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In previous lessons, the trigonometric ratios and their properties were studied. 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 $θ.$

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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"]}},"text":"<span class=\"mlmath-simple\"><\/span>"},"formTextBefore":"<span class=\"mlmath-simple\"><span class=\"variable\">m<\/span>\u22201<span class=\"space big-before big-after\">=<\/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.

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{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":false,"useShortLog":false,"variables":["x"],"constants":["PI"]}},"text":"<span class=\"mlmath-simple\"><\/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.

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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?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?

Round the answer to the first decimal place.{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]}},"text":"<span class=\"mlmath-simple\"><\/span>"},"formTextBefore":"<span class=\"mlmath-simple\"><span class=\"text\">Height<\/span><span class=\"space big-before big-after\">=<\/span><\/span>","formTextAfter":"m","answer":{"text":["27.5"]}}

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?

Start by drawing an angle at which the man is looking at the top of the building. Then identify a right triangle.

The length of the$tanθtan70_{∘} =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

$h≈27.5$

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? ### Hint

### Solution

Therefore, in order to land successfully, the plane should descend at an angle of $7_{∘}.$

Round the answer to the closest degree.

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]}},"text":"<span class=\"mlmath-simple\"><\/span>"},"formTextBefore":"<span class=\"mlmath-simple\"><span class=\"text\">Angle of Descent<\/span><span class=\"space big-before big-after\">=<\/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"]}}

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.

First, draw a right triangle, whose hypotenuse shows the expected path of descent of the plane and label the given distances on the diagram.

Since the lengths of the opposite side and the hypotenuse are given, the angle of descent can be calculated by using the sine ratio.$sinθsinθ =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_{-1}14017 $

UseCalc

Use a calculator

$θ=6.974556…$

RoundDec

Round to

$θ≈7_{∘}$

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.

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.

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.

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?

Round the answer to the closest integer.{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]}},"text":"<span class=\"mlmath-simple\"><\/span>"},"formTextBefore":"<span class=\"mlmath-simple\"><span class=\"text\">Height<\/span><span class=\"space big-before big-after\">=<\/span><\/span>","formTextAfter":"<span class=\"mlmath-simple\"><span class=\"text\">ft<\/span><\/span>","answer":{"text":["181"]}}

Which trigonometric ratio describes the ratio between the height and the length of the ski hill in relation to the angle of depression?

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.

It is also known that the length of the ski hill is 870 feet. Let h represent the height of the hill.

The ratio between height h and the length of the ski hill is described by the sine of m∠1.$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

$h≈181$

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. Use the Pythagorean Theorem to calculate the distance that the ball travels. Round the answer to the closest integer. ### Hint

### Solution

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.
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.
Now, the equation needs to be solved for c.
It was derived that if Mark takes a shot 17.3 feet away from the goal post, the ball will travel approximately 20 feet.

Find the farthest Mark's position from the goal post at which he can still score. Round the answer to the first decimal place.

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{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]}},"text":"<span class=\"mlmath-simple\"><\/span>"},"formTextBefore":null,"formTextAfter":"<span class=\"mlmath-simple\"><span class=\"text\">ft<\/span><\/span>","answer":{"text":["20"]}}

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.

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.

Using the tangent of $θ,$ the distance d can be calculated.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

$c≈20$

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 $75_{∘}.$

Find the length of the awning they should buy. Round the answer to the first decimal place.{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]}},"text":"<span class=\"mlmath-simple\"><\/span>"},"formTextBefore":"<span class=\"mlmath-simple\"><span class=\"text\">Length<\/span><span class=\"space big-before big-after\">=<\/span><\/span>","formTextAfter":"ft","answer":{"text":["1.9"]}}

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?

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.

Let $ℓ$ represent the length of the awning. In order to find its value, the cotangent ratio can be used.$cotm∠1cot75_{∘} =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

$ℓ≈1.9$

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.

The radius of the Earth is 3963 miles and the height of Denali is 3.8 miles. Use the given information to determine whether the curvature of the earth will block her line of sight.{"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"]}},"text":"<span class=\"mlmath-simple\"><\/span>"},"formTextBefore":"<span class=\"mlmath-simple\"><span class=\"text\">Angle of Depression<\/span><span class=\"space big-before big-after\">=<\/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"]}}

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.

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.

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.

Note that the length of the hypotenuse of the right triangle is equal to the sum of the Earth's radius and the height of the mountain. Applying the Pythagorean Theorem to the triangle, the following equation can be obtained.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+h_{2} $

Next, to find the angle of depression from the top of the mountain to the horizon, any of the trigonometric ratios can be used.

The tangent of $θ$ is equal to the ratio of the length of the opposite side r to the length of the adjacent side d.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"]}},"text":"<span class=\"mlmath-simple\"><\/span>"},"formTextBefore":"<span class=\"mlmath-simple\"><span class=\"text\">Length<\/span><span class=\"space big-before big-after\">=<\/span><\/span>","formTextAfter":"ft","answer":{"text":["26"]}}

Use one of the trigonometric ratios to determine the horizontal distance from the submarine to the whale's nose and tale.

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_{∘}=d_{1}9 $

$d_{1}≈10$

$tan14_{∘}=d_{2}9 $

$d_{2}≈36$

$Length =36−10=26ft $

Therefore, the whale is approximately 26 feet long.
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