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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?

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.

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

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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Hint

Solution

Hint

Solution