8. Application of Trigonometric Ratios
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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.
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| | 10 Theory slides |
| | 9 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Application of Trigonometric Ratios
Slide of 10
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.
Here are a few recommended readings before getting started with this lesson.
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.
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c Write the given expressions in terms of sine or cosine. Write your answer without the degree symbol.
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d Determine the value of x that makes the equation true.
cos 28^(∘)=sin x^(∘)
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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?
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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
Round the answer to the first decimal place.
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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
The length of the adjacent side of an acute angle of 70^(∘) is known, and the length of the opposite side should be found. Therefore, the tangent ratio can be applied. tan θ &=Opposite/Adjacent & ⇓ tan 70^(∘) &= h/10 By solving this equation, the value of h can be found.
tan 70^(∘) = h/10
MultEqn
LHS * 10=RHS* 10
10tan 70^(∘) = h
RearrangeEqn
Rearrange equation
h = 10tan 70^(∘)
UseCalc
Use a calculator
h = 27.474774...
RoundDec
Round to 1 decimal place(s)
h ≈ 27.5
The height of the office building is approximately 27.5 meters.
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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?
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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.
Since the lengths of the opposite side and the hypotenuse are given, the angle of descent can be calculated by using the sine ratio. sin θ &=Opposite/Hypotenuse & ⇓ sin θ &= 34/280 To solve this equation for θ, the inverse of sine can be used.
sin θ = 34/280
ReduceFrac
a/b=.a /2./.b /2.
sin θ = 17/140
sin^(-1)(LHS) = sin^(-1)(RHS)
θ = sin^(-1) 17/140
UseCalc
Use a calculator
θ = 6.974556...
RoundDec
Round to 1 decimal place(s)
θ ≈ 7^(∘)
Therefore, in order to land successfully, the plane should descend at an angle of 7^(∘).
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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.
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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?
Round the answer to the closest integer.
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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
The ratio between height h and the length of the ski hill is described by the sine of m∠ 1. sin m∠ 1 = h/870 By substituting m∠ 1 with 12^(∘), the value of h can be calculated.
sin m∠ 1 = h/870
Substitute
m∠ 1= 12^(∘)
sin 12^(∘) = h/870
▼
Solve for h
MultEqn
LHS * 870=RHS* 870
870sin 12^(∘) = h
RearrangeEqn
Rearrange equation
h = 870sin 12^(∘)
UseCalc
Use a calculator
h = 180.883171...
RoundInt
Round to nearest integer
h ≈ 181
It can be concluded that the height of the hill is approximately 181 feet.
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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.
External credits: @macrovector
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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
Using the tangent of θ, the distance d can be calculated. tan θ = 10/d 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
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. a^2+b^2=c^2 ⇓ 10^2+17.3^2=c^2 Now, the equation needs to be solved for c.
10^2+17.3^2=c^2
▼
Solve for c
CalcPow
Calculate power
100+299.29=c^2
AddTerms
Add terms
399.29=c^2
RearrangeEqn
Rearrange equation
c^2=399.29
SqrtEqn
sqrt(LHS)=sqrt(RHS)
c=19.982242...
RoundInt
Round to nearest integer
c≈ 20
It was derived that if Mark takes a shot 17.3 feet away from the goal post, the ball will travel approximately 20 feet.
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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^(∘).
Find the length of the awning they should buy. Round the answer to the first decimal place.
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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.
Let l represent the length of the awning. In order to find its value, the cotangent ratio can be used. cot m∠ 1 &=Adjacent/Opposite & ⇓ cot 75^(∘) &= l/7 By solving the above equation, the value of l can be found.
cot 75^(∘) = l/7
MultEqn
LHS * 7=RHS* 7
7cot 75^(∘) = l
RearrangeEqn
Rearrange equation
l = 7cot 75^(∘)
UseCalc
Use a calculator
l = 1.875644...
RoundDec
Round to 1 decimal place(s)
l ≈ 1.9
The length of the awning should be at least 1.9 feet long.
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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.
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.
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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.
