14-6 Angle Identities

Objectives

To verify and use angle identities

To verify and use sum and difference identities

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The fact that ( ± x , ± y ) open plus minus x comma plus minus y close and ( ± y , ± x ) open plus minus y comma plus minus x close can represent eight different points suggests that you can derive several trigonometric identities directly from the unit circle.

Essential Understanding Several trigonometric identities involve a single angle. Other trigonometric identities involve two angles. No important trigonometric identity is additive; for example, sin ( A + B ) ≠ sin A + sin B . sine open eh plus b close not equal to sine eh plus sine b .

Here's Why It Works In the figure below, angles θ theta and − θ negative theta have the same amount of rotation, but the rotations are in opposite directions.

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Point Q is a reflection of P across the x-axis. The x-coordinates of P and Q (cosine) are the same and their y-coordinates (sine)are opposites. So cos ( − θ ) = cos θ cosine open negative theta close equals cosine theta and sin ( − θ ) = − sin θ . sine open negative theta close equals negative sine theta .

Similarly, S is the reflection of R across the x-axis. So tan ( − θ ) = − tan θ . tangent open negative theta close equals negative tangent theta .

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