Definition of the Trig Functions
Right triangle definition
For this definition θ is any angle.
For this definition we assume that Unit circle definition
π
0 <θ < or 0° < θ < 90° .
2 y
( x, y )
om
θ
hypotenuse 1
y
opposite x
x
θ
.c
adjacent
sin θ = csc θ = sin θ== = y= csc θ =
opposite hypotenuse y 1
ce
hypotenuse opposite 1 y
cos θ = sec θ = cos θ== = x= sec θ =
adjacent hypotenuse 1
hypotenuse adjacent 1 x
tan θ = cot θ = tan θ = cot θ =
opposite adjacent y x
ra
adjacent opposite x y
Facts and Properties
m
Domain
The domain is all the values of θ hat Period
T, such that f (θ + T ) =f (θ ) . So, if ω
can be plugged into the function. The period of a function is the number,
xa
sin θ , θ can be any angle is a fixed number and θ is any angle we
cos θ , θ can be any angle
⎛ 1⎞
have the following periods.
tan θ , θ ≠ ⎜ n + ⎟ π , n = = 0, ±±1, ±±2,…
.e
⎝ 2⎠ 2π
sin (ωθ ) → T=
csc θ , θ ≠ n π , n = 0, ± 1, ± 2,… ω
2π
⎛ cos (ωθ ) →
1⎞
w
sec θ , θ ≠ ⎜ n + ⎟ π , n = = 0, ±±1, ±±2,… T=
⎝ 2⎠ ω
π
cot θ , θ ≠ n π , n = 0, ± 1, ± 2,… tan (ωθ ) → =
w
ω
T
2π
Range csc (ωθ ) → =
w
ω
T
T e range is all possible values to get
2π
csc θ ≥ 1 and csc θ ≤ −1 sec (ωθ ) → =
out of the function.
−1 ≤ sin θ ≤ 1 T
ω
−1 ≤ cos θ ≤ 1 sec θ ≥ 1 and sec θ ≤ −1 π
−∞ ≤ tan θ ≤ ∞ −∞ ≤ cot θ ≤ ∞ cot (ωθ ) → =
ω
T
Right triangle definition
For this definition θ is any angle.
For this definition we assume that Unit circle definition
π
0 <θ < or 0° < θ < 90° .
2 y
( x, y )
om
θ
hypotenuse 1
y
opposite x
x
θ
.c
adjacent
sin θ = csc θ = sin θ== = y= csc θ =
opposite hypotenuse y 1
ce
hypotenuse opposite 1 y
cos θ = sec θ = cos θ== = x= sec θ =
adjacent hypotenuse 1
hypotenuse adjacent 1 x
tan θ = cot θ = tan θ = cot θ =
opposite adjacent y x
ra
adjacent opposite x y
Facts and Properties
m
Domain
The domain is all the values of θ hat Period
T, such that f (θ + T ) =f (θ ) . So, if ω
can be plugged into the function. The period of a function is the number,
xa
sin θ , θ can be any angle is a fixed number and θ is any angle we
cos θ , θ can be any angle
⎛ 1⎞
have the following periods.
tan θ , θ ≠ ⎜ n + ⎟ π , n = = 0, ±±1, ±±2,…
.e
⎝ 2⎠ 2π
sin (ωθ ) → T=
csc θ , θ ≠ n π , n = 0, ± 1, ± 2,… ω
2π
⎛ cos (ωθ ) →
1⎞
w
sec θ , θ ≠ ⎜ n + ⎟ π , n = = 0, ±±1, ±±2,… T=
⎝ 2⎠ ω
π
cot θ , θ ≠ n π , n = 0, ± 1, ± 2,… tan (ωθ ) → =
w
ω
T
2π
Range csc (ωθ ) → =
w
ω
T
T e range is all possible values to get
2π
csc θ ≥ 1 and csc θ ≤ −1 sec (ωθ ) → =
out of the function.
−1 ≤ sin θ ≤ 1 T
ω
−1 ≤ cos θ ≤ 1 sec θ ≥ 1 and sec θ ≤ −1 π
−∞ ≤ tan θ ≤ ∞ −∞ ≤ cot θ ≤ ∞ cot (ωθ ) → =
ω
T
