Exploring Trigonometry Applications in Real Life Scenarios

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 .

A right triangle with given side lengths

b Find the measure of acute angle

∠ 1 .

Round your answer to the closest degree.

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.

cos 2 8^{\circ} = sin x^{\circ}

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 $4 2^{\circ}$ and $1 4^{\circ},$ respectively.

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 $10$ meters from the building, looking up toward the rooftop at an approximate angle of $7 0^{\circ} .$ 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.

External credits: @freepik

The length of the adjacent side of an acute angle of

7 0^{\circ}

is known, and the length of the opposite side should be found. Therefore, the tangent ratio can be applied.

tan θ tan 7 0^{\circ} = \frac{Opposite}{Adjacent} ⇓ = \frac{h}{1 0}

By solving this equation, the value of

h

can be found.

tan 7 0^{\circ} = \frac{h}{1 0}

MultEqn

$LHS \cdot 10 = RHS \cdot 10$

10 tan 7 0^{\circ} = h

RearrangeEqn

Rearrange equation

h = 10 tan 7 0^{\circ}

UseCalc

Use a calculator

h = 27.474774 \dots

RoundDec

Round to $1$ decimal place(s)

h \approx 27.5

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?

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.

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 sin θ = \frac{Opposite}{Hypotenuse} ⇓ = \frac{3 4}{2 8 0}

To solve this equation for

θ,

the inverse of sine can be used.

sin θ = \frac{3 4}{2 8 0}

ReduceFrac

$\frac{a}{b} = \frac{a / 2}{b / 2}$

sin θ = \frac{1 7}{1 4 0}

$sin^{- 1} (LHS) = sin^{- 1} (RHS)$

θ = sin^{- 1} \frac{1 7}{1 4 0}

UseCalc

Use a calculator

θ = 6.974556 \dots

RoundDec

Round to $1$ decimal place(s)

θ \approx 7^{\circ}

Therefore, in order to land successfully, the plane should descend at an angle of

7^{\circ} .

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.

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

∠ θ .

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

∠ θ .

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 $1 2^{\circ} .$ What is the height of the 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 $1 2^{\circ} .$ 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 = \frac{h}{8 7 0}

By substituting

m ∠ 1

with

1 2^{\circ},

the value of

h

can be calculated.

sin m ∠ 1 = \frac{h}{8 7 0}

Substitute

$m ∠ 1 = 1 2^{\circ}$

sin 1 2^{\circ} = \frac{h}{8 7 0}

Solve for

h

MultEqn

$LHS \cdot 870 = RHS \cdot 870$

870 sin 1 2^{\circ} = h

RearrangeEqn

Rearrange equation

h = 870 sin 1 2^{\circ}

UseCalc

Use a calculator

h = 180.883171 \dots

RoundInt

Round to nearest integer

h \approx 181

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.

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 $10$ 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 = \frac{1 0}{d}

Substitute

θ

with

3 0^{\circ}

and solve the equation for

d .

tan θ = \frac{1 0}{d}

Substitute

$θ = 3 0^{\circ}$

tan 3 0^{\circ} = \frac{1 0}{d}

Solve for

d

MultEqn

$LHS \cdot d = RHS \cdot d$

d tan 3 0^{\circ} = 10

DivEqn

$LHS / tan 3 0^{\circ} = RHS / tan 3 0^{\circ}$

d = \frac{1 0}{tan 3 0 ^{\circ}}

UseCalc

Use a calculator

d = 17.320508 \dots

RoundDec

Round to $1$ decimal place(s)

d \approx 17.3

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} ⇓ 1 0^{2} + 17 . 3^{2} = c^{2}

Now, the equation needs to be solved for

c .

1 0^{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

$LHS = RHS$

c = 19.982242 \dots

RoundInt

Round to nearest integer

c \approx 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.

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 $7 5^{\circ} .$

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.

