📐 Pre-Calculus

Trigonometry
Master Quiz

20 exam-style problems across all core units · Detailed AI-verified solutions · Modeled after SAT, ACT & college placement standards

20
Problems
8
Units
30min
Suggested
⏱ Elapsed Time
00:00
Score
0/0

Review each concept, memorize key formulas, then study the worked examples before attempting the quiz.

1
Angles & Radian Measure

Angles can be measured in degrees or radians. One full revolution = 360° = 2π radians.

Radians → Degrees
× (180/π)
Degrees → Radians
× (π/180)
Arc Length
s = rθ
Coterminal Angles
θ + 360n°
⚡ Must Memorize
  • 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2
  • 120° = 2π/3, 135° = 3π/4, 150° = 5π/6, 180° = π
  • 270° = 3π/2, 360° = 2π
📝 Example
Convert 240° to radians.
240 × π/180 = 240π/180 = 4π/3
2
The Unit Circle

The unit circle has radius 1, centered at origin. For angle θ, the point on the circle is (cos θ, sin θ).

⚡ Key Points on Unit Circle
  • 0°(0): (1, 0)
  • 30°(π/6): (√3/2, 1/2)
  • 45°(π/4): (√2/2, √2/2)
  • 60°(π/3): (1/2, √3/2)
  • 90°(π/2): (0, 1)
  • ASTC Rule: All → Sine → Tangent → Cosine (positive per quadrant)
📝 Example
Find sin(5π/6).
5π/6 is in Q2 (reference angle π/6). sin(5π/6) = sin(π/6) = 1/2
3
Trig Ratios & Right Triangles

SOH-CAH-TOA defines the basic trig ratios in a right triangle.

sin θ
opp / hyp
cos θ
adj / hyp
tan θ
opp / adj
csc θ
hyp / opp
sec θ
hyp / adj
cot θ
adj / opp
⚡ Special Triangles
  • 30-60-90: sides = 1, √3, 2
  • 45-45-90: sides = 1, 1, √2
  • Pythagorean triples: 3-4-5, 5-12-13, 8-15-17
4
Trigonometric Identities
Pythagorean
sin²θ + cos²θ = 1
Pythagorean
1 + tan²θ = sec²θ
Pythagorean
1 + cot²θ = csc²θ
Quotient
tan θ = sin/cos
⚡ Even/Odd Identities
  • cos(−θ) = cos θ  [Even]
  • sin(−θ) = −sin θ  [Odd]
  • tan(−θ) = −tan θ  [Odd]
📝 Example
Simplify: (sin²θ − 1) / cos θ
= −cos²θ / cos θ = −cos θ (since sin²θ−1 = −cos²θ)
5
Graphs of Trig Functions

General form: y = A·sin(Bx + C) + D or y = A·cos(Bx + C) + D

Amplitude
|A|
Period
2π / |B|
Phase Shift
−C / B
Vertical Shift
D
⚡ Key Graph Facts
  • sin(0) = 0, cos(0) = 1 (starting values)
  • sin and cos: period = 2π, range [−1, 1]
  • tan: period = π, vertical asymptotes at π/2 + nπ
6
Sum, Difference & Double Angle
sin(A±B)
sinA cosB ± cosA sinB
cos(A±B)
cosA cosB ∓ sinA sinB
sin 2θ
2 sinθ cosθ
cos 2θ
cos²θ − sin²θ
cos 2θ alt
2cos²θ − 1
cos 2θ alt
1 − 2sin²θ
7
Inverse Trig Functions
arcsin domain
[−1, 1]
arcsin range
[−π/2, π/2]
arccos range
[0, π]
arctan range
(−π/2, π/2)
📝 Example
Find arctan(−1).
arctan(−1) = −π/4 (since tan(−π/4) = −1, in range (−π/2, π/2))
8
Law of Sines & Cosines
Law of Sines
a/sinA = b/sinB = c/sinC
Law of Cosines
c² = a² + b² − 2ab cosC
Area of Triangle
½ ab sinC
Use LoS when
AAS, ASA, SSA
⚡ When to Use Which
  • Two angles + one side → Law of Sines
  • Two sides + included angle → Law of Cosines
  • Three sides → Law of Cosines
  • SSA → Law of Sines (watch for ambiguous case!)