📐 Precalculus Series

Trigonometry
Mastery Quiz

20 Essential Problems · All Core Concepts

⏱ 40 min recommended 📊 Multiple Choice 🎯 Exam Style

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Unit 1

Angles & Radian Measure

📌 Core Concepts to Memorize

Conversionrad = deg × π/180

Full circle2π rad = 360°

Arc lengths = rθ (θ in rad)

Sector areaA = ½r²θ

Coterminalθ ± 360° (or ± 2π)

✦ Worked Example

Convert 150° to radians.

150° × π/180 = 5π/6 rad

Radian Conversion

Convert 225° to radians.

A 3π/4

B 5π/4

C 7π/6

D 4π/3

Arc Length

A circle has radius r = 6. Find the arc length subtended by a central angle of π/3 radians.

A 2π

B π

C 3π

D 4π

Unit 2

Unit Circle & Trig Functions

📌 Key Values — Memorize These

sin 0°0

sin 30°1/2

sin 45°√2/2

sin 60°√3/2

sin 90°1

ASTC ruleQ1:All Q2:Sin Q3:Tan Q4:Cos

★ Reciprocal Identities

csc θ

1/sin θ

sec θ

1/cos θ

cot θ

1/tan θ = cos/sin

tan θ

sin θ/cos θ

✦ Worked Example

Find tan(π/3).

sin(π/3)/cos(π/3) = (√3/2)/(1/2) = √3

Unit Circle Values

Evaluate cos(5π/6).

A √3/2

B 1/2

C −√3/2

D −1/2

ASTC / Quadrant Signs

If sin θ < 0 and cos θ > 0, in which quadrant does θ lie?

A Quadrant I

B Quadrant II

C Quadrant III

D Quadrant IV

Reference Angle

Find the exact value of sin(210°).

A 1/2

B −√3/2

C −1/2

D √3/2

Unit 3

Pythagorean Identities

★ The Big Three — Must Memorize

Primary

sin²θ + cos²θ = 1

Tangent form

tan²θ + 1 = sec²θ

Cotangent form

cot²θ + 1 = csc²θ

✦ Worked Example

If sin θ = 3/5 and θ is in Q1, find cos θ.

cos²θ = 1 − (9/25) = 16/25 → cos θ = 4/5

Pythagorean Identity

If cos θ = −5/13 and θ is in Quadrant III, find sin θ.

A 12/13

B −12/13

C 5/13

D −5/13

Identity Simplification

Simplify: (1 − sin²θ)/cos θ

A sin θ

B cos θ

C tan θ

D sec θ

Unit 4

Graphing Trig Functions

📌 y = A·sin(Bx + C) + D

Amplitude|A|

Period2π/|B|

Phase shift−C/B (left if +, right if −)

Vertical shiftD (midline)

Amplitude & Period

For y = −3 sin(2x − π/2) + 1, what is the amplitude and period?

A Amplitude = 3, Period = π

B Amplitude = −3, Period = π

C Amplitude = 3, Period = 2π

D Amplitude = 3, Period = 4π

Phase Shift

What is the phase shift of y = cos(3x + π)?

A π to the right

B π/3 to the left

C π to the left

D π/3 to the right

Unit 5

Sum & Difference Formulas

★ Must Know Cold

sin(A ± B)

sinA cosB ± cosA sinB

cos(A ± B)

cosA cosB ∓ sinA sinB

tan(A ± B)

(tanA ± tanB)/(1 ∓ tanA tanB)

✦ Worked Example

Find cos(75°) using sum formula.

cos(45°+30°) = cos45·cos30 − sin45·sin30
= (√2/2)(√3/2) − (√2/2)(1/2) = (√6−√2)/4

Sum Formula

Use the sum formula to find sin(105°). Express as a fraction.

A (√6 − √2)/4

B (√6 + √2)/4

C (√3 + 1)/4

D (√3 − 1)/4

Unit 6

Double & Half Angle Formulas

★ Double Angle Formulas

sin 2θ

2 sinθ cosθ

cos 2θ

cos²θ − sin²θ

cos 2θ alt

2cos²θ − 1

cos 2θ alt2

1 − 2sin²θ

Double Angle

If sin θ = 3/5 (θ in Q1), find sin(2θ).

A 24/25

B 7/25

C 6/25

D 12/25

Double Angle — cos

If cos θ = −1/3 (θ in Q2), find cos(2θ).

A −7/9

B 7/9

C −1/9

D 1/9

Unit 7

Inverse Trig Functions

📌 Ranges of Inverse Functions

arcsin range[−π/2, π/2]

arccos range[0, π]

arctan range(−π/2, π/2)

Inverse Trig

Evaluate arctan(−√3). Give the answer in radians.

A −π/3

B 2π/3

C −π/6

D 4π/3

Composition of Inverse

Evaluate sin(arccos(5/13)).

A 5/13

B 13/12

C 12/13

D 5/12

Unit 8

Solving Trig Equations

📌 General Solutions

sin θ = kθ = arcsin(k) + 2nπ or π−arcsin(k) + 2nπ

cos θ = kθ = ±arccos(k) + 2nπ

tan θ = kθ = arctan(k) + nπ

Trig Equation

Solve 2sin²x − sinx − 1 = 0 on [0, 2π).

A x = π/6, 5π/6

B x = 7π/6, 11π/6, π/2

C x = 7π/6, 11π/6, 3π/2

D x = π/6, 5π/6, 3π/2

Basic Trig Equation

Find all solutions of tan x = 1 on [0, 2π).

A x = π/4 only

B x = π/4, 5π/4

C x = π/4, 3π/4

D x = π/4, 3π/4, 5π/4, 7π/4

Unit 9

Laws of Sines & Cosines

★ Triangle Laws

Law of Sines

a/sinA = b/sinB = c/sinC

Law of Cosines

c² = a² + b² − 2ab cosC

Law of Cosines

In triangle ABC, a = 5, b = 7, C = 60°. Find side c.

A √39

B √74

C √49

D 6

Law of Sines

In triangle ABC, A = 30°, a = 6, b = 8. Which is true about angle B?

A B = 30°

B sin B = 2/3

C sin B = 2/3, two triangles possible

D No triangle exists

Unit 10

Trig Identity Proofs & Applications

Identity Verification

Which expression is equivalent to sec²θ − 1?

A sin²θ

B tan²θ

C cot²θ

D cos²θ

Application — Height Problem

From a point 50 m from the base of a building, the angle of elevation to the top is 60°. What is the height of the building?

A 50/√3 m

B 25√3 m

C 50√3 m

D 100 m

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