Download Math 65A Trig “Cheat Sheet”

Math 65B “Trig Toolbox”
Definition of the Six Trigonometric Functions
Right triangle definitions, 0 < θ < π/2.
Hypotenuse
Opposite
Trigonometric Functions on the Unit Circle
On the unit circle, for any ordered pair ( x, y ) ,
( x, y ) = (cosθ , sin θ ) where θ is the indicated
angle.
θ
Adjacent
opposite
hypotenuse
adjacent
cosθ =
hypotenuse
opposite
tan θ =
adjacent
hypotenuse
opposite
hypotenuse
secθ =
adjacent
adjacent
cot θ =
opposite
sin θ =
cscθ =
Rectangular coordinate system definitions, where θ is any
angle.
y
y
r
sin θ =
cscθ =
2
2
r
y
r= x +y
x
r
cosθ =
(x, y)
secθ =
r
x
r
θ
y
y
x
tan θ =
x
cot θ =
x
x
y
Special Triangles
30°
45°
2
3
2
1
45°
60°
1
1
Reduction Formulas
Reciprocal Identities
1
1
sin x =
sec x =
csc x
cos x
1
1
csc x =
cos x =
sin x
sec x
Recall: y = sin x is an odd function and
y = cos x is an even function.
1
cot x
1
cot x =
tan x
tan x =
sin(− x) = − sin x
cos(− x) = cos x
tan(− x) = − tan x
csc(− x) = − csc x
sec(− x) = sec x
cot(− x) = − cot x
Sum and Difference Formulas
Ratio Identities
sin x
tan x =
cos x
cot x =
cos x
sin x
Pythagorean Identities
sin 2 x + cos 2 x = 1,
cos( A ± B) = cos A cos B ∓ sin A sin B
Double Angle Formulas
1 + tan 2 x = sec 2 x
1 + cot 2 x = csc2 x
Cofunction Identities
⎛π
⎞
sin ⎜ − x ⎟ = cos x
⎝2
⎠
⎛π
⎞
tan ⎜ − x ⎟ = cot x
⎝2
⎠
sin( A ± B) = sin A cos B ± cos A sin B
sin 2 x = 2sin x cos x
cos 2 x = cos 2 x − sin 2 x = 2cos 2 x − 1 = 1 − 2sin 2 x
2 tan x
tan 2 x =
1 − tan 2 x
Power Reduction (Half Angle) Formulas
⎛π
⎞
cos ⎜ − x ⎟ = sin x
⎝2
⎠
⎛π
⎞
cot ⎜ − x ⎟ = tan x
⎝2
⎠
1 − cos 2 x
2
1 + cos 2 x
cos 2 x =
2
sin 2 x =