Trigonometric Functions
Trigonometric Identities
Formulas for the tangent function can be derived from similar formulas involving the sine and cosine. The sum identity for tangent is derived as follows:
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To determine the difference identity for tangent, use the fact that tan(−β) = −tanβ.
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Example 1: Find the exact value of tan 75°.
Because 75° = 45° + 30°
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Example 2: Verify that tan (180° − x) = −tan x.
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Example 3: Verify that tan (180° + x) = tan x.
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Example 4: Verify that tan (360° − x) = − tan x.
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The preceding three examples verify three formulas known as the reduction identities for tangent. These reduction formulas are useful in rewriting tangents of angles that are larger than 90° as functions of acute angles.
The double-angle identity for tangent is obtained by using the sum identity for tangent.
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The half-angle identity for tangent can be written in three different forms.
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In the first form, the sign is determined by the quadrant in which the angle α/2 is located.
Example 5: Verify the identity
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Example 6: Verify the identity tan (α/2) = (1 − cos α)/sin α.
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Example 7: Verify the identity tan (α − 2) = sin π/(1 + cos α).
Begin with the identity in Example 6.
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Example 8: Use a half-angle identity for the tangent to find the exact value for tan 15°.
What follows are two alternative solutions.
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