Trig Identities- Unit 5 HPreCalc

Tips on verifying identities

• convert everything to sin cosine if you can’t spot any identities off the bat or anything remotely close

• Multiple and factor out stuff when there’s no identity to make one

• Combine and split factors

• Work with both sides (consider both sides, and do the arrow thing)

Reciprocal identities

• cscx = 1/sinx

• cotx = 1/tanx

• secx = 1/cosx

Quotient identities

• tanx = sinx/cosx

• cotx = cosx/sinx

Negative identities

odd identities

• sin(-x) = -sinx

• csc(-x) = -cscx

• tan(-x) = -tanx

• cot(-x) = -cotx

Even identities

• cos(-x) = cosx

• sec(-x) = secx

Cos and sec are even functions as shown by graphs.

Pythagorean Identities

• sin²x + cos²x = 1

• tan²x + 1 = sec²x

• cot²x + 1 = csc²x

How to solve trig equations

Watch domain restrictions as well

1. move everything to one side

2. factor so you have separate functions = 0, then solve.

Also you have to keep all functions in, no dividing out, bc you lose solutions

Example: sinx = 2sinxcosx

0 = 2sinxcosx - sinx

0 = sinx(2cosx - 1)

sinx = 0 & cosx = ½

Come up with solutions in the given domain

ALL solutions

• write out any solutions in 1 rev + 2pi.

• Say you hv pi/3, and 5pi/3, write as x = pi/3 + 2npi & x = 5pi/3 + 2npi

Tips on solving trig equations

• Isolate functions so you can use 0 property

• Combine like terms

• Take square roots (USE A PLUS/MINUS). This is a last resort bc it creates extraneous solutions

• Factor out a GCF & factor in general

• Substitute identities in to simplify

How to use sum diff identities:

• come up with 2 angles ON THE UNIT CIRCLE that add/subtract to the angle you’re trying to find.

• Plug into the corresponding formula. Use the signs based on if you decided to add or subtract

Double angle Identities

• sin4x = 2 sin2x cos2x

• sin6x = 2 sin3x cos3x

• cos4x = cos²2x sin²2x