Negative Angle Identities

Negative Angle Identities in Trigonometry

Theory

In trigonometry, angles are not limited to positive values. Angles can also be negative, meaning they are measured clockwise on the unit circle. To work efficiently with negative angles, trigonometry provides special identities known as negative angle identities. These identities help us understand how sine, cosine, and tangent behave when the angle sign changes.

Concept Explanation

Negative angle identities are based on the symmetry of the unit circle. When an angle changes from positive to negative, its point on the unit circle is reflected across the x-axis. Because of this reflection:

• Sine values change sign because sine represents the y-coordinate.
• Cosine values stay the same because cosine represents the x-coordinate.
• Tangent values change sign because tangent is the ratio of sine to cosine.

Steps or Formula

The main negative angle identities are:

• \(\sin(-\theta) = -\sin\theta\)
• \(\cos(-\theta) = \cos\theta\)
• \(\tan(-\theta) = -\tan\theta\)

These formulas allow us to convert negative angles into positive ones, making calculations easier.

Example Problem

Consider the angle \(\theta = 30^\circ\). On the unit circle:

• The point for \(30^\circ\) is \((\cos 30^\circ, \sin 30^\circ)\).
• The point for \(-30^\circ\) is \((\cos(-30^\circ), \sin(-30^\circ))\).

Using the identities:

• \(\sin(-30^\circ) = -\sin 30^\circ = -\frac{1}{2}\)
• \(\cos(-30^\circ) = \cos 30^\circ = \frac{\sqrt{3}}{2}\)
• \(\tan(-30^\circ) = -\tan 30^\circ = -\frac{1}{\sqrt{3}}\)

Now apply the identities to solve this expression:

\[ \sin(-45^\circ) + \cos(-45^\circ) \]

Step by step:

\[ = -\sin 45^\circ + \cos 45^\circ \]

\[ = -\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} \]

Final Answer

\[ 0 \]

Negative angle identities simplify trigonometric problems by using symmetry on the unit circle. By recognizing which functions change sign and which remain the same, calculations become faster and more intuitive.