Summary Ultimate College Trigonometry Study Guide: 150 Practice Problems with Step-by-Step Solutions

📘 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: (cos⁡θ,sin⁡θ)(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
sin⁡θ=yr,cos⁡θ=xr,tan⁡θ=yxsinθ=ry​,cosθ=rx​,tanθ=xy​

Reciprocal Functions
csc⁡θ=1sin⁡θ,sec⁡θ=1cos⁡θ,cot⁡θ=1tan⁡θcscθ=sinθ1​,secθ=cosθ1​,cotθ=tanθ1​



4. Fundamental Identities
Pythagorean Identity
sin⁡2θ+cos⁡2θ=1sin2θ+cos2θ=1
θθ
sin⁡2θ≈0.329, cos⁡2θ≈0.671sin2θ≈0.329,cos2θ≈0.671
sin⁡2θ+cos⁡2θ≈1sin2θ+cos2θ≈1
θ = 35°|cos θ| = 0.819|sin θ| = 0.574cos² θsin² θ0.671 + 0.329 = 1

Derived identities:

●​ 1+tan⁡2θ=sec⁡2θ1+tan2θ=sec2θ
●​ 1+cot⁡2θ=csc⁡2θ1+cot2θ=csc2θ




Quotient Identities

●​ tan⁡θ=sin⁡θcos⁡θtanθ=cosθsinθ​
●​ cot⁡θ=cos⁡θsin⁡θcotθ=sinθcosθ​




Even/Odd Identities

●​ cos⁡(−θ)=cos⁡θcos(−θ)=cosθ
●​ sin⁡(−θ)=−sin⁡θsin(−θ)=−sinθ
●​ tan⁡(−θ)=−tan⁡θtan(−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