Finding unknown sides or angles

Trigonometric ratios are used to find the unknown side or angle of a right-angled triangle. There are three different trigonometric ratios, \(\sin(\theta), \cos(\theta)\), and \(\tan (\theta)\). Each is defined according to a different pair of sides in the triangle.

Use this page to revise the following concepts within Finding unknown sides or angles:

Trigonometric Ratios

Trigonometric ratios are defined in terms of the sides of a right-angled triangle. The sides are labelled as Opposite (Opp), Adjacent (Adj) and Hypotenuse (Hyp).   The hypotenuse  is the longest side of a right-angled triangle and will always be opposite the right angle. The definitions of the opposite and adjacent sides depend on the position of the angle in question. The adjacent side is the side that is directly next to the angle, excluding the hypotenuse. The opposite side is the one across from the angle.

Each ratio corresponds to a specific relationship between two of those sides in relation to the unknown angle, \(\theta\).

Check your understanding by labelling the following triangles and naming the trigonometric ratio:

Finding the unknown side of a right-angled triangle

To solve for an unknown side of a right-angled triangle, you must already know either:

  • One side and one non-right angle, in which case you can use trigonometry
  • Two sides, where you can apply the Pythagorean theorem.

To solve using trigonometry:

1. Label the right-angled triangle with Opp, Hyp, Adj in relation to the known angle.

2. Identify the trigonometric ratio

\(\sin(\theta)=\frac{Opp}{Hyp},\ \tan(\theta)=\frac{Opp}{Adj}\textsf{ or } \cos(\theta)=\frac{Adj}{Hyp}\)

3. Substitute the known values into the equation, and then rearrange the equation to solve for the unknown side

e.g. \(Hyp\times \sin(\theta)=Opp\) or \(Hyp=\frac{Opp}{\sin(\theta)}\)

Worked Example: Finding the unknown side (numerator)

For the triangle below, find the length of the unknown side, \(x\), to one decimal place.

A right angled triangle with the internal angle labelled 52°, hypotenuse labelled 65 and the adjacent side labelled x.

Steps:

1. Label triangle with Opp, Hyp, Adj given the location of the known angle \(\theta\) in the right-angled triangle.

2. Identify the trigonometric ratio.

In this case, we are trying to find the length of the Adjacent side, and we already know the length of the Hypotenuse, so the ratio we’re interested in is:

\(\cos(\theta)=\frac{Adj}{Hyp}\)

3. Substitute the known values into the equation, and then rearrange the equation to solve for the unknown side:

\(\begin{align} \cos(52^{\circ})& =\frac{x}{65} \\ x&=65\times \cos(52)\\ &\approx40.0\text{ m} \end{align}\)

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Worked Example: Finding the unknown side (denominator)

For the triangle below, find the length of the unknown side, \(x\), to one decimal place.

Right-angled triangle. Internal acute angle is labelled 30°. Opposite side is labelled 20m and Hypotenuse is labelled x.

Steps:

1. Label triangle with Opp, Hyp, Adj given the location of the known angle in the right-angled triangle.

Right-angled triangle. Internal acute angle is labelled 30°. Opposite side is labelled 20m and Hypotenuse is labelled x.

2. Identify the trigonometric ratio:

\(\sin(\theta)=\frac{Opp}{Hyp}\)

3. Substitute the known values into the equation, and then rearrange the equation to solve for the unknown side:

\(\begin{align} Hyp & =\frac{Opp}{\sin(\theta)}\\ x &=\frac{20}{\sin(30)}\\   &=40.0\text{ m} \end{align}\)

Note: your calculator needs to be set degrees mode, rather than radians to reach this answer.

Finding the unknown angle of a right angled triangle

To solve for an unknown angle of a right-angled triangle when you know the length of two sides, the following steps are taken:

1. Label triangle with Opp, Hyp, Adj given the location of the unknown angle in the right-angled triangle.

2. Identify the trigonometric ratio

\(\sin(\theta)=\frac{Opp}{Hyp},\ \tan(\theta)=\frac{Opp}{Adj}\textsf{ or } \cos(\theta)=\frac{Adj}{Hyp}\)

3. Substitute and take the inverse operation of the trigonometric ratio, that is:

\(\theta=\sin^{-1}(\frac{Opp}{Hyp}),\ \theta=\tan^{-1}(\frac{Opp}{Adj})\textsf{ or } \theta=\cos^{-1}(\frac{Adj}{Hyp})\)

The inverse operation is one that ‘reverses’ the original operation, \(\sin^{-1}\) does not mean sine to the power of negative 1. It represents the inverse of sine, allowing you to find the angle from the sine value. This is the same for the inverse functions \(\cos^{-1}\) and \(\tan^{-1}\).

Worked Example

Find the unknown angle, \(\theta\), to the closest degree.

Right-angled triangle. Angle of interest is labelled θ. Opposite has a length of 6 and Hypotenuse has a length of 10.

Steps:

1. Label the triangle and identify the trigonometric ratio

Right-angled triangle. Angle of interest is labelled θ. Opposite has a length of 6 and Hypotenuse has a length of 10. The sides have been labelled Hyp and Opp.

2. \(\sin(\theta)=\frac{Opp}{Hyp}\)

3. Substitute the known values and use a calculator to find the inverse of the trigonometric function

\(\begin{align} \sin(\theta)&=\frac{6}{10} \\ \theta &=\sin^{-1}(\frac{6}{10}) \\ &=36.8699^{\circ}\approx 37^{\circ} \end{align}\)