Chapter Seven: Trigonometric Identities

Question 1

pythagorean identity

Answer 1

cos²x + sin²x = 1

Question 2

tangent version of pythagorean identity

Answer 2

1 + tan²x = sec²x

Question 3

cotangent version of pythagorean identity

Answer 3

1 + cot²x = csc²x

Question 4

sine of a sum of angles

Answer 4

sin(x₁ + x₂) = sinx₁cosx₂ + cosx₁sinx₂

Question 5

sine of a difference of angles

Answer 5

sin(x₁ - x₂) = sinx₁cosx₂ - cosx₁sinx₂

Question 6

cosine of a sum of angles

Answer 6

cos(x₁ + x₂) = cosx₁cosx₂ - sinx₁sinx₂

Question 7

cosine of a difference of angles

Answer 7

cos(x₁ - x₂) = cosx₁cosx₂ + sinx₁sinx₂

Question 8

tangent of a sum of angles

Answer 8

tan(x₁ + x₂) = (tanx₁ + tanx₂) / (1 - tanx₁tanx₂)

• plus sign on top, minus sign on bottom

Question 9

tangent of difference of angles

Answer 9

tan(x₁ - x₂) = (tanx₁ - tanx₂) / (1 + tanx₁tanx₂)

• minus on top plus on bottom

Question 10

double angle identity for sine

Answer 10

sin(2x) = 2sinxcosx

Question 11

double angle identity for cosine

Answer 11

cos2x = cos²x - sin²x

cos2x = 2cos²x - 1

cos2x = 1 - 2sin²x

Question 12

double angle identity for tangent

Answer 12

tan2x = (2tanx) / (1 - tan²x)

Question 13

how do you prove cofunction identities?

Answer 13

use the sum and difference identities

Question 14

cofunction identity for cosine

Answer 14

cos(x - π/2) = sinx

Question 15

cofunction identity for sine

Answer 15

sin(x - π/2) = -cosx

Question 16

cofunction identity for tangent

Answer 16

tan(x - π/2) = -cotx

Question 17

power-reducing identity for cosine

Answer 17

cos²x = (1 + cos2x) / 2

Question 18

power reducing identity for sine

Answer 18

sin²x = (1 - cos2x) / 2

Question 19

half-angle identity for cosine

Answer 19

Question 20

half-angle identity for sine

Answer 20

Question 21

half-angle identity for tangent

Answer 21