9.05 Applications of trigonometry

To find missing angles in right-angled triangles using trigonometry, we require at least two known sides and the trigonometric ratios.

Remember that the trigonometric ratios for a right-angled triangle are:

\sin \theta =\dfrac{\text{Opposite }}{\text{Hypotenuse }} \quad \quad \cos \theta =\dfrac{\text{Adjacent }}{\text{Hypotenuse }} \quad \quad \tan \theta =\dfrac{\text{Opposite }}{\text{Adjacent }}

To isolate the angle in each of these relationships, we can apply the inverse trigonometric function to each side of the equation. This will give us:

\theta =\sin^{-1}\left(\dfrac{\text{Opposite }}{\text{Hypotenuse }}\right) \quad \quad \theta =\cos^{-1}\left(\dfrac{\text{Adjacent }}{\text{Hypotenuse }}\right) \quad \quad \theta =\tan^{-1}\left(\dfrac{\text{Opposite }}{\text{Adjacent }}\right)

Any of these relationships can be used to find \theta depending on which side lengths of the triangle are known.

While we may represent the inverse trigonometric functions using an index of -1, they are not reciprocals of the original functions.

For example: \sin ^{-1}\theta \neq \dfrac{1}{\sin \theta }

To find missing sides in right-angled triangles using trigonometry, we require at least one angle (other than the right angle) and at least one other side length.

We can find a missing side length using one of the trigonometric ratios where the missing length the only unknown value. Doing this allows us to solve the equation and find that value.

For example:

In this right-angled triangle, we are given one angle and one side. Using this, we want to find the length x.

Since the given side is adjacent to the given angle and x is the length of the hypotenuse, we can express their relationship using the trigonometric ratio: \cos 26\degree =\frac{5}{x}

We can isolate x in this equation by multiplying both sides by x and then dividing both sides by \cos 26\degree . This will give us: x=\frac{5}{\cos 26\degree }

Solving this tells us that x=5.56, rounded to two decimal places.

Depending on the given angle, given side and missing side, we will need to use one of the three trigonometric ratios so that all relevant information is in the equation.

Examples

Example 1

The person in the picture sights a pigeon above him. If the angle the person is looking at is \alpha, find \alpha in degrees.

Round your answer to two decimal places.

Worked Solution

Create a strategy

Use the inverse trigonometric function that relates the two given sides of the right-angled triangle.

Apply the idea

With respect to the given angle of \alpha, the side length of 13\, \text{m} is adjacent and the side length of 19\, \text{m} is opposite, so we can use the inverse tangent ratio.

\displaystyle \alpha	\displaystyle =	\displaystyle \tan^{-1}\left(\frac{\text{Opposite }}{\text{Adjacent}}\right)	Use the inverse tan function
	\displaystyle =	\displaystyle \tan^{-1}\left(\frac{19}{13}\right)	Substitute the values
	\displaystyle =	\displaystyle 55.62 \degree	Evaluate using calculator and round

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Example 2

A helicopter is 344 metres away from its landing pad.

If the angle of depression to the landing pad is 32, what is the height, x, of the helicopter above the ground?

Round your answer to the nearest metre.

Worked Solution

Create a strategy

Use the trigonometric ratio that relates the two sides and substitute the given values.

Apply the idea

With respect to the given angle of 32\degree, the side of length x is opposite and the side of length 344\,\text{m} is the hypotenuse, so we can use the sine ratio.

\displaystyle \sin \theta	\displaystyle =	\displaystyle \frac{\text{Opposite }}{\text{Hypotenuse}}	Use the sine ratio
\displaystyle \sin 32 \degree	\displaystyle =	\displaystyle \frac{x}{344}	Substitute the values and x
\displaystyle x	\displaystyle =	\displaystyle 344 \sin 32 \degree	Multiply both sides by 344
	\displaystyle =	\displaystyle 182\ \text{m}	Evaluate using calculator

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Example 3

A 27.3\, \text{m} long string of lights joins the top of a tree to a point on the ground.

If the tree is 7.4\, \text{m} tall, find \theta, the angle the string of lights would make with the tree to the nearest two decimal places.

Worked Solution

Create a strategy

Sketch the situation to identify the needed trigonometric ratio, then solve for the angle \theta.

Apply the idea

The situation has been sketched below:

With respect to the given angle of \theta, 7.4 is the adjacent side and 27.3 is the hypotenuse, so we can use the cosine ratio.

\displaystyle \cos \theta	\displaystyle =	\displaystyle \frac{\text{Adjacent }}{\text{Hypotenuse}}	Use the cos ratio
\displaystyle \cos \theta	\displaystyle =	\displaystyle \frac{7.4}{27.3}	Substitute the values
\displaystyle \theta	\displaystyle =	\displaystyle \cos ^{-1}\left(\frac{7.4}{27.3}\right)	Apply the inverse
	\displaystyle =	\displaystyle 74.27\degree	Evaluate using a calculator

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