Formulas and Identities

Trig Cheat Sheet Definition of the Trig Functions Right triangle definition For this definition we assume that 0 <θ <

π

2

or 0° < θ < 90° .

Unit circle definition For this definition θ is any angle. y

( x, y ) hypotenuse

y

opposite

1

θ x

x

θ

tan θ + 1 = sec θ 2

sin θ =

Tangent and Cotangent Identities sin θ cos θ tan θ = cot θ = cos θ sin θ Reciprocal Identities 1 1 csc θ = sin θ = sin θ csc θ 1 1 sec θ = cos θ = cos θ sec θ 1 1 cot θ = tan θ = tan θ cot θ Pythagorean Identities sin 2 θ + cos 2 θ = 1

hypotenuse opposite hypotenuse sec θ = adjacent adjacent cot θ = opposite csc θ =

y =y 1 x cos θ = = x 1 y tan θ = x

sin θ =

1 y 1 sec θ = x x cot θ = y csc θ =

Facts and Properties Domain The domain is all the values of θ that can be plugged into the function. sin θ , θ can be any angle cos θ , θ can be any angle 1⎞ ⎛ tan θ , θ ≠ ⎜ n + ⎟ π , n = 0, ± 1, ± 2,… 2⎠ ⎝ θ ≠ n π , n = 0, ± 1, ± 2,… csc θ , 1⎞ ⎛ sec θ , θ ≠ ⎜ n + ⎟ π , n = 0, ± 1, ± 2,… 2⎠ ⎝ θ ≠ n π , n = 0, ± 1, ± 2,… cot θ , Range The range is all possible values to get out of the function. csc θ ≥ 1 and csc θ ≤ −1 −1 ≤ sin θ ≤ 1 −1 ≤ cos θ ≤ 1 sec θ ≥ 1 and sec θ ≤ −1 −∞ ≤ tan θ ≤ ∞ −∞ ≤ cot θ ≤ ∞

Period The period of a function is the number, T, such that f (θ + T ) = f (θ ) . So, if ω is a fixed number and θ is any angle we have the following periods. sin (ωθ ) →

T=

cos (ω θ ) →

T

tan ( ωθ ) →

T

csc (ωθ ) →

T

sec (ω θ ) →

T

cot (ωθ ) →

T

ω 2π = ω π = ω 2π = ω 2π = ω π = ω

2

1 + cot 2 θ = csc 2 θ Even/Odd Formulas sin ( −θ ) = − sin θ csc ( −θ ) = − cscθ cos ( −θ ) = cos θ

sec ( −θ ) = sec θ

tan ( −θ ) = − tan θ

cot ( −θ ) = − cot θ

Periodic Formulas If n is an integer. sin (θ + 2π n ) = sin θ

csc (θ + 2π n ) = csc θ

cos (θ + 2π n ) = cos θ sec (θ + 2π n ) = sec θ tan (θ + π n ) = tan θ cot (θ + π n ) = cot θ Double Angle Formulas sin ( 2θ ) = 2sin θ cos θ cos ( 2θ ) = cos 2 θ − sin 2 θ = 2 cos 2 θ − 1 = 1 − 2sin 2 θ 2 tan θ tan ( 2θ ) = 1 − tan 2 θ Degrees to Radians Formulas If x is an angle in degrees and t is an angle in radians then π πx 180t t = ⇒ t= and x = π 180 x 180

Half Angle Formulas 1 sin 2 θ = (1 − cos ( 2θ ) ) 2 1 cos 2 θ = (1 + cos ( 2θ ) ) 2 1 − cos ( 2θ ) 2 tan θ = 1 + cos ( 2θ ) Sum and Difference Formulas sin (α ± β ) = sin α cos β ± cos α sin β cos (α ± β ) = cos α cos β ∓ sin α sin β tan α ± tan β 1 ∓ tan α tan β Product to Sum Formulas 1 sin α sin β = ⎡⎣cos (α − β ) − cos (α + β ) ⎤⎦ 2 1 cos α cos β = ⎡⎣cos (α − β ) + cos (α + β ) ⎤⎦ 2 1 sin α cos β = ⎣⎡sin (α + β ) + sin (α − β ) ⎦⎤ 2 1 cos α sin β = ⎡⎣sin (α + β ) − sin (α − β ) ⎤⎦ 2 Sum to Product Formulas ⎛α + β ⎞ ⎛α −β ⎞ sin α + sin β = 2sin ⎜ ⎟ cos ⎜ ⎟ 2 ⎝ ⎠ ⎝ 2 ⎠ ⎛α + β ⎞ ⎛α − β ⎞ sin α − sin β = 2 cos ⎜ ⎟ ⎟ sin ⎜ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎛α + β ⎞ ⎛α −β ⎞ cos α + cos β = 2 cos ⎜ ⎟ cos ⎜ ⎟ 2 ⎝ ⎠ ⎝ 2 ⎠ ⎛α + β ⎞ ⎛α − β ⎞ cos α − cos β = −2sin ⎜ ⎟ sin ⎜ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ Cofunction Formulas tan (α ± β ) =

