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Similarity, Proof, and Trigonometry

Application of Trigonometric Ratios

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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 ∠1. 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 x that makes the equation true.

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 and 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 What is the height of the building?

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.

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 h can be found.
The height of the office building is approximately 27.5 meters.

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?
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.

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 treetop
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 870 feet long. Its angle of depression is What is the height of the ski hill?

Snowboarder sliding down a ski hill
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 ∠1 are congruent angles.

Snowboarder sliding down a slope

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

Snowboarder sliding down a slope
The ratio between height h and the length of the ski hill is described by the sine of m∠1.
By substituting m∠1 with the value of h can be calculated.
Solve for h
It can be concluded that the height of the hill is approximately 181 feet.

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.
A football player kicking a ball
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 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 the distance d can be calculated.
Substitute with and solve the equation for d.
Solve 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.
102+17.32=c2
Solve for c
100+299.29=c2
399.29=c2
c2=399.29
It was derived that if Mark takes a shot 17.3 feet away from the goal post, the ball will travel approximately 20 feet.

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

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, ∠1 and the 75-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.
The length of the awning should be at least 1.9 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 660 miles away from Denali.

Mountain on the globe of Earth
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.
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 C 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 d.
r2+d2=(r+h)2
Solve for d
d2=(r+h)2r2
d2=r2+2rh+h2r2
d2=2rh+h2
The radius of Earth is 3963 miles and the height of Denali is 3.8 miles (20310 feet). By substituting these values for r and h, respectively, the value of d can be calculated.
Simplify right-hand side
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.

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 r to the length of the adjacent side d.
By substituting r with 3963 and d with 173.6, the measure of can be determined.
Solve for

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 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 and respectively.

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.

Using the tangent ratio, the horizontal distance from the submarine to the whale labeled as d1 can be found.
Solve for d1
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 d2 can be formed and solved.
Solve for d2
Finally, by calculating the difference between d2 and d1, the whale's length can be found.
Therefore, the whale is approximately 26 feet long.
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