The document is a cheat sheet for trigonometry identities and functions. It lists important trigonometric identities for basic functions, Pythagorean identities, double angle identities, sum and difference identities, product to sum identities, and triple angle identities. It also provides the function ranges and some key functional values for sin, cos, tan, and cot.

1. Symbolab Trigonometry Cheat Sheet
Basic Identities:
 tan(𝑥) =
sin(𝑥)
cos(𝑥)
 tan(𝑥) =
1
cot(𝑥)
 cot(𝑥) =
1
tan(𝑥)
 cot(𝑥) =
cos(𝑥)
sin(𝑥)
 sec(𝑥) =
1
cos(𝑥)
 csc(𝑥) =
1
sin(𝑥)
Pythagorean Identities
 cos2(𝑥) + sin2(𝑥) = 1
 sec2(𝑥) − tan2(𝑥) = 1
 csc2(𝑥) − cot2(𝑥) = 1
Double Angle Identities
 sin(2𝑥) = 2 sin(𝑥) cos(𝑥)
 cos(2𝑥) = 1 − 2 sin2(𝑥)
 cos(2𝑥) = 2 cos2(𝑥) − 1
 cos(2𝑥) = cos2(𝑥) − sin2(𝑥)
 tan(2𝑥) =
2 tan(𝑥)
1−tan2(𝑥)
Sum Difference Identities
 sin(𝑠 + 𝑡) = sin(𝑠) cos(𝑡) + cos(𝑠) sin(𝑡)
 sin(𝑠 − 𝑡) = sin(𝑠) cos(𝑡) − cos(𝑠) sin(𝑡)
 cos(𝑠 + 𝑡) = cos(𝑠) cos(𝑡) − sin(𝑠) sin(𝑡)
 cos(𝑠 − 𝑡) = cos(𝑠) cos(𝑡) + sin(𝑠) sin(𝑡)
 tan(𝑠 + 𝑡) =
tan(𝑠)+tan(𝑡)
1−tan(𝑠) tan(𝑡)
 tan(𝑠 − 𝑡) =
tan(𝑠)−tan(𝑡)
1+tan(𝑠) tan(𝑡)
Product To Sum Identities
 cos(𝑠) cos(𝑡) =
cos(𝑠−𝑡)+cos(𝑠+𝑡)
2
 sin(𝑠) sin(𝑡) =
cos(𝑠−𝑡)−cos(𝑠+𝑡)
2
 sin(𝑠) cos(𝑡) =
sin(𝑠+𝑡)+sin(𝑠−𝑡)
2
 cos(𝑠) sin(𝑡) =
sin(𝑠+𝑡)−sin(𝑠−𝑡)
2
Triple Angle Identities
 sin(3𝑥) = − sin3(𝑥) + 3 cos2(𝑥) sin(𝑥)
 sin(3𝑥) = −4 sin3(𝑥) + 3 sin(𝑥)
 cos(3𝑥) = cos3(𝑥) − 3 sin2(𝑥) cos(𝑥)
 cos(3𝑥) = 4 cos3(𝑥) − 3 cos(𝑥)
 tan(3𝑥) =
3 tan(𝑥)−tan3(𝑥)
1−3 tan2(𝑥)
 cot(3𝑥) =
3 cot(𝑥)−cot3(𝑥)
1−3 cot2(𝑥)

2. Function Ranges:
sin(𝑥) −1 ≤ 𝑦 ≤ 1 arcsin(𝑥) −
𝜋
2
≤ 𝑦 ≤
𝜋
2
cos(𝑥) −1 ≤ 𝑦 ≤ 1 arccos(𝑥) 0 ≤ 𝑦 ≤ 𝜋
tan(𝑥) ∞ < 𝑦 < ∞ arctan(𝑥) −
𝜋
2
< 𝑦 <
𝜋
2
cot(𝑥) ∞ < 𝑦 < ∞ arccot(𝑥) 0 < 𝑦 < 𝜋
csc(𝑥) −∞ < 𝑦 ≤ 1
∪ 1 ≤ 𝑦 < ∞
arccsc(𝑥)
0 ≤ 𝑦 <
𝜋
2
∪ 𝜋 ≤ 𝑦 <
3𝜋
2
sec(𝑥) −∞ < 𝑦 ≤ 1 ∪
1 ≤ 𝑦 < ∞
arcsec(𝑥) −𝜋 < 𝑦 ≤ −
𝜋
2
∪ 0 < 𝑦 <
𝜋
2
Function Values:
𝐬𝐢𝐧(𝒙) 𝐜𝐨𝐬(𝒙) 𝐭𝐚𝐧(𝒙) 𝐜𝐨𝐭(𝒙)
𝟎 0 1 0 Undefined
𝝅
𝟔
1
2
√3
2
1
√3
√3
𝝅
𝟒
1
√2
1
√2
1 1
𝝅
𝟑
√3
2
1
√3
√3 1
√3
𝝅
𝟐
1 0 Undefined 0
𝟐𝝅
𝟑
√3
2
−
1
2
−√3 −
1
√3
𝟑𝝅
𝟒
1
√2
−
1
√2
−
1
√3
−√3
𝟓𝝅
𝟔
1
2 −
√3
2
−1 −1
𝝅 0 −1 0 Undefined
𝟕𝝅
𝟔
−
1
2 −
√3
2
𝟓𝝅
𝟒
−
1
√2
−
1
√2
𝟒𝝅
𝟑 −
√3
2
−
1
2
𝟑𝝅
𝟐
−1 0
𝟓𝝅
𝟑 −
√3
2
1
2
𝟕𝝅
𝟒
−
1
√2
1
√2
𝟏𝟏𝝅
𝟔
−
1
2
√3
2