Pre-Calculus Trigonometry | 20 Essential Problems

Angle Measures: Degrees & Radians

An angle can be measured in degrees or radians.

Conversion Formulas Degrees → Radians: multiply by π/180
Radians → Degrees: multiply by 180/π
Full circle: 360° = 2π rad | Half: 180° = π rad

Key values: 30°= π/6, 45°= π/4, 60°= π/3, 90°= π/2, 120°= 2π/3, 150°= 5π/6

✦ Worked Example

Convert 150° to radians.

150 × (π/180) = 5π/6

The Unit Circle

On the unit circle (radius = 1), a point at angle θ has coordinates (cos θ, sin θ). Memorize these key values:

θ = 0° (0)cos=1, sin=0

θ = 30° (π/6)cos=√3/2, sin=1/2

θ = 45° (π/4)cos=√2/2, sin=√2/2

θ = 60° (π/3)cos=1/2, sin=√3/2

θ = 90° (π/2)cos=0, sin=1

θ = 180° (π)cos=−1, sin=0

θ = 270° (3π/2)cos=0, sin=−1

θ = 120° (2π/3)cos=−1/2, sin=√3/2

CAST Rule (Signs by Quadrant) Q1 (0–90°): All positive
Q2 (90–180°): Sin positive
Q3 (180–270°): Tan positive
Q4 (270–360°): Cos positive

Six Trigonometric Functions

For a right triangle with opposite (opp), adjacent (adj), hypotenuse (hyp):

SOH-CAH-TOA + Reciprocals sin θ = opp/hyp    csc θ = hyp/opp
cos θ = adj/hyp    sec θ = hyp/adj
tan θ = opp/adj    cot θ = adj/opp
tan θ = sin θ / cos θ

✦ Worked Example

In a right triangle, opp = 3, hyp = 5. Find sin θ, cos θ, tan θ.

sin θ = 3/5, cos θ = 4/5, tan θ = 3/4

Pythagorean & Fundamental Identities

Must Memorize — Pythagorean Identities sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ

Even / Odd Identities cos(−θ) = cos θ [even]
sin(−θ) = −sin θ [odd]
tan(−θ) = −tan θ [odd]

Sum & Difference / Double Angle Formulas

Sum & Difference sin(A±B) = sinA cosB ± cosA sinB
cos(A±B) = cosA cosB ∓ sinA sinB
tan(A±B) = (tanA ± tanB)/(1 ∓ tanA·tanB)

Double Angle Formulas sin(2θ) = 2 sinθ cosθ
cos(2θ) = cos²θ − sin²θ = 1−2sin²θ = 2cos²θ−1
tan(2θ) = 2tanθ/(1 − tan²θ)

✦ Worked Example

Find the exact value of sin(75°) using sin(45°+30°).

sin75° = sin45°cos30° + cos45°sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6+√2)/4

Graphs of Trig Functions

For y = A·sin(Bx + C) + D:

Key Parameters Amplitude = |A| (not for tan/cot)
Period = 2π/B (sin, cos) | π/B (tan, cot)
Phase Shift = −C/B (left if neg, right if pos)
Vertical Shift = D

Domain/Range: sin & cos domain: all reals; range: [−|A|+D, |A|+D]
tan domain: x ≠ π/2 + nπ; range: all reals

Law of Sines & Law of Cosines

Law of Sines (AAS, ASA, SSA) a/sin A = b/sin B = c/sin C

Law of Cosines (SAS, SSS) a² = b² + c² − 2bc·cos A
b² = a² + c² − 2ac·cos B
c² = a² + b² − 2ab·cos C

Area of a triangle: Area = (1/2)ab·sin C

✦ Worked Example

In triangle ABC, a = 8, b = 5, C = 60°. Find c.

c² = 64 + 25 − 2(8)(5)(0.5) = 89 − 40 = 49, so c = 7

Inverse Trig & Solving Equations

Inverse Trig Ranges arcsin: domain [−1,1], range [−π/2, π/2]
arccos: domain [−1,1], range [0, π]
arctan: domain all reals, range (−π/2, π/2)

When solving sin θ = k over [0, 2π):
Find reference angle α = arcsin|k|, then use CAST to find both solutions.
General solution: θ = α + 2nπ or θ = (π − α) + 2nπ

✦ Worked Example

Solve: 2cos θ − 1 = 0 on [0, 2π)

cos θ = 1/2 → θ = π/3 or θ = 5π/3

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