5.02 Trigonometric ratios

The trigonometric ratios describe the relationship between a given angle and two sides of a right-angled triangle. These rely on the trigonometric functions \sin,\,\cos, and \tan.

The trigonometric ratios 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}

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)

If we are given any two values in a right-angled triangle, either angles or side lengths, we can use these ratios to find any other angles or side lengths in the triangle.

Examples

Example 1

Consider the angle \theta. What is the value of the ratio \dfrac{\text{Opposite }}{\text{Hypotenuse }}?

Worked Solution

Create a strategy

Identify the opposite side using the angle, and the hypotenuse using the right angle.

Apply the idea

With respect to the angle \theta, the opposite side is 12, and the hypotenuse is the longest side of 13.

\displaystyle \frac{\text{Opposite }}{\text{Hypotenuse }}

\displaystyle =

\displaystyle \frac{12}{13}

Substitute the sides

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

We can then rearrange that ratio to make our unknown value the subject of an equation and then evaluate to find its value.

Examples

Example 2

Find the value of f correct to two decimal places.

Worked Solution

Create a strategy

Use the trigonometric ratio of cosine and substitute the given values.

Apply the idea

With respect to the angle 47\degree, the length of the adjacent side is f, and the length of the hypotenuse is 8, so we can use the cosine ratio.

\displaystyle \cos \theta	\displaystyle =	\displaystyle \frac{\text{Adjacent }}{\text{Hypotenuse}}	Use the cosine ratio
\displaystyle \cos 47\degree	\displaystyle =	\displaystyle \frac{f}{8}	Substitute the values and f
\displaystyle f	\displaystyle =	\displaystyle \cos 47\degree \times 8	Multiply both sides by 8
\displaystyle f	\displaystyle \approx	\displaystyle 5.46	Evaluate and round

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Example 3

Find the value of g correct to two decimal places.

Worked Solution

Create a strategy

Use the trigonometric ratio of sine and substitute the given values.

Apply the idea

With respect to the angle 42\degree, the length of the opposite side is 11, and the length of the hypotenuse is g, so we can use the sine ratio.

\displaystyle \sin \theta	\displaystyle =	\displaystyle \frac{\text{opposite }}{\text{Hypotenuse}}	Use the sine ratio
\displaystyle \sin 42\degree	\displaystyle =	\displaystyle \frac{11}{g}	Substitute the values and f
\displaystyle g\sin 42\degree	\displaystyle =	\displaystyle 11	Multiply both sides by g
\displaystyle g	\displaystyle =	\displaystyle \dfrac{11}{\sin 42\degree}	Divide both sides by \sin 42\degree
\displaystyle g	\displaystyle \approx	\displaystyle 16.44	Evaluate and round

Watch question walkthrough

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 4

Find the value of x to the nearest degree.

Worked Solution

Create a strategy

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

Apply the idea

With respect to the angle of x, the length of the adjacent side is 7, and the length of the hypotenuse side is 25, so we can use the cosine ratio.

\displaystyle \cos x	\displaystyle =	\displaystyle \frac{\text{Adjacent }}{\text{Hypotenuse}}	Use the cosine ratio
\displaystyle \cos x	\displaystyle =	\displaystyle \frac{7}{25}	Substitute the values
\displaystyle x	\displaystyle =	\displaystyle \cos^{-1}\left(\frac{7}{25}\right)	Apply the inverse ratio
	\displaystyle \approx	\displaystyle 74 \degree	Evaluate and round

Reflect and check

After identifying which sides we were given, we chose the inverse trigonometric ratio that matched those sides. We then solved the equation to find our unknown angle size.

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