9.03 Finding a side

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 \sin 42\degree and \cos 71\degree 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.

When multiplying, \sin 42\degree \times 9 \neq \sin \left(42\degree \times 9\right) since the former multiplies the actual value while the latter multiples only the angle.

Similarly when dividing, \dfrac{\cos 71\degree}{9}\neq \cos \dfrac{71}{9}\degree 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\sin 42\degree and \dfrac{\cos 71\degree }{9}.

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.

Examples

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 that relates the two sides and substitute the given values.

Apply the idea

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

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

Reflect and check

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.

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Example 3

Find the value of h correct to two decimal places.

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 21\degree, the side of length 16 is opposite and the side of length h\ 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 21 \degree	\displaystyle =	\displaystyle \frac{16}{h}	Substitute the values and h
\displaystyle h \sin 21 \degree	\displaystyle =	\displaystyle 16	Multiply both sides by h
\displaystyle h	\displaystyle =	\displaystyle \frac{16}{\sin 21 \degree}	Divide both sides by \sin 21 \degree
	\displaystyle =	\displaystyle 44.65\ \text{m}	Evaluate using a calculator

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Example 4

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

Worked Solution

Create a strategy

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

Apply the idea

The length AB is the distance d that we need to find.

With respect to the given angle of 13\degree, the side of length 50 is opposite and the side of length AB=d is adjacent, so we can use the tangent ratio.

\displaystyle \tan \theta	\displaystyle =	\displaystyle \frac{\text{Opposite }}{\text{Adjacent}}	Use the tan ratio
\displaystyle \tan 13 \degree	\displaystyle =	\displaystyle \frac{50}{d}	Substitute the values and d
\displaystyle d \tan 13 \degree	\displaystyle =	\displaystyle 50	Multiply both sides by d
\displaystyle d	\displaystyle =	\displaystyle \frac{50}{\tan 13 \degree}	Divide both sides by \tan 13 \degree
	\displaystyle =	\displaystyle 216.57\text{ km}	Evaluate using a calculator

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