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. r^2+d^2=(r+h)^2 First, it should be solved for d.
r^2+d^2=(r+h)^2
▼
Solve for d
SubEqn
LHS-r^2=RHS-r^2
d^2=(r+h)^2-r^2
ExpandPosPerfectSquare
(a+b)^2=a^2+2ab+b^2
d^2=r^2+2rh+h^2-r^2
SubTerm
Subtract term
d^2=2rh+h^2
SqrtEqn
sqrt(LHS)=sqrt(RHS)
d=sqrt(2rh+h^2)
The radius of Earth is 3963 miles and the height of Denali is 3.8 miles (20 310 feet). By substituting these values for r and h, respectively, the value of d can be calculated.
This means that when standing on the peak of Denali, it is possible to see locations up to 173.6 miles away. However, the most eastern point of Russia is located 660 miles away. Therefore, it is not visible from Denali. Hence, the curvature of the Earth blocks the mountaineer's line of sight.
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.
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. tan θ = r/d By substituting r with 3963 and d with 173.6, the measure of ∠ θ can be determined.
The angle of depression measures approximately 87^(∘).
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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?
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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 d_1 can be found.
tan 42^(∘) = 9/d_1
d_1≈ 10
Next, the angle of depression from the submarine to the whale's tale can be used. LIke before, draw a corresponding right triangle.
Again, using the tangent ratio, the equation for the unknown distance d_2 can be formed and solved.
tan 14^(∘) = 9/d_2
d_2≈ 36
Finally, by calculating the difference between d_2 and d_1, the whale's length can be found. Length&= 36- 10 &=26 ft Therefore, the whale is approximately 26 feet long.
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Application of Trigonometric Ratios
Exercises
While on a field trip, a classmate of yours is looking out from the crown of the Statue of Liberty. He sees a ship coming into the harbor at Liberty Island. What is the distance between the foot of the Statue of Liberty and the ship if the angle of depression by which your classmate sees the ship is 20^(∘) and the statue is 250 feet high. Round to the nearest foot.
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From the exercise, we know that the classmate is looking down at the harbor with an angle of depression of 20^(∘). We also know that the Statue of Liberty is 250 feet high. We can illustrate this with a diagram.
Notice that the angle of depression and one of the acute angles in the right triangle are complementary. This means the sum of their measures are 90^(∘). This information will help us determine the measure of the acute angle. m∠ θ+20^(∘) = 90^(∘) ⇓ m∠ θ = 70^(∘) The acute angle is 70^(∘). Let's add this to the diagram.
Now we can use the tangent ratio to determine d.
The distance from the base of the statue to the ship is approximately 687 feet.
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Solution
Regulations state that a slide on a playground can have an angle of depression of at most 40^(∘) for it to be safe for children. If a slide is to be 7 feet high, what is the minimum length of the slide in order for it to comply with the safety criterion? Round to one decimal.
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We can illustrate this situation with a right triangle. We know that the slide is 7 feet high, and the angle of depression can, at most, be 40^(∘). We are looking for the minimum length of the slide, which we will label as a.
From the diagram, we see that the angle of depression and one of the acute angles in the right triangle are complementary. This means their measures sum to 90^(∘). With this information, we can determine the measure of the acute angle. m∠ θ+40^(∘) = 90^(∘) ⇓ m∠ θ = 50^(∘) The acute angle is 50^(∘). Let's add this to the diagram.
To determine the length of the slide, we can use the cosine ratio.
The minimum length of the slide is 10.9 feet.
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Solution
Davontay wants to measure the height of the building he lives in. At a distance of 55 feet from his home, he places a laser pointer on a mount. When the laser pointer is tilted 53^(∘) upward, the laser beam is visible on the peak of the roof. How high is the building? Round the answer to the nearest foot.
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Let's begin by drawing a diagram of the situation.
We know that the triangle is a right triangle, and we know the measurement of the adjacent side. We want to determine the building height — the length of the 53^(∘) angle's opposite side h. The tangent ratio makes it possible to solve the problem.
Together with Davontay, we have solved the height of the building to be about 73 feet.
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Solution
Magdalena is standing on the top of a hill looking down at her house. The angle of depression she is looking with is 20^(∘).
What angle of elevation would she have if she was standing at her house looking at the top of the hill?
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The angle of depression is the angle between the horizontal plane and the line with which Magdalena is looking down. We know that this is 20^(∘). Let's draw this situation.
Similar to the angle of depression, the angle of elevation is the angle between the horizontal plane and the line with which someone looks up. Since both angles are measured with respect to the horizontal plane, they must be congruent.
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Application of Trigonometric Ratios
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