>Introduction>What Is the Definition of Supplementary Angles?

What Is the Definition of Supplementary Angles?

Theory

Supplementary Angles

If the sum of two angles $u$ and $v$ is $180$ $^{\circ}$ , they’re known as supplementary angles. For supplementary angles, the following is true:

\sin u = \sin v \cos u = - \cos v

This is important to remember when you solve trigonometric equations with sine and cosine. More than one angle can give the same sine or cosine value, so equations with sine and cosine often have more than one solution. Here’s an example.

Example 1

Solve the equation $2 \sin x = 1$ for $x \in [0^{\circ}, 360^{\circ})$

You want $\sin x$ by itself:

\begin{array}{l} 2 \sin x & = 1 \\ \sin x & = \frac{1}{2} \end{array}

You can now use the unit circle to see that $\sin 30^{\circ}$ is $\frac{1}{2}$ . Since $\sin x$ has the same value for the supplementary angle to $30^{\circ}$ , you must remember to also include the angle $180^{\circ} - 30^{\circ} = 150^{\circ}$ in your answer. The equation has the two solutions:

x_{1} = 30^{\circ} x_{2} = 150^{\circ}

You can find more examples like this in basic trigonometric equations.

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