Lesson

To find the value of a missing side in a right-angled triangle using trigonometry, we need to know the value of at least one angle (other than the right angle) and at least one side length.

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 pair of sides and a given angle, we can use one of these ratios to write the relationship between those three values.

Although we can write our values in some relationship using a trigonometric ratio, we still need to be able to turn our trigonometric function into a number.

We can evaluate trigonometric function expressions like $\sin42^\circ$`s``i``n`42° and $\cos71^\circ$`c``o``s`71° using the trigonometric functions on our calculator and entering the desired angle.

Once we have input a trigonometric function with some angle it is now a single term that we can multiply or divide by. To make sure that we are treating the trigonometric function as a single term, we need to keep an eye on our brackets.

For example:

When multiplying, $\sin42^\circ\times9$`s``i``n`42°×9 $\ne$≠ $\sin\left(42^\circ\times9\right)$`s``i``n`(42°×9) since the former multiplies the actual value while the latter multiples only the angle.

Similarly when dividing, $\frac{\cos71^\circ}{9}$`c``o``s`71°9 $\ne$≠ $\cos\frac{71}{9}^\circ$`c``o``s`719° since the former divides the actual value while the latter divides only the angle.

To avoid confusions, we try to always multiply on the left of a trigonometric function as a coefficient and express division using fractions. This gives us clearer expressions of the form $9\sin42^\circ$9`s``i``n`42° and $\frac{\cos71^\circ}{9}$`c``o``s`71°9.

Caution

When evaluating trigonometric function expressions, make sure that your calculator is in degrees mode.

There is another way to refer to angle size called radians, but we are not using that for our calculations.

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.

Find the value of $x$`x`.

**Think:** With respect to the given angle, the side of length $5$5 is adjacent and the side of length $x$`x` is opposite. This means that we should choose the trigonometric ratio $\tan\theta=\frac{\text{Opposite }}{\text{Adjacent }}$`t``a``n``θ`=Opposite Adjacent to relate our given values.

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

$\tan38^\circ=\frac{x}{5}$`t``a``n`38°=`x`5

We can multiply both sides of the equation by $5$5 to make $x$`x` the subject and then evaluate to find its value (rounded to two decimal places):

$x=5\tan38^\circ$`x`=5`t``a``n`38°$=$=$3.91$3.91

**Reflect:** After identifying which sides we were working with, we chose the trigonometric ratio that matched those sides. We then solved the equation to find our unknown side length.

Evaluate $7\cos77^\circ$7`c``o``s`77° to two decimal places.

Find the value of $h$`h` correct to two decimal places.

A lighthouse is positioned at point $A$`A`, and a boat is at point $B$`B`. If $d$`d` is the distance between the lighthouse and the boat, find $d$`d` to two decimal places.

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