Trig Identities

Question 1

What are the Pythagorean identities?

Answer 1

sin²x + cos²x = 1

tan²x + 1 = sec²x

1 + cot²x = csc²x

Question 2

What are the reciprocal identities?

Answer 2

sec x = 1 / cos x

csc x = 1 / sin x

tan x = sin x / cos x

cot x = cos x / sin x

cot x = 1 / tan x

Question 3

What are the double-angle formulas?

Answer 3

Question 4

What is the double-angle formula for sine?

Answer 4

sin 2x = 2 sinx cosx

Question 5

What are the double-angle formulas for cosine?

Answer 5

cos 2x = cos²x - sin²x

        = 2cos<sup>2</sup>x - 1

        = 1 - sin<sup>2</sup>x

Question 6

What are the Even-Odd Identities?

Answer 6

sin (-x) = - sin x

cos (-x) = cos x

tan (-x) = - tan x

Question 7

What are the Cofunction Identities?

Answer 7

sin( (π/2) - u) = cos u

cos( (π/2) - u) = sin u

tan( (π/2) - u) = cot u

cot( (π/2) - u) = tan u

sec( (π/2) - u) = csc u

csc( (π/2) - u) = sec u

Question 8

What are the Addition and Subtraction Formulas for Sine?

Answer 8

sin(s + t) = sin s cos t + cos s sin t

sin(s - t) = sin s cos t - cos s sin t

Question 9

What are the Addition and Subtraction Formulas for Cosine?

Answer 9

cos(s + t) = cos s cos t - sin s sin t

cos(s - t) = cos s cos t + sin s sin t

Question 10

What are the Addition and Subtraction Formulas for Tangent?

Answer 10

tan(s + t) = (tan s + tan t) / (1 - tan s tan t)

tan(s - t) = (tan s - tan t) / (1 + tan s tan t)

Question 11

What are the formulas for lowering powers?

Answer 11

sin²x = (1 - cos 2x) / 2

cos²x = (1 + cos 2x) / 2

tan²x = (1 - cos 2x) / (1 + cos 2x)

Question 12

What are the half-angle formulas?

Answer 12

sin(u/2) = ± ((1 - cos u)/2)^½ note this is a square root

cos(u/2) = ± ((1 + cos u)/2)^½ note this is a square root

tan(u/2) = (1 - cos u) / sin u

tan(u/2) = sin u / (1 + cos u)

Question 13

What are the product-to-sum formulas?

Answer 13

sin u cos v = ½[sin(u + v) + sin (u - v)]

cos u sin v = ½[sin(u + v) - sin (u - v)]

cos u cos v = ½[cos(u + v) + cos(u - v)]

sin u sin v = ½[cos(u - v) - cos(u + v)]

Question 14

What are the sum-to-product formulas?

Answer 14

sin x + sin y = 2 sin ((x + y) / 2) cos ((x - y) / 2)

sin x - sin y = 2 cos ((x + y) / 2) sin ((x - y) / 2)

cos x + cos y = 2 cos ((x + y) / 2) cos ((x - y) / 2)

cos x - cos y = -2 sin ((x + y) / 2) sin ((x - y) / 2)

Question 15

What are the sums of sines and cosines?

Answer 15

If A and B are real numbers, then

A sin x + B cos x = k sin (x + Ø)

where

k = (A² + B²)^½ note this is a square root

and Ø satisfies

cos Ø = A / ((A² + B²)^½) note this is a square root

sin Ø = B / ((A² + B²)^½) note this is a square root