Cheatography

# Solving Trigonometric Proofs Cheat Sheet by lwaites

A quick reference guide for solving trigonometric proofs and some basic trig identities to use.

### Solving Trigon­ometric Proofs

 Method 1 Pick one side of the equation (usually the most compli­cated side) and work with that side until it is equal to the other side Method 2 Work with both sides simult­ane­ously until they are both equal to the same expres­sion.

 Rewrite Rewrite the expression in terms of sine and cosine only Multiply by One Multiply the numerator and denomi­nator of a rational expression by a carefully chosen "­one­" Combine fractions Write sums and differ­ences of rational expres­sions as a single fraction Factor Factor trigon­ometric expres­sions, using "­u-s­ubs­tit­uti­on" as needed Pythag­orean Theorem Look for combin­ations or portions of Pythag­orean Identi­ties. Remember that you can multiply, divide, add or subtract the identity to get a new version. Goal Always keep the goal in mind. As you manipulate one side of the equation, keep the other side in mind. Look for common­alities

### Pythag­orean Identities

 sin2x + cos2x = 1   sin2x = 1 - cos2x   cos2x = 1 - sin2x 1 + cot2x = csc2x   cot2x = csc2x - 1 tan2x + 1 = sec2x   tan2x = sec2x - 1
Divide original Pythag­orean identity by sin2x or cos2x to get other identi­ties, subtract to get even more.

### Basic Trigon­ometric Functions

 sin (-x) = -sin (x) tan (x) = sin (x) / cos (x) csc (x) = 1/sin (x) cos (-x) = cos (x) cot (x) = cos (x) / sin (x) sec (x) = 1/cos (x) tan (-x) = -tan (x) cot (x) = 1/tan (x)

### Comple­mentary Angle Identities

 sin (pi/2 - x) = cos (x) cos (pi/2 - x) = sin (x) tan (pi/2 - x) = cot (x)

### Sum and Difference Identities

 sin (a +/- b) = sin (a) cos (b) +/- cos (a) sin (b) cos (a +/- b) = cos (a) cos (b) -/+ sin (a) sin (b)
Use sin (a +/- b) / cos (a +/- b) to find tan (a +/- b)

### Double Angle Identities

 sin (2x) cos (2x) 2 sin (x) cos (x) cos2 (x) - sin2 (x) 1 - 2 sin2 (x) 2 cos2 (x) - 1

Good stuff