Finding an Angle in a Right-Angled Triangle (VCE SSCE General Mathematics): Revision Notes

Finding an Angle in a Right-Angled Triangle

Introduction to inverse trigonometric functions

When working with right-angled triangles, you sometimes need to work backwards from a trigonometric ratio to find the angle that produced it. This process uses inverse trigonometric functions.

Think of it this way: if someone tells you that $\sin \theta = 0.8480$, and challenges you to discover what angle $\theta$ was used, you need a way to reverse or "undo" the sine operation.

The reverse operation for sine is called the inverse of sine, written as $\sin^{-1}$. The superscript $-1$ is not a power or exponent. It simply indicates that we are undoing or reversing the sine function.

Understanding the cycle

The relationship between a trigonometric function and its inverse can be visualised as a cycle:

The cycle shows how sine and its inverse work together:

• The forward arrow shows: given an angle $\theta$, use sine to find $\sin \theta$
• The backward arrow shows: given $\sin \theta$, use $\sin^{-1}$ to find the angle $\theta$

For example:

• Forward: $\sin(58°) = 0.8480$
• Backward: $\sin^{-1}(0.8480) = 58°$

The three inverse functions

Just as there are three main trigonometric functions, there are three inverse functions:

• $\sin^{-1}$ (inverse sine): used to find an angle when you know the sine ratio
• $\cos^{-1}$ (inverse cosine): used to find an angle when you know the cosine ratio
• $\tan^{-1}$ (inverse tangent): used to find an angle when you know the tangent ratio

Finding an angle from a trigonometric ratio value

When you know the value of a trigonometric ratio and need to find the corresponding angle, you use your calculator's inverse function keys.

Using your calculator

On a CAS calculator, you'll find the inverse function keys labelled as $\sin^{-1}$, $\cos^{-1}$, and $\tan^{-1}$. The general process is straightforward: access the appropriate inverse function, enter the ratio value, and calculate.

Understanding the language

The terminology used with inverse trigonometric functions can be confusing at first. Here's how to read and interpret the notation:

When you see:	Think:
$\sin(58°) = 0.8480$	"The sine of the angle 58° equals 0.8480"
$\sin^{-1}(0.8480) = 58°$	"The angle whose sine is 0.8480 equals 58°"
$\cos(60°) = 0.5$	"The cosine of the angle 60° equals 0.5"
$\cos^{-1}(0.5) = 60°$	"The angle whose cosine is 0.5 equals 60°"
$\tan(59.1°) = 1.67$	"The tangent of the angle 59.1° equals 1.67"
$\tan^{-1}(1.67) = 59.1°$	"The angle whose tangent is 1.67 equals 59.1°"

Finding an angle when given two sides

When you know two sides of a right-angled triangle and need to find an unknown angle, you combine your knowledge of trigonometric ratios with inverse functions.

Step-by-step method

Step 1: Identify which two sides you know (opposite, adjacent, or hypotenuse relative to the angle you're finding)

Step 2: Choose the appropriate trigonometric ratio:

• If you know the opposite and hypotenuse: use $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
• If you know the adjacent and hypotenuse: use $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
• If you know the opposite and adjacent: use $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

Step 3: Substitute the known side lengths into the ratio formula

Step 4: Use the appropriate inverse function to find the angle

Worked example

Finding the other acute angle

Remember that the three angles in any triangle sum to $180°$. In a right-angled triangle, one angle is $90°$, so the other two acute angles must sum to $90°$.

Once you've found one acute angle, you can find the other by subtracting from $90°$.

In the example above, the other acute angle would be: $90° - 26.9° = 63.1°$

Method summary

Exam tips

Remember!