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Trigonometry Cheat Sheet

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Symbolab Trigonometry Cheat Sheet
Basic Identities:
sin(π‘₯)
ο‚· tan(π‘₯) = cos(π‘₯)
cos(π‘₯)
ο‚·
cot(π‘₯) =
1
tan(π‘₯) = cot(π‘₯)
ο‚·
1
sec(π‘₯) = cos(π‘₯)
cot(π‘₯) = tan(π‘₯)
ο‚·
csc(π‘₯) = sin(π‘₯)
Pythagorean Identities
ο‚· cos 2 (π‘₯) + sin2 (π‘₯) = 1
ο‚· sec 2 (π‘₯) βˆ’ tan2 (π‘₯) = 1
ο‚·
csc 2 (π‘₯) βˆ’ cot 2 (π‘₯) = 1
Double Angle Identities
ο‚· sin(2π‘₯) = 2 sin(π‘₯) cos(π‘₯)
ο‚· cos(2π‘₯) = 1 βˆ’ 2 sin2(π‘₯)
ο‚· cos(2π‘₯) = 2 cos 2 (π‘₯) βˆ’ 1
ο‚·
ο‚·
cos(2π‘₯) = cos2 (π‘₯) βˆ’ sin2 (π‘₯)
2 tan(π‘₯)
tan(2π‘₯) = 1βˆ’tan2(π‘₯)
ο‚·
sin(𝑠) cos(𝑑) =
ο‚·
cos(𝑠) sin(𝑑) =
ο‚·
ο‚·
sin(π‘₯)
1
1
Sum Difference Identities
ο‚· sin(𝑠 + 𝑑) = sin(𝑠) cos(𝑑) + cos(𝑠) sin(𝑑)
ο‚· sin(𝑠 βˆ’ 𝑑) = sin(𝑠) cos(𝑑) βˆ’ cos(𝑠) sin(𝑑)
ο‚· cos(𝑠 + 𝑑) = cos(𝑠) cos(𝑑) βˆ’ sin(𝑠) sin(𝑑)
ο‚· cos(𝑠 βˆ’ 𝑑) = cos(𝑠) cos(𝑑) + sin(𝑠) sin(𝑑)
tan(𝑠)+tan(𝑑)
ο‚· tan(𝑠 + 𝑑) = 1βˆ’tan(𝑠) tan(𝑑)
ο‚·
tan(𝑠)βˆ’tan(𝑑)
tan(𝑠 βˆ’ 𝑑) = 1+tan(𝑠) tan(𝑑)
Product To Sum Identities
cos(π‘ βˆ’π‘‘)+cos(𝑠+𝑑)
ο‚· cos(𝑠) cos(𝑑) =
2
ο‚·
sin(𝑠) sin(𝑑) =
cos(π‘ βˆ’π‘‘)βˆ’cos(𝑠+𝑑)
2
Triple Angle Identities
ο‚· sin(3π‘₯) = βˆ’ sin3 (π‘₯) + 3 cos2 (π‘₯) sin(π‘₯)
ο‚· sin(3π‘₯) = βˆ’4 sin3 (π‘₯) + 3 sin(π‘₯)
ο‚· cos(3π‘₯) = cos 3 (π‘₯) βˆ’ 3 sin2 (π‘₯) cos(π‘₯)
ο‚· cos(3π‘₯) = 4 cos 3 (π‘₯) βˆ’ 3 cos(π‘₯)
ο‚·
tan(3π‘₯) =
ο‚·
cot(3π‘₯) =
3 tan(π‘₯)βˆ’tan3 (π‘₯)
1βˆ’3 tan2 (π‘₯)
3 cot(π‘₯)βˆ’cot3 (π‘₯)
1βˆ’3 cot2 (π‘₯)
sin(𝑠+𝑑)+sin(π‘ βˆ’π‘‘)
2
sin(𝑠+𝑑)βˆ’sin(π‘ βˆ’π‘‘)
2
Function Ranges:
sin(π‘₯)
βˆ’1 ≀ 𝑦 ≀ 1
cos(π‘₯)
tan(π‘₯)
cot(π‘₯)
csc(π‘₯)
sec(π‘₯)
βˆ’1 ≀ 𝑦 ≀ 1
∞<𝑦<∞
∞<𝑦<∞
βˆ’βˆž < 𝑦 ≀ 1
βˆͺ1≀𝑦<∞
βˆ’βˆž < 𝑦 ≀ 1 βˆͺ
1≀𝑦<∞
arcsin(π‘₯)
arccos(π‘₯)
arctan(π‘₯)
arccot(π‘₯)
arccsc(π‘₯)
arcsec(π‘₯)
πœ‹
πœ‹
≀𝑦≀
2
2
0β‰€π‘¦β‰€πœ‹
πœ‹
πœ‹
βˆ’ <𝑦<
2
2
0<𝑦<πœ‹
πœ‹
3πœ‹
0≀𝑦 < βˆͺπœ‹ ≀𝑦 <
2
2
πœ‹
πœ‹
βˆ’πœ‹ < 𝑦 ≀ βˆ’ βˆͺ 0 < 𝑦 <
2
2
βˆ’
Function Values:
𝟎
𝝅
πŸ”
𝝅
πŸ’
𝝅
πŸ‘
𝝅
𝟐
πŸπ…
πŸ‘
πŸ‘π…
πŸ’
πŸ“π…
πŸ”
𝝅
πŸ•π…
πŸ”
πŸ“π…
πŸ’
πŸ’π…
πŸ‘
πŸ‘π…
𝟐
πŸ“π…
πŸ‘
πŸ•π…
πŸ’
πŸπŸπ…
πŸ”
𝐬𝐒𝐧(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
βˆ’
2
√2
βˆ’
2
√3
βˆ’
2
βˆ’1
1
2
√2
βˆ’
2
√3
βˆ’
2
βˆ’1
√3
βˆ’
2
√2
βˆ’
2
1
βˆ’
2
0
√3
2
√2
βˆ’
2
1
βˆ’
2
1
2
√2
2
√3
2
βˆ’
βˆ’
Undefined
βˆ’βˆš3
√3
3
βˆ’1
βˆ’
0
𝐜𝐨𝐭(𝒙)
1
√3
3
βˆ’βˆš3
βˆ’
βˆ’1
Undefined
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