Pre-Calculus Trigonometry | 20 Essential Questions

Angles & Radian Measure

Key Concepts to Memorize

Radian = Arc Length / Radius → θ = s/r
180° = π radians
Degrees → Radians: multiply by π/180
Radians → Degrees: multiply by 180/π
Arc length: s = rθ (θ in radians)
Sector area: A = ½r²θ

To convert DEG→RAD: think "Divide by 180, multiply by π." For RAD→DEG: flip it.

Example
Convert 135° to radians:
135 × (π/180) = 3π/4
Answer: 3π/4 ≈ 2.356 rad

The Unit Circle

Critical Values to Know

θ (degrees)	θ (radians)	sin θ	cos θ	tan θ
0°	0	0	1	0
30°	π/6	1/2	√3/2	1/√3
45°	π/4	√2/2	√2/2	1
60°	π/3	√3/2	1/2	√3
90°	π/2	1	0	undef.

ASTC rule: "All Students Take Calculus" — quadrants I, II, III, IV show which functions are positive (All, Sin, Tan, Cos).

Trig Functions & Graphs

Sine & Cosine — General Form

y = A sin(Bx − C) + D
Amplitude: |A|
Period: 2π / |B|
Phase shift: C / B (right if positive)
Vertical shift: D

Range of sin and cos: [−1, 1]
Range of tan: (−∞, +∞)

Example
y = 3 sin(2x − π) + 1
Amplitude = 3, Period = 2π/2 = π, Phase shift = π/2 right, Vertical shift = 1 up

Trigonometric Identities

Must-Know Identities

Pythagorean: sin²θ + cos²θ = 1
               1 + tan²θ = sec²θ
               1 + cot²θ = csc²θ

Reciprocal: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ

Quotient: tan θ = sin θ/cos θ, cot θ = cos θ/sin θ

Double Angle: sin 2θ = 2 sin θ cos θ
                cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1

Sum: sin(A+B) = sinA cosB + cosA sinB
       cos(A+B) = cosA cosB − sinA sinB

For cos 2θ, remember THREE forms. Pick the one that uses only sin OR only cos for simplification.

Inverse Trig & Equations

Inverse Function Domains & Ranges

arcsin: domain [−1,1], range [−π/2, π/2]
arccos: domain [−1,1], range [0, π]
arctan: domain (−∞,∞), range (−π/2, π/2)

General solutions (n ∈ ℤ):
sin θ = k → θ = arcsin(k) + 2nπ OR π − arcsin(k) + 2nπ
cos θ = k → θ = ±arccos(k) + 2nπ
tan θ = k → θ = arctan(k) + nπ

Example
Solve sin θ = √3/2 on [0, 2π):
θ = π/3 and θ = 2π/3

Law of Sines & Law of Cosines

Triangle Solving

Law of Sines: a/sin A = b/sin B = c/sin C

Law of Cosines: c² = a² + b² − 2ab cos C
(also: cos C = (a² + b² − c²) / 2ab)

Area of Triangle: (1/2) ab sin C

Use Law of Sines for AAS or ASA cases. Use Law of Cosines for SSS or SAS cases (when you have the included angle).

⬇ Practice Exam — 20 Questions

Question 01 / 20

Radian Measure

An angle measures 240°. What is its equivalent measure in radians?

3π/4

4π/3

5π/6

7π/6

Solution Multiply by π/180:
240 × (π/180) = 240π/180 = 4π/3
✓ Correct Answer: (B) 4π/3

Question 02 / 20

Arc Length

A circle has a radius of 6 cm. A central angle subtends an arc of length 9π cm. What is the measure of the central angle in radians?

π/2

3π/2

2π/3

3π/4

Solution Use s = rθ, so θ = s/r = 9π/6 = 3π/2
✓ Correct Answer: (B) 3π/2

Question 03 / 20

Unit Circle

What are the exact coordinates of the point on the unit circle at θ = 5π/6?

