6.06 Problem solving with trigonometry

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

Remember that the trigonometric ratios for a right-angled triangle are:\sin \theta =\dfrac{\text{Opposite }}{\text{Hypotenuse }} \quad \quad \cos \theta =\dfrac{\text{Adjacent }}{\text{Hypotenuse }} \quad \quad \tan \theta =\dfrac{\text{Opposite }}{\text{Adjacent }}

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)

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}

To find missing sides is right-angled triangles using trigonometry, we require at least one angle (other than the right angle) and at least one other side length.

We can find a missing side length by expressing it in a trigonometric ratio as the only unknown value, and then solve the equation to find that value.

Depending on the given angle, given side and missing side, we will need to use one of the three trigonometric ratios so that all relevant information is in the equation.

Examples

Example 1

AB is a tangent to a circle with centre O. OB is 31 cm long and cuts the circle at C.

Find the length of BC to two decimal places.

Worked Solution

Create a strategy

Use the cosine ratio.

Apply the idea

Notice that we can find the length of BC by subtracting the length of OC from OB. Also, we can see that OC has the same length as OA as they are both radii. So we need to calculate the OA first.

With respect to the given angle of 63\degree, the adjacent side is OA, and the hypotenuse is OB=31, so we can use the cosine ratio.

\displaystyle \cos \theta	\displaystyle =	\displaystyle \frac{\text{Adjacent }}{\text{Hypotenuse}}	Use the cosine ratio
\displaystyle \cos 63\degree	\displaystyle =	\displaystyle \frac{OA}{31}	Substitute the values
\displaystyle OA	\displaystyle =	\displaystyle 31\cos 63\degree	Multiply both sides by 31

\displaystyle BC	\displaystyle =	\displaystyle OB-OC	Subtract OC from OB
	\displaystyle =	\displaystyle 31-31\cos 63\degree	Substitute the values
	\displaystyle \approx	\displaystyle 16.93 \text{ cm}	Evaluate using a calculator

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In some cases, a single application of trigonometry on a single right-angled triangle will not be enough to find the values we are looking for.

In these cases, we can use trigonometry once to find a new value, and then use trigonometry again with our new value to find another new value. We can repeat this process as many times as is necessary to find the value that we are looking for.

Examples

Example 2

Find the size of angle z to two decimal places.

Worked Solution

Create a strategy

Use the inverse tangent function.

Apply the idea

With respect to the angle of x, the length of the opposite side is 5, and the length of the adjacent side is 8, so we can use the inverse tangent function.

\displaystyle \theta	\displaystyle =	\displaystyle \tan^{-1}\left(\dfrac{\text{Opposite }}{\text{Adjacent }}\right)	Use the inverse \tan function
\displaystyle x	\displaystyle =	\displaystyle \tan^{-1}\left(\dfrac{5}{8}\right)	Substitute the values
	\displaystyle \approx	\displaystyle 32.0054	Evaluate

We can do the same process to find the angle (x+z) in the larger triangle. This angle has an opposite side length of 6+5=11, and an adjacent side length of 8.

\displaystyle x+z	\displaystyle =	\displaystyle \tan^{-1}\left(\dfrac{11}{8}\right)	Substitute the values
	\displaystyle \approx	\displaystyle 53.9726	Evaluate

To find z we can just subtract the size of x from the size of (x+z).

\displaystyle z	\displaystyle =	\displaystyle x+z-x	Subtract x, from x+z
	\displaystyle =	\displaystyle 53.9726-32.0054	Substitute x+z and x
	\displaystyle \approx	\displaystyle 21.97	Evaluate and round

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In other cases, multiple applications of trigonometry need to be used at the same time in order to solve for a common factor.

These cases arise when multiple trigonometric ratios involve the same missing value. If some relationship between these ratios is known then the missing value can be solved for.

Examples

Example 3

Joshua and Catherine are looking up at the top of a statue with angles of elevation of 29\degree and 43\degree respectively. If Catherine is standing 3 m closer to the statue than Joshua, what is the height h of the statue?

Worked Solution

Create a strategy

Use the tangent ratio.

Apply the idea

We know the angles of elevation of 29\degree and 43\degree for Joshua and Catherine, respectively. We can find the distance, the adjacent side, from Catherine and Joshua to the statue's base in terms of h, the opposite side.

We can use the rearranged trigonometric ratio of tangent to find both distances and substitue the values: \text{Adjacent}=\dfrac{\text{Opposite}}{\tan \theta}

\text{Joshua to the statue}=\dfrac{h}{\tan 29\degree} \quad \quad \text{Catherine to the statue}=\dfrac{h}{\tan 43\degree}

Since Catherine is 3 m closer to the statue's base, we know that:

\displaystyle \dfrac{h}{\tan 29\degree}-\dfrac{h}{\tan 43\degree}	\displaystyle =	\displaystyle 3	Subtract the distances
\displaystyle h\left(\dfrac{1}{\tan 29\degree}-\dfrac{1}{\tan 43\degree}\right)	\displaystyle =	\displaystyle 3	Factor out h
\displaystyle 0.731679h	\displaystyle \approx	\displaystyle 3	Evaluate the brackets
\displaystyle h	\displaystyle =	\displaystyle \dfrac{3}{0.731679}	Divide both sides by 0.731679
	\displaystyle \approx	\displaystyle 4.10 \text{ m}	Evaluate using a calculator

The height of the statue is approximately 4.10 \text{ m}.