⎛π ⎞ sin ⎜ − θ ⎟ = cos θ ⎝2 ⎠

⎛π ⎞ cos ⎜ − θ ⎟ = sin θ ⎝2 ⎠

⎛π ⎞ csc ⎜ − θ ⎟ = sec θ ⎝2 ⎠

⎛π ⎞ sec ⎜ − θ ⎟ = csc θ ⎝2 ⎠

⎛π ⎞ tan ⎜ − θ ⎟ = cot θ 2 ⎝ ⎠

⎛π ⎞ cot ⎜ − θ ⎟ = tan θ 2 ⎝ ⎠

Unit Circle y

⎛ 3 1⎞ ⎜− , ⎟ ⎝ 2 2⎠

3π 4

( 0,1)

π

⎛ 1 3⎞ ⎜− , ⎟ ⎝ 2 2 ⎠ ⎛ 2 2⎞ , ⎜− ⎟ ⎝ 2 2 ⎠

Inverse Trig Functions

2 2π 3

π

90° 120°

3

sin ( sin −1 ( x ) ) = x

sin −1 ( sin (θ ) ) = θ

tan ( tan −1 ( x ) ) = x

tan −1 ( tan (θ ) ) = θ

π

45° 30°

Domain and Range Function Domain

⎛ 3 1⎞ ⎜⎜ 2 , 2 ⎟⎟ ⎝ ⎠

y = sin −1 x

6

−1 ≤ x ≤ 1

−1

150° ( −1,0 )

y = cos −1 x is equivalent to x = cos y y = tan x is equivalent to x = tan y

⎛ 2 2⎞ , ⎜⎜ ⎟⎟ ⎝ 2 2 ⎠

4

π 180°

Inverse Properties cos ( cos −1 ( x ) ) = x cos−1 ( cos (θ ) ) = θ

−1

π

60°

135°

5π 6

⎛1 3⎞ ⎜⎜ , ⎟⎟ ⎝2 2 ⎠

Definition y = sin −1 x is equivalent to x = sin y

0

360°

y = cos x

−1 ≤ x ≤ 1

y = tan −1 x

−∞ < x < ∞

(1,0 )

Range

π

⎛ 3 1⎞ ⎜− ,− ⎟ 2⎠ ⎝ 2

⎛ 2 2⎞ ,− ⎜− ⎟ 2 ⎠ ⎝ 2

210°

5π 4

330° 225°

4π 3

240°

⎛ 1 3⎞ ⎜ − ,− ⎟ ⎝ 2 2 ⎠

315° 7π 300° 270° 4 5π 3π 3 2 ⎛ ( 0,−1)

11π 6

π 2

< y<

For any ordered pair on the unit circle ( x, y ) : cos θ = x and sin θ = y

2

β

a

γ

Law of Sines sin α sin β sin γ = = a b c

Law of Tangents a − b tan 12 (α − β ) = a + b tan 12 (α + β )

Law of Cosines a 2 = b 2 + c 2 − 2bc cos α

b − c tan 12 ( β − γ ) = b + c tan 12 ( β + γ )

c = a + b − 2ab cos γ 2

⎛ 5π ⎞ 1 cos ⎜ ⎟ = ⎝ 3 ⎠ 2

π

b

b 2 = a 2 + c 2 − 2ac cos β

Example

tan −1 x = arctan x

α

⎛ 3 1⎞ ⎜ ,− ⎟ ⎝ 2 2⎠

1 3⎞ ⎜ ,− ⎟ ⎝2 2 ⎠

cos −1 x = arccos x

Law of Sines, Cosines and Tangents

x

⎛ 2 2⎞ ,− ⎜ ⎟ 2 ⎠ ⎝ 2

π

≤ y≤ 2 2 0≤ y ≤π

c 7π 6

Alternate Notation sin −1 x = arcsin x

2

2

a − c tan 12 (α − γ ) = a + c tan 12 (α + γ )

Mollweide’s Formula a + b cos 12 (α − β ) = c sin 12 γ

3 ⎛ 5π ⎞ sin ⎜ ⎟ = − 2 ⎝ 3 ⎠