(−√3/2, 1/2)

(√3/2, −1/2)

(−1/2, √3/2)

(−√2/2, √2/2)

Solution 5π/6 is in Quadrant II. Reference angle = π − 5π/6 = π/6.
cos(π/6) = √3/2 → cos(5π/6) = −√3/2 (negative in QII)
sin(π/6) = 1/2 → sin(5π/6) = +1/2 (positive in QII)
Coordinates (cos θ, sin θ) = (−√3/2, 1/2)
✓ Correct Answer: (A) (−√3/2, 1/2)

Question 04 / 20

ASTC / Sign of Trig Functions

If sin θ < 0 and cos θ > 0, in which quadrant does the terminal side of θ lie?

Quadrant I

Quadrant II

Quadrant III

Quadrant IV

Solution Using ASTC: cos is positive in QI and QIV. Sin is negative in QIII and QIV. The only quadrant where both conditions hold (sin < 0 AND cos > 0) is Quadrant IV.
✓ Correct Answer: (D) Quadrant IV

Question 05 / 20

Amplitude & Period

For the function f(x) = −4 cos(3x) + 2, what is the amplitude and period?

Amplitude = −4, Period = 3π

Amplitude = 4, Period = 2π/3

Amplitude = 4, Period = 3π

Amplitude = 2, Period = 2π/3

Solution For y = A cos(Bx) + D:
Amplitude = |A| = |−4| = 4 (always positive)
Period = 2π / |B| = 2π / 3
The vertical shift +2 does NOT affect amplitude or period.
✓ Correct Answer: (B) Amplitude = 4, Period = 2π/3

Question 06 / 20

Phase Shift

What is the phase shift of y = sin(2x − π/2)?

π/2 to the left

π/4 to the right

π/2 to the right

π to the right

Solution Phase shift = C/B where the form is sin(Bx − C).
Here B = 2, C = π/2.
Phase shift = (π/2) / 2 = π/4 to the RIGHT (positive means rightward shift).
✓ Correct Answer: (B) π/4 to the right

Question 07 / 20

Pythagorean Identity

If sin θ = 5/13 and θ is in Quadrant I, what is the value of cos θ?

12/13

5/12

−12/13

13/12

Solution Using sin²θ + cos²θ = 1:
(5/13)² + cos²θ = 1
25/169 + cos²θ = 1
cos²θ = 144/169
cos θ = ±12/13. Since QI, cos θ = +12/13.
✓ Correct Answer: (A) 12/13

Question 08 / 20

Reciprocal Identities

Simplify: (sin θ)(csc θ) − cos²θ

1 − cos²θ

sin²θ

tan²θ

Solution sin θ · csc θ = sin θ · (1/sin θ) = 1
So the expression = 1 − cos²θ = sin²θ (by Pythagorean identity)
Both (B) and (C) are equivalent, but (C) sin²θ is the fully simplified form.
✓ Correct Answer: (C) sin²θ

Question 09 / 20

Double Angle Formula

If cos θ = 3/5 and 0 < θ < π/2, find sin(2θ).

7/25

24/25

12/25

−24/25

Solution cos θ = 3/5, so sin θ = 4/5 (QI, using 3-4-5 Pythagorean triple).
sin(2θ) = 2 sin θ cos θ = 2 · (4/5) · (3/5) = 24/25
✓ Correct Answer: (B) 24/25

Question 10 / 20

Sum Formula

Using the angle addition formula, find the exact value of cos(75°).

(√6 − √2) / 4

(√6 + √2) / 4

(√3 − 1) / 4

√3 / 2

Solution cos(75°) = cos(45° + 30°) = cos45°cos30° − sin45°sin30°
= (√2/2)(√3/2) − (√2/2)(1/2)
= √6/4 − √2/4
= (√6 − √2) / 4
✓ Correct Answer: (A) (√6 − √2) / 4

Question 11 / 20

Inverse Trig Functions

What is the exact value of arcsin(−√2/2)?

