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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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Lesson Settings & Tools
10 Theory slides
9 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
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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.

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
A right triangle with given side lengths
b Find the measure of acute angle Round your answer to the closest degree.
A right triangle with given side lengths
c Write the given expressions in terms of sine or cosine. Write your answer without the degree symbol.
d Determine the value of that makes the equation true.
Challenge

Measuring Depths Using Trigonometry

A submarine sailed underwater near a whale that was swimming feet deeper than the submarine. A sonar on that submarine measured the angles of depression to the whale's mouth and tail to be and respectively.

A submarine near a whale
External credits: @brgfx, @freepik, @upklyak
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 meters from the building, looking up toward the rooftop at an approximate angle of What is the height of the building?

A man standing 10 meters away from the base of the building where he works, looking up at an approximate angle of 70 degrees towards the rooftop.
External credits: @freepik
Round the answer to the first decimal place.

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.

Man and a building
External credits: @freepik
The length of the adjacent side of an acute angle of is known, and the length of the opposite side should be found. Therefore, the tangent ratio can be applied.
By solving this equation, the value of can be found.
MultEqn

RearrangeEqn

Rearrange equation

UseCalc

Use a calculator

RoundDec

Round to decimal place(s)

The height of the office building is approximately meters.
Example

Measuring Angles Using Trigonometry

A plane arriving at O'Hare International Airport is meters above the ground and meters from the expected touchdown point on a runway. At what angle is the plane supposed to descend to land successfully?
Plane landing on a runway
Round the answer to the closest degree.

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.

Plane landing on a runway
Since the lengths of the opposite side and the hypotenuse are given, the angle of descent can be calculated by using the sine ratio.
To solve this equation for the inverse of sine can be used.
ReduceFrac

UseCalc

Use a calculator

RoundDec

Round to decimal place(s)

Therefore, in order to land successfully, the plane should descend at an angle of
Plane landing on a runway
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.

Man looking at the treetop, demonstrating the concept of angle of elevation in a natural setting.
The angle between the horizon and the viewer's gaze as they look up is an angle of elevation. In the diagram, it is marked as
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.

The angle between the horizon and the viewer's gaze as they look down is an angle of depression. In the diagram, it is labeled as
It can be noted that the names of these angles indicate the position of an object in relation to the position of the viewer.
Example

Using Angles of Depressions to Solve Problems

A snowboarder is sliding down a ski hill that is feet long. Its angle of depression is What is the height of the ski hill?

Snowboarder.jpg

Round the answer to the closest integer.

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 That angle can be identified on the diagram as To determine the height of the hill, a line parallel to can be drawn at the bottom of the hill. By the Alternate Interior Angles Theorem, and are congruent angles.

Snowboarder sliding down a slope
External credits: @macrovector

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

Snowboarder sliding down a slope
External credits: @macrovector
The ratio between height and the length of the ski hill is described by the sine of
By substituting with the value of can be calculated.
Substitute

▼
Solve for
MultEqn

RearrangeEqn

Rearrange equation

UseCalc

Use a calculator

RoundInt

Round to nearest integer

It can be concluded that the height of the hill is approximately feet.
Example

Using Angles of Elevations to Solve Problems

Mark, who plays fútbol for South High School, takes a shot into a foot tall goal post. The ball travels at a degree angle of elevation toward the center of the goal.
A football player kicking a ball
External credits: @macrovector
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.

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 feet. Therefore, to find the farthest position of Mark from the goal post, let one leg of the triangle be feet long.

A football player kicking a ball
External credits: @macrovector
Using the tangent of the distance can be calculated.
Substitute with and solve the equation for
Substitute

▼
Solve for
MultEqn

DivEqn

UseCalc

Use a calculator

RoundDec

Round to decimal place(s)

Therefore, standing feet away from the goal post is the farthest position from which Mark can score a goal.
A football player kicking a ball
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.
Now, the equation needs to be solved for
▼
Solve for
CalcPow

Calculate power

AddTerms

Add terms

RearrangeEqn

Rearrange equation

SqrtEqn

RoundInt

Round to nearest integer

It was derived that if Mark takes a shot feet away from the goal post, the ball will travel approximately feet.
Example

Real Life Applications of Trigonometry

A family has a 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

The building with an awning
Find the length of the awning they should buy. Round the answer to the first decimal place.

