Dominika is running across the diagonal of a rectangular parking lot.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]},"rendered":false},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.674115em;vertical-align:0em;\"><\/span><span class=\"mord\"><span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.674115em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","answer":{"text":["26.6"]}}
security
report
code
security
report
code
The diagonal can be seen as the hypotenuse of a right triangle where the parking lot's width is the triangle's shorter leg. We also know that the hypotenuse is 123 % greater than the shorter leg. This means that its length is 2.23 times the length of the leg. Therefore, if we label the shorter leg x, the hypotenuse becomes 2.23x.
With the given information, we can determine ∠ θ by using the sine ratio.
security
report
code
Application of Trigonometric Ratios
Recommended exercises
To get personalized recommended exercises, please login first.
Loading content
Loading content