📘 College Trigonometry Study Guide
1. Foundations of Trigonometry
Angles
● Measured in degrees or radians
● Conversion:
○ Degrees → Radians:
θrad=θdeg⋅π180θrad=θdeg⋅180π
○ Radians → Degrees:
θdeg=θrad⋅180πθdeg=θrad⋅π180
Standard Position
● Vertex at origin
● Initial side on positive x-axis
● Counterclockwise = positive angle
2. The Unit Circle (MOST IMPORTANT)
x2+y2=1x2+y2=1
θθ
P(θ)=(cos(θ), sin(θ))=(22, 22)P(θ)=(cos(θ),sin(θ))=(22,22)
(22, 22)(22,22)
● Defines trig functions for all angles
● Coordinates: (cosθ,sinθ)(cosθ,sinθ)
Key Values (Memorize!)
Angle sin cos tan
0° 0 1 0
30° 1/2 √3/2 √3/3
45° √2/2 √2/2 1
60° √3/2 1/2 √3
, 90° 1 0 undefine
d
3. Trigonometric Functions
Basic Functions
sinθ=yr,cosθ=xr,tanθ=yxsinθ=ry,cosθ=rx,tanθ=xy
Reciprocal Functions
cscθ=1sinθ,secθ=1cosθ,cotθ=1tanθcscθ=sinθ1,secθ=cosθ1,cotθ=tanθ1
4. Fundamental Identities
Pythagorean Identity
sin2θ+cos2θ=1sin2θ+cos2θ=1
θθ
sin2θ≈0.329, cos2θ≈0.671sin2θ≈0.329,cos2θ≈0.671
sin2θ+cos2θ≈1sin2θ+cos2θ≈1
θ = 35°|cos θ| = 0.819|sin θ| = 0.574cos² θsin² θ0.671 + 0.329 = 1
Derived identities:
● 1+tan2θ=sec2θ1+tan2θ=sec2θ
● 1+cot2θ=csc2θ1+cot2θ=csc2θ
Quotient Identities
● tanθ=sinθcosθtanθ=cosθsinθ
● cotθ=cosθsinθcotθ=sinθcosθ
Even/Odd Identities
● cos(−θ)=cosθcos(−θ)=cosθ
● sin(−θ)=−sinθsin(−θ)=−sinθ
● tan(−θ)=−tanθtan(−θ)=−tanθ
, 5. Graphs of Trig Functions
Sine & Cosine
y=Asin(Bx−C)+Dy=Asin(Bx−C)+D
aa
bb
cc
dd
-10-8-6-4-2246810-0.50.511.52
● Amplitude = |A|
● Period = 2πBB2π
● Phase shift = CBBC
● Vertical shift = D
Tangent
y=tan(x)y=tan(x)
-10-8-6-4-2246810-55
● Period = π
● Vertical asymptotes at:
x=π2+kπx=2π+kπ
6. Inverse Trigonometric Functions
● sin−1(x)sin−1(x) (arcsin)
● cos−1(x)cos−1(x) (arccos)
● tan−1(x)tan−1(x) (arctan)
Key Ranges
● arcsin → [−π2,π2][−2π,2π]
● arccos → [0,π][0,π]
● arctan → (−π2,π2)(−2π,2π)
7. Trigonometric Equations
1. Foundations of Trigonometry
Angles
● Measured in degrees or radians
● Conversion:
○ Degrees → Radians:
θrad=θdeg⋅π180θrad=θdeg⋅180π
○ Radians → Degrees:
θdeg=θrad⋅180πθdeg=θrad⋅π180
Standard Position
● Vertex at origin
● Initial side on positive x-axis
● Counterclockwise = positive angle
2. The Unit Circle (MOST IMPORTANT)
x2+y2=1x2+y2=1
θθ
P(θ)=(cos(θ), sin(θ))=(22, 22)P(θ)=(cos(θ),sin(θ))=(22,22)
(22, 22)(22,22)
● Defines trig functions for all angles
● Coordinates: (cosθ,sinθ)(cosθ,sinθ)
Key Values (Memorize!)
Angle sin cos tan
0° 0 1 0
30° 1/2 √3/2 √3/3
45° √2/2 √2/2 1
60° √3/2 1/2 √3
, 90° 1 0 undefine
d
3. Trigonometric Functions
Basic Functions
sinθ=yr,cosθ=xr,tanθ=yxsinθ=ry,cosθ=rx,tanθ=xy
Reciprocal Functions
cscθ=1sinθ,secθ=1cosθ,cotθ=1tanθcscθ=sinθ1,secθ=cosθ1,cotθ=tanθ1
4. Fundamental Identities
Pythagorean Identity
sin2θ+cos2θ=1sin2θ+cos2θ=1
θθ
sin2θ≈0.329, cos2θ≈0.671sin2θ≈0.329,cos2θ≈0.671
sin2θ+cos2θ≈1sin2θ+cos2θ≈1
θ = 35°|cos θ| = 0.819|sin θ| = 0.574cos² θsin² θ0.671 + 0.329 = 1
Derived identities:
● 1+tan2θ=sec2θ1+tan2θ=sec2θ
● 1+cot2θ=csc2θ1+cot2θ=csc2θ
Quotient Identities
● tanθ=sinθcosθtanθ=cosθsinθ
● cotθ=cosθsinθcotθ=sinθcosθ
Even/Odd Identities
● cos(−θ)=cosθcos(−θ)=cosθ
● sin(−θ)=−sinθsin(−θ)=−sinθ
● tan(−θ)=−tanθtan(−θ)=−tanθ
, 5. Graphs of Trig Functions
Sine & Cosine
y=Asin(Bx−C)+Dy=Asin(Bx−C)+D
aa
bb
cc
dd
-10-8-6-4-2246810-0.50.511.52
● Amplitude = |A|
● Period = 2πBB2π
● Phase shift = CBBC
● Vertical shift = D
Tangent
y=tan(x)y=tan(x)
-10-8-6-4-2246810-55
● Period = π
● Vertical asymptotes at:
x=π2+kπx=2π+kπ
6. Inverse Trigonometric Functions
● sin−1(x)sin−1(x) (arcsin)
● cos−1(x)cos−1(x) (arccos)
● tan−1(x)tan−1(x) (arctan)
Key Ranges
● arcsin → [−π2,π2][−2π,2π]
● arccos → [0,π][0,π]
● arctan → (−π2,π2)(−2π,2π)
7. Trigonometric Equations