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, and the degree angle are congruent angles.

The building with an awning
Let represent the length of the awning. In order to find its value, the cotangent ratio can be used.
By solving the above equation, the value of can be found.
MultEqn

RearrangeEqn

Rearrange equation

UseCalc

Use a calculator

RoundDec

Round to decimal place(s)

The length of the awning should be at least feet long.
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 miles away from Denali.

Mountain on the globe of Earth
The radius of the Earth is miles and the height of Denali is miles. Use the given information to determine whether the curvature of the earth will block her line of sight.
Find the angle of depression at which a person on top of Denali will look at the horizon. Round the answer to the nearest degree.

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 represent the center of the globe.

A right triangle formed by the center of Earth, person on the mountain and horizon
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.
First, it should be solved for
▼
Solve for
SubEqn

ExpandPosPerfectSquare

SubTerm

Subtract term

SqrtEqn

The radius of Earth is miles and the height of Denali is miles ( feet). By substituting these values for and respectively, the value of can be calculated.
SubstituteII

,

▼
Simplify right-hand side
CalcPow

Calculate power

Multiply

Multiply

AddTerms

Add terms

CalcRoot

Calculate root

RoundDec

Round to decimal place(s)

This means that when standing on the peak of Denali, it is possible to see locations up to miles away. However, the most eastern point of Russia is located 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.

A right triangle formed by the center of Earth, person on the mountain and horizon
The tangent of is equal to the ratio of the length of the opposite side to the length of the adjacent side
By substituting with and with the measure of can be determined.
SubstituteII

,

▼
Solve for

UseCalc

Use a calculator

RoundInt

Round to nearest integer

The angle of depression measures approximately
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 feet deeper than the submarine. A sonar on that submarine measured the angles of depression to the whale's mouth and tail to be and respectively.

A submarine is positioned at the top left of an underwater landscape while a whale is positioned at the bottom right side.
External credits: @brgfx, @freepik, @upklyak
How can the sailors use this information to measure the length of the whale?

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 Mark this information on the diagram and form a corresponding right triangle.

A submarine and a whale
External credits: @brgfx, @freepik, @upklyak
Using the tangent ratio, the horizontal distance from the submarine to the whale labeled as can be found.
▼
Solve for
MultEqn

DivEqn

UseCalc

Use a calculator

RoundInt

Round to nearest integer

Next, the angle of depression from the submarine to the whale's tale can be used. LIke before, draw a corresponding right triangle.
A submarine and a whale
External credits: @brgfx, @freepik, @upklyak
Again, using the tangent ratio, the equation for the unknown distance can be formed and solved.
▼
Solve for
MultEqn

DivEqn

UseCalc

Use a calculator

RoundInt

Round to nearest integer

Finally, by calculating the difference between and the whale's length can be found.
Therefore, the whale is approximately feet long.
A submarine and a whale
External credits: @brgfx, @freepik, @upklyak



Level 1
Level 2
Level 3
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 and the statue is feet high. Round to the nearest foot.

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.

Regulations state that a slide on a playground can have an angle of depression of at most for it to be safe for children. If a slide is to be feet high, what is the minimum length of the slide in order for it to comply with the safety criterion? Round to one decimal.

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.

Davontay wants to measure the height of the building he lives in. At a distance of feet from his home, he places a laser pointer on a mount. When the laser pointer is tilted upward, the laser beam is visible on the peak of the roof. How high is the building? Round the answer to the nearest foot.

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.

Magdalena is standing on the top of a hill looking down at her house. The angle of depression she is looking with is

an angle of depression of 20 degrees
What angle of elevation would she have if she was standing at her house looking at the top of the hill?

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.

Application of Trigonometric Ratios

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