8.04 Finding an angle

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

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

\begin{aligned} \sin \theta &=\dfrac{\text{Opposite }}{\text{Hypotenuse }} \\ \cos \theta &=\dfrac{\text{Adjacent }}{\text{Hypotenuse }} \\ \tan \theta &=\dfrac{\text{Opposite }}{\text{Adjacent }} \end{aligned}

For any unknown angle and a pair of side lengths, we can use one of these ratios to write the relationship between those three values.

Once we have chosen a trigonometric ratio relating our unknown angle to two known side lengths, we need to isolate the angle. In order to do this, we need to reverse the effect of the trigonometric function.

To reverse the effect of a function on some value, we need to apply the inverse of that function.

Trigonometric functions have dedicated inverse functions of the form \sin ^{-1}, \cos ^{-1} and \tan ^{-1} which reverse their respective trigonometric functions. For example: \sin ^{-1}\left(\sin \theta \right) =\theta

Applying the inverse trigonometric functions to our trigonometric ratios gives us three new ratios that can be used to find angles in a right-angled triangle:

\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 }

We evaluate inverse trigonometric functions in the same way that we evaluate normal trigonometric functions.

To evaluate inverse trigonometric functions, we can input the inverse function on our calculators and enter the ratio corresponding to the angle to get expressions of the form \sin ^{-1}\left(\dfrac{4}{5}\right)and \tan ^{-1}\left(\dfrac{4}{3}\right).

As was with the normal trigonometric functions, we want to treat the inverse trigonometric function and its included ratio as a single term. This means that when we want to multiply or divide by it, we should perform that operation on the whole expression.

When evaluating inverse trigonometric function expressions, make sure that your calculator is in degrees mode. This will tell our calculator to evaluate inverse trigonometric functions in terms of degrees.

Examples

Based on where the angle is in the triangle and which pair of sides we are given, we can choose one of the trigonometric ratios to describe the relationship between those values.

We can then rearrange that ratio (or choose the corresponding inverse ratio) to make our unknown angle the subject of an equation and then solve for it.

Examples

Example 2

Use the tangent ratio to find the size of the angle marked y, correct to the nearest degree.

Worked Solution

Create a strategy

Use the tangent ratio to set up the equation then use the inverse ratio to find the angle.

Apply the idea

With respect to the given angle of y, the side of length 55.4 is adjacent and the side of length 59.6 is opposite.

\displaystyle \tan y	\displaystyle =	\displaystyle \frac{\text{Opposite }}{\text{Adjacent}}	Use the tangent ratio
\displaystyle \tan y	\displaystyle =	\displaystyle \frac{59.6}{55.4}	Substitute the values
\displaystyle y	\displaystyle =	\displaystyle \tan^{-1}\left(\frac{59.6}{55.4}\right)	Apply the inverse ratio
	\displaystyle =	\displaystyle 47 \degree	Evaluate and round

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Example 3

Find the value of \theta to the nearest degree.

Worked Solution

Create a strategy

Use the trigonometric ratio that relates the two sides then apply the inverse ratio.

Apply the idea

With respect to the given angle of \theta, the side AC=8 is the opposite side and the side AB=10 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 \theta	\displaystyle =	\displaystyle \frac{8}{10}	Substitute the values
\displaystyle \theta	\displaystyle =	\displaystyle \sin^{-1}\left(\frac{8}{10}\right)	Applying inverse ratio
	\displaystyle =	\displaystyle 53 \degree	Evaluate and round

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