Lesson

The trigonometric ratios give us the trigonometric function of an angle in terms of two of the side lengths. We can use these ratios to find angles in right-angled triangles if we can isolate the angles in these equations.

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:

- $\sin\theta=\frac{\text{Opposite }}{\text{Hypotenuse }}$
`s``i``n``θ`=Opposite Hypotenuse - $\cos\theta=\frac{\text{Adjacent }}{\text{Hypotenuse }}$
`c``o``s``θ`=Adjacent Hypotenuse - $\tan\theta=\frac{\text{Opposite }}{\text{Adjacent }}$
`t``a``n``θ`=Opposite Adjacent

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 $\sin^{-1}$`s``i``n`−1, $\cos^{-1}$`c``o``s`−1 and $\tan^{-1}$`t``a``n`−1 which reverse their respective trigonometric functions. For example: $\sin^{-1}\left(\sin\theta\right)=\theta$`s``i``n`−1(`s``i``n``θ`)=`θ`.

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:

- $\theta=\sin^{-1}\left(\frac{\text{Opposite }}{\text{Hypotenuse }}\right)$
`θ`=`s``i``n`−1(Opposite Hypotenuse ) - $\theta=\cos^{-1}\left(\frac{\text{Adjacent }}{\text{Hypotenuse }}\right)$
`θ`=`c``o``s`−1(Adjacent Hypotenuse ) - $\theta=\tan^{-1}\left(\frac{\text{Opposite }}{\text{Adjacent }}\right)$
`θ`=`t``a``n`−1(Opposite Adjacent )

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

Caution

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

For example: $\sin^{-1}\theta$`s``i``n`−1`θ` $\ne$≠ $\frac{1}{\sin\theta}$1`s``i``n``θ`

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 $\sin^{-1}\left(\frac{4}{5}\right)$`s``i``n`−1(45) and $\tan^{-1}\left(\frac{4}{3}\right)$`t``a``n`−1(43).

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.

Caution

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.

There is another way to refer to angle size called radians, but we do not want those to be our calculator outputs.

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.

Find the value of $x$`x`.

**Think:** With respect to the angle $x$`x`, the side of length $4$4 is opposite and the side of length $5$5 is adjacent. This means that we can use the inverse trigonometric ratio $\theta=\tan^{-1}\left(\frac{\text{Opposite }}{\text{Adjacent }}\right)$`θ`=`t``a``n`−1(Opposite Adjacent ).

**Do:** Substituting our values into the inverse trigonometric ratio gives us:

$x=\tan^{-1}\frac{4}{5}$`x`=`t``a``n`−145

Evaluating $x$`x` (and rounding to two decimal places) gives us:

$x=38.66$`x`=38.66

**Reflect:** 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.

If $\cos\theta=\frac{31}{53}$`c``o``s``θ`=3153, find $\theta$`θ`, writing your answer to the nearest degree.

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

Find the value of $\theta$`θ` to the nearest degree.

applies trigonometry, given diagrams, to solve problems, including problems involving angles of elevation and depression