MATH 1316
Chapter 5 Identities
MEMORIZE!!
Reciprocal Identities
1
cot 𝜃 = tan 𝜃
1
sec 𝜃 = cos 𝜃
1
csc 𝜃 = sin 𝜃
Quotient Identities
sin 𝜃
tan 𝜃 = cos 𝜃
cot 𝜃 =
cos 𝜃
sin 𝜃
Pythagorean Identities
𝐬𝐢𝐧𝟐 𝜽 + 𝐜𝐨𝐬𝟐 𝜽 = 𝟏
𝐭𝐚𝐧𝟐 𝜽 + 𝟏 = 𝐬𝐞𝐜 𝟐 𝜽
𝟏 + 𝐜𝐨𝐭 𝟐 𝜽 = 𝐜𝐬𝐜 𝟐 𝜽
Alternative Forms of the Pythagorean Identities
𝐬𝐢𝐧𝟐 𝜽 = 1 − cos2 𝜃
𝐭𝐚𝐧𝟐 𝜽 = sec 2 𝜃 − 1
𝐜𝐨𝐭 𝟐 𝜽 = csc 2 𝜃 − 1
𝐜𝐨𝐬𝟐 𝜽 = 1 − sin2 𝜃
𝐬𝐞𝐜 𝟐 𝜽 = tan2 𝜃 + 1
𝐜𝐬𝐜 𝟐 𝜽 = 1 + cot 2 𝜃
Negative Angle Identities
cos(−𝜃) = cos 𝜃
sin(−𝜃) = − sin 𝜃
tan(−𝜃) = − tan 𝜃
sec(−𝜃) = sec 𝜃
csc(−𝜃) = − csc 𝜃
cot(−𝜃) = − cot 𝜃
Cofunction Identities
cos(90° − 𝜃) = sin 𝜃
sin(90° − 𝜃) = cos 𝜃
sec(90° − 𝜃) = csc 𝜃
csc(90° − 𝜃) = sec 𝜃
tan(90° − 𝜃) = cot 𝜃
cot(90° − 𝜃) = tan 𝜃
Sum and Difference Identities
NOTE: Angles A and B can be measured in degrees or radians.
tan 𝐴 + tan 𝐵
cos(𝐴 + 𝐵) = cos 𝐴 cos 𝐵 − sin 𝐴 sin 𝐵
tan(𝐴 + 𝐵) = 1−tan 𝐴 tan 𝐵
cos(𝐴 − 𝐵) = cos 𝐴 cos 𝐵 + sin 𝐴 sin 𝐵
tan 𝐴 − tan 𝐵
sin(𝐴 + 𝐵) = sin 𝐴 cos 𝐵 + cos 𝐴 sin 𝐵
tan(𝐴 − 𝐵) = 1+tan 𝐴 tan 𝐵
sin(𝐴 − 𝐵) = sin 𝐴 cos 𝐵 − cos 𝐴 sin 𝐵
Double Angle Identities
cos 2𝐴 = cos 2 𝐴 − sin2 𝐴
sin 2𝐴 = 2 sin 𝐴 cos 𝐴
cos 2𝐴 = 1 − 2 sin2 𝐴
2 tan 𝐴
cos 2𝐴 = 2 cos 2 𝐴 − 1
tan 2𝐴 = 1 −
tan2 𝐴
Half Angle Identities *NOTE: Either + or – is used depending upon the quadrant for the angle
cos
tan
𝐴
2
𝐴
2
1+cos 𝐴
2
= ±√
1−cos 𝐴
= ±√
1+cos 𝐴
sin
tan
𝐴
2
=
𝐴
2
= ±√
sin 𝐴
1 + cos 𝐴
𝐴
2
1−cos 𝐴
2
tan
𝐴
2
=
1 − cos 𝐴
sin 𝐴