SINE 
COSINE 
TANGENT 
*These formulas are the most commonly used multiple-angle formulas. Like other trigonometric identities, they are used to rewrite trigonometric functions in more convenient forms.
Proof of the SINE formula:  "This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.")
Proof of the COSINE formula:  "This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.")
Proof of the TANGENT formula:  "This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.")
*Compared to others, the proofs for the double-angle formulas are simple. Their basis are the Sum & Difference Formulas for sine, cosine, and tangent (Section 5.4, page 404).
Practice Problem (page 411): Solving a Double-Angle Equation
Find all solutions of 2cosx + sin2x = 0
1. 2cosx + 2sinxcosx = 0 (substitute the SINE double-angle formula for sin2x)
2. 2cosx(1 + sinx) = 0 (factor out a 2cosx)
3. 2cosx = 0, 1 + sinx = 0 (set both factors equal to 0)
4. cosx = 0, sin = -1 (simplify both factors)
5. x = 
, x = 
(use prior knowledge of the unit circle to find all solutions for x on the interval  "This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.")
)
6. x = (eliminate duplicate solutions)
*Depending on the period of the equation, there may be additional answers.
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