−π/4

3π/4

5π/4

−π/3

Solution arcsin has range [−π/2, π/2]. We need sin(θ) = −√2/2 in this range.
sin(π/4) = √2/2, so sin(−π/4) = −√2/2.
arcsin(−√2/2) = −π/4.
Note: 3π/4 also has sin = √2/2, but it's outside arcsin's range.
✓ Correct Answer: (A) −π/4

Question 12 / 20

Solving Trig Equations

Find all solutions of 2 cos²x − cos x − 1 = 0 on [0, 2π).

x = π/3, 5π/3

x = π/3, π, 5π/3

x = 0, 2π/3, 4π/3

x = π/2, 3π/2

Solution Factor: 2cos²x − cos x − 1 = (2cos x + 1)(cos x − 1) = 0
Case 1: 2cos x + 1 = 0 → cos x = −1/2 → x = 2π/3, 4π/3 ... wait, check options.
Case 2: cos x − 1 = 0 → cos x = 1 → x = 0
Case 1: cos x = −1/2 → x = 2π/3, 4π/3
Combined: x = 0, 2π/3, 4π/3
✓ Correct Answer: (C) x = 0, 2π/3, 4π/3

Question 13 / 20

Law of Sines

In triangle ABC, angle A = 30°, angle B = 45°, and side a = 8. Find side b (the side opposite angle B).

8√2

4√6

8√6/3

4√2

Solution Law of Sines: a/sin A = b/sin B
8/sin30° = b/sin45°
8/(1/2) = b/(√2/2)
16 = b/(√2/2)
b = 16 × (√2/2) = 8√2
✓ Correct Answer: (A) 8√2

Question 14 / 20

Law of Cosines

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

√39

√74

√34

√49

Solution c² = a² + b² − 2ab cos C
c² = 25 + 49 − 2(5)(7)cos60°
c² = 74 − 70(1/2)
c² = 74 − 35 = 39
c = √39
✓ Correct Answer: (A) √39

Question 15 / 20

Trig Identity Simplification

Simplify the expression: (1 − sin²θ) / cos θ

sin θ

cos θ

tan θ

sec θ

Solution 1 − sin²θ = cos²θ (Pythagorean identity)
So: cos²θ / cos θ = cos θ
✓ Correct Answer: (B) cos θ

Question 16 / 20

Tangent Function

What is the period of the function y = tan(πx/2)?

2π

π/2

Solution The standard period of tan(x) is π.
For y = tan(Bx), period = π / |B|.
Here B = π/2, so period = π ÷ (π/2) = π × (2/π) = 2.
✓ Correct Answer: (C) 2

Question 17 / 20

Co-Function Identities

Which expression is equal to cos(π/2 − θ)?

−sin θ

cos θ

sin θ

−cos θ

Solution Using the co-function identity: cos(π/2 − θ) = sin θ
Proof via addition formula: cos(π/2)cos θ + sin(π/2)sin θ = (0)(cos θ) + (1)(sin θ) = sin θ
✓ Correct Answer: (C) sin θ

Question 18 / 20

Triangle Area

A triangle has sides a = 6, b = 10, and the included angle C = 30°. What is the area of the triangle?

15√3

30√3

Solution Area = (1/2) ab sin C = (1/2)(6)(10) sin 30°
= (1/2)(60)(1/2) = 30/2 = 15
✓ Correct Answer: (B) 15

Question 19 / 20

Trig Identity Proof

Which identity is equivalent to cos(2θ) when expressed only in terms of sin θ?

2sin²θ − 1

1 − 2sin²θ

2cos²θ − 1

sin²θ − cos²θ

Solution Start from: cos(2θ) = cos²θ − sin²θ
Replace cos²θ with (1 − sin²θ):
= (1 − sin²θ) − sin²θ = 1 − 2sin²θ
Note: Option (C) is in terms of cos only. Option (A) is the NEGATIVE of the answer.
✓ Correct Answer: (B) 1 − 2sin²θ

Question 20 / 20

Combined Concepts

The function f(x) = 2 sin(πx − π/2) has which of the following properties?
Select the answer that correctly states the amplitude, period, and phase shift.

