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