# Trigonometry Cheat Sheet

```Symbolab Trigonometry Cheat Sheet
Basic Identities:
sin(𝑥)
 tan(𝑥) = cos(𝑥)
cos(𝑥)

cot(𝑥) =
1
tan(𝑥) = cot(𝑥)

1
sec(𝑥) = cos(𝑥)
cot(𝑥) = tan(𝑥)

csc(𝑥) = sin(𝑥)
Pythagorean Identities
 cos 2 (𝑥) + sin2 (𝑥) = 1
 sec 2 (𝑥) − tan2 (𝑥) = 1

csc 2 (𝑥) − cot 2 (𝑥) = 1
Double Angle Identities
 sin(2𝑥) = 2 sin(𝑥) cos(𝑥)
 cos(2𝑥) = 1 − 2 sin2(𝑥)
 cos(2𝑥) = 2 cos 2 (𝑥) − 1


cos(2𝑥) = cos2 (𝑥) − sin2 (𝑥)
2 tan(𝑥)
tan(2𝑥) = 1−tan2(𝑥)

sin(𝑠) cos(𝑡) =

cos(𝑠) sin(𝑡) =


sin(𝑥)
1
1
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)+tan(𝑡)
 tan(𝑠 + 𝑡) = 1−tan(𝑠) tan(𝑡)

tan(𝑠)−tan(𝑡)
tan(𝑠 − 𝑡) = 1+tan(𝑠) tan(𝑡)
Product To Sum Identities
cos(𝑠−𝑡)+cos(𝑠+𝑡)
 cos(𝑠) cos(𝑡) =
2

sin(𝑠) sin(𝑡) =
cos(𝑠−𝑡)−cos(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) = cos 3 (𝑥) − 3 sin2 (𝑥) cos(𝑥)
 cos(3𝑥) = 4 cos 3 (𝑥) − 3 cos(𝑥)

tan(3𝑥) =

cot(3𝑥) =
3 tan(𝑥)−tan3 (𝑥)
1−3 tan2 (𝑥)
3 cot(𝑥)−cot3 (𝑥)
1−3 cot2 (𝑥)
sin(𝑠+𝑡)+sin(𝑠−𝑡)
2
sin(𝑠+𝑡)−sin(𝑠−𝑡)
2
Function Ranges:
sin(𝑥)
−1 ≤ 𝑦 ≤ 1
cos(𝑥)
tan(𝑥)
cot(𝑥)
csc(𝑥)
sec(𝑥)
−1 ≤ 𝑦 ≤ 1
∞&lt;𝑦&lt;∞
∞&lt;𝑦&lt;∞
−∞ &lt; 𝑦 ≤ 1
∪1≤𝑦&lt;∞
−∞ &lt; 𝑦 ≤ 1 ∪
1≤𝑦&lt;∞
arcsin(𝑥)
arccos(𝑥)
arctan(𝑥)
arccot(𝑥)
arccsc(𝑥)
arcsec(𝑥)
𝜋
𝜋
≤𝑦≤
2
2
0≤𝑦≤𝜋
𝜋
𝜋
− &lt;𝑦&lt;
2
2
0&lt;𝑦&lt;𝜋
𝜋
3𝜋
0≤𝑦 &lt; ∪𝜋 ≤𝑦 &lt;
2
2
𝜋
𝜋
−𝜋 &lt; 𝑦 ≤ − ∪ 0 &lt; 𝑦 &lt;
2
2
−
Function Values:
𝟎
𝝅
𝟔
𝝅
𝟒
𝝅
𝟑
𝝅
𝟐
𝟐𝝅
𝟑
𝟑𝝅
𝟒
𝟓𝝅
𝟔
𝝅
𝟕𝝅
𝟔
𝟓𝝅
𝟒
𝟒𝝅
𝟑
𝟑𝝅
𝟐
𝟓𝝅
𝟑
𝟕𝝅
𝟒
𝟏𝟏𝝅
𝟔
𝐬𝐢𝐧(𝒙)
𝐜𝐨𝐬(𝒙)
𝐭𝐚𝐧(𝒙)
0
1
2
√2
2
√3
2
1
1
√3
2
√2
2
1
2
0
0
√3
3
1
Undefined
√3
√3
√3
3
0
√3
2
√2
2
1
2
0
1
−
2
√2
−
2
√3
−
2
−1
1
2
√2
−
2
√3
−
2
−1
√3
−
2
√2
−
2
1
−
2
0
√3
2
√2
−
2
1
−
2
1
2
√2
2
√3
2
−
−
Undefined
−√3
√3
3
−1
−
0
𝐜𝐨𝐭(𝒙)
1
√3
3
−√3
−
−1
Undefined
```