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 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 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.

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Catch-Up and Review

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Solution

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