Angle Sum and Difference, Double Angle and Half Angle Formulas

Angle Sum and Difference, Double Angle and Half Angle Formulas

Hipparchus, considered to be the most eminent of Greek astronomers (born 160 B.C.),
derived the formulas for
The following formulas (or formulae, in Latin) are trigonometric identities.

Sum and Difference Formulas:

Sum, Difference, Double, Half Angle Formulae-1

Double Angle Formulas:

Sum, Difference, Double, Half Angle Formulae-2

Half Angle Formulas

Sum, Difference, Double, Half Angle Formulae-3

Example 1:

Sum, Difference, Double, Half Angle Formulae-4

Solution: The given information produces the triangles shown at the right. The Pythagorean Theorem, or a Pythagorean triple, is used to find the missing sides. Using the information from the triangles, find the answers to parts a and b.

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Sum, Difference, Double, Half Angle Formulae-6
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Example 2:

Using the half angle formula, find the exact value of cos 15º.

Solution:
The positive square root is chosen because cos 15º lies in Quadrant I.

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Example 3:
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