Chapter 7: Identities

Cosine Sum Identity:

cos(u+v) = (cosu*cov)-(sinu*sinv)

Cosine Difference Identity:

cos(u-v) = (cosu*cosv)+(sinu*sinv)

Cosine Cofunction Identity

cos(x+/-pi/2)=+/-sinx

Sine Double Angle Identity:

sin 2x = 2 sin x cos x

Sine Sum Identity:

sin(u+v) = (sinu*cov)+(cosu*sinv)

Sine Difference Identity:

sin(u-v) = (sinu*cov)-(cosu*sinv)

Sine Cofunction Identity

sin(x+/-pi/2)=+/-cosx

Tangent Double Angle Identity:

tan(2x) = (2tanx)/(1-tan^2x)

Tangent Difference Identity:

tan(u-v) = (tanu-tanv)/(1+tanu*tanv)

Pythagorean Identity #3:

tan^2s+1= sec^2s

Unit Circle equation:

x^2+y^2=1

Pythagorean Identity #1:

(cos s)^2+(sin s)^2=1

Arcsin:

- Exists in Q1 - Exists in Q4 - Negative angles/radians - Does include endpoints 𝞹/2 and -𝞹/2

Arctangent:

- Exists in Q1 - Exists in Q4 - Negative angles/radians - Does not include endpoints 𝞹/2 and -𝞹/2

Composition:

- cancels each other out - Does not happen when outside of the range of the inverse - Mostly in radians in the textbook, but can be in degrees - All simplifying

Pythagorean Identity #2:

1 + cot^2s = csc^2s

Steps to finding most simplified form:

1. Write expression in terms of sines and cosines 2. Think in equivalent terms 3. Find pythagorean identities 4. Put in most simplified form

Cofunction Identities

1. tan (90° - x) = cot x 2. cot (90° - x) = tan x 3. sec (90° - x) = csc x 4. csc (90° - x) = sec x

Tangent Sum Identity:

6: tan(u+v) = (tanu+tanv)/(1-tanu*tanv)

Cosine Double Angle Identity:

Cos (2x) = 1 - 2 sin^2 x Cos (2x) = 2 cos^2 x - 1

Arccosine:

Exists in Q1 and Q2

Inverse Trig Functions

Input is value, output is an angle

Proving:

Two columns Do it for only one side Work with the more complex side first Convert all expressions to sines and cosines Shortcuts: 3 dot triangle for therefore QED Proof Resolved

Verify:

Use both sides to turn each side into an equivalent expression

Proof:

Use one side to turn it into the other side