Let

ℓ

represent the length of the awning. In order to find its value, the cotangent ratio can be used.

cot m ∠ 1 cot 7 5^{\circ} = \frac{Adjacent}{Opposite} ⇓ = \frac{ℓ}{7}

By solving the above equation, the value of

ℓ

can be found.

cot 7 5^{\circ} = \frac{ℓ}{7}

MultEqn

$LHS \cdot 7 = RHS \cdot 7$

7 cot 7 5^{\circ} = ℓ

RearrangeEqn

Rearrange equation

ℓ = 7 cot 7 5^{\circ}

UseCalc

Use a calculator

ℓ = 1.875644 \dots

RoundDec

Round to $1$ decimal place(s)

ℓ \approx 1.9

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.

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.

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

$(a + b)^{2} = a^{2} + 2 a b + b^{2}$

d^{2} = r^{2} + 2 r h + h^{2} - r^{2}

SubTerm

Subtract term

d^{2} = 2 r h + h^{2}

SqrtEqn

$LHS = RHS$

d = 2 r h + h^{2}

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.

d = 2 r h + h^{2}

SubstituteII

$r = 3963$ , $h = 3.8$

d = 2 (3963) (3.8) + (3.8)^{2}

Simplify right-hand side

CalcPow

Calculate power

d = 2 (3963) (3.8) + 14.44

Multiply

d = 30118.8 + 14.44

AddTerms

Add terms

d = 30133.24

CalcRoot

Calculate root

d = 173.589285 \dots

RoundDec

Round to $1$ decimal place(s)

d \approx 173.6

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 = \frac{r}{d}

By substituting

r

with

3963

and

d

with

173.6,

the measure of

∠ θ

can be determined.

tan θ = \frac{r}{d}

SubstituteII

$r = 3963$ , $d = 173.6$

tan θ = \frac{3 9 6 3}{1 7 3 . 6}

Solve for

θ

$tan^{- 1} (LHS) = tan^{- 1} (RHS)$

θ = tan^{- 1} \frac{3 9 6 3}{1 7 3 . 6}

UseCalc

Use a calculator

θ = 87.491750 \dots

RoundInt

Round to nearest integer

θ \approx 87

The angle of depression measures approximately

8 7^{\circ} .

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 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 $4 2^{\circ} .$ Mark this information on the diagram and form a corresponding right triangle.

External credits: @brgfx, @freepik, @upklyak

Using the tangent ratio, the horizontal distance from the submarine to the whale labeled as

d_{1}

can be found.

tan 4 2^{\circ} = \frac{9}{d _{1}}

Solve for

d_{1}

MultEqn

$LHS \cdot d_{1} = RHS \cdot d_{1}$

d_{1} tan 4 2^{\circ} = 9

DivEqn

$LHS / tan 4 2^{\circ} = RHS / tan 4 2^{\circ}$

d_{1} = \frac{9}{tan 4 2 ^{\circ}}

UseCalc

Use a calculator

d_{1} = 9.995512 \dots

RoundInt

Round to nearest integer

d_{1} \approx 10

Next, the angle of depression from the submarine to the whale's tale can be used. LIke before, draw a corresponding right triangle.

External credits: @brgfx, @freepik, @upklyak

Again, using the tangent ratio, the equation for the unknown distance

d_{2}

can be formed and solved.

tan 1 4^{\circ} = \frac{9}{d _{2}}

Solve for

d_{2}

MultEqn

$LHS \cdot d_{2} = RHS \cdot d_{2}$

d_{2} tan 1 4^{\circ} = 9

DivEqn

$LHS / tan 1 4^{\circ} = RHS / tan 1 4^{\circ}$

d_{2} = \frac{9}{tan 1 4 ^{\circ}}

UseCalc

Use a calculator

d_{2} = 36.097028 \dots

RoundInt

Round to nearest integer

d_{2} \approx 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.

External credits: @brgfx, @freepik, @upklyak

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Catch-Up and Review

Hint

Solution

Hint

Solution

Hint

Solution

Hint

Solution

Hint

Solution

Hint

Solution

Finding the Distance to the Horizon

Finding the Angle of Depression

Hint

Solution