Amplitude = 2, Period = 2, Phase shift = 1/2 right

Amplitude = 2, Period = π, Phase shift = π/2 right

Amplitude = 2, Period = 2, Phase shift = π right

Amplitude = 1, Period = 2, Phase shift = 1/2 left

Solution f(x) = 2 sin(πx − π/2). Compare to A sin(Bx − C) + D:
A = 2 → Amplitude = |2| = 2
B = π → Period = 2π / π = 2
C = π/2 → Phase shift = C/B = (π/2)/π = 1/2 to the RIGHT
✓ Correct Answer: (A) Amplitude = 2, Period = 2, Phase shift = 1/2 right

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Answer Key & Full Solutions

Pre-Calculus

Trigonometry — 20 Essential Questions

Name: _________________________________ Date: _____________ Score: _____/20

Q1 · Radian Measure

1. An angle measures 240°. What is its equivalent in radians?

A) 3π/4 B) 4π/3 C) 5π/6 D) 7π/6

Q2 · Arc Length

2. A circle of radius 6 cm has an arc of 9π cm. Find the central angle in radians.

A) π/2 B) 3π/2 C) 2π/3 D) 3π/4

Q3 · Unit Circle

3. Find the coordinates on the unit circle at θ = 5π/6.

A) (−√3/2, 1/2) B) (√3/2, −1/2) C) (−1/2, √3/2) D) (−√2/2, √2/2)

Q4 · ASTC

4. If sin θ < 0 and cos θ > 0, θ lies in which quadrant?

A) I B) II C) III D) IV

Q5 · Amplitude & Period

5. For f(x) = −4 cos(3x) + 2, find amplitude and period.

A) −4, 3π B) 4, 2π/3 C) 4, 3π D) 2, 2π/3

Q6 · Phase Shift

6. What is the phase shift of y = sin(2x − π/2)?

A) π/2 left B) π/4 right C) π/2 right D) π right

Q7 · Pythagorean Identity

7. If sin θ = 5/13 and θ is in QI, find cos θ.

A) 12/13 B) 5/12 C) −12/13 D) 13/12

Q8 · Reciprocal Identities

8. Simplify: sin(θ)·csc(θ) − cos²θ

A) 0 B) 1 − cos²θ C) sin²θ D) tan²θ

Q9 · Double Angle

9. If cos θ = 3/5 and 0 < θ < π/2, find sin(2θ).

A) 7/25 B) 24/25 C) 12/25 D) −24/25

Q10 · Sum Formula

10. Find the exact value of cos(75°).

A) (√6−√2)/4 B) (√6+√2)/4 C) (√3−1)/4 D) √3/2

Q11 · Inverse Trig

11. Find the exact value of arcsin(−√2/2).

A) −π/4 B) 3π/4 C) 5π/4 D) −π/3

Q12 · Trig Equations

12. Solve 2cos²x − cos x − 1 = 0 on [0, 2π).

A) π/3, 5π/3 B) π/3, π, 5π/3 C) 0, 2π/3, 4π/3 D) π/2, 3π/2

Q13 · Law of Sines

13. In triangle ABC, A=30°, B=45°, a=8. Find b.

A) 8√2 B) 4√6 C) 8√6/3 D) 4√2

Q14 · Law of Cosines

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

A) √39 B) √74 C) √34 D) 7

Q15 · Identity Simplification

15. Simplify (1 − sin²θ) / cos θ.

A) sin θ B) cos θ C) tan θ D) sec θ

Q16 · Tangent Period

16. Find the period of y = tan(πx/2).

A) π B) 2π C) 2 D) π/2

Q17 · Co-function Identity

17. Which expression equals cos(π/2 − θ)?

A) −sin θ B) cos θ C) sin θ D) −cos θ

Q18 · Triangle Area

18. Triangle with a=6, b=10, C=30°. Find area.

A) 30 B) 15 C) 15√3 D) 30√3

Q19 · Double Angle Identity

19. Express cos(2θ) only in terms of sin θ.

A) 2sin²θ−1 B) 1−2sin²θ C) 2cos²θ−1 D) sin²θ−cos²θ

Q20 · Combined Concepts

20. For f(x) = 2 sin(πx − π/2), identify amplitude, period, and phase shift.

A) A=2, T=2, shift=1/2 right B) A=2, T=π, shift=π/2 right C) A=2, T=2, shift=π right D) A=1, T=2, shift=1/2 left

Answer Key & Solutions

1. B — 240 × π/180 = 4π/3
2. B — θ = s/r = 9π/6 = 3π/2
3. A — QII ref.angle π/6: cos(5π/6)=−√3/2, sin(5π/6)=1/2
4. D — ASTC: sin− and cos+ only in Quadrant IV
5. B — |A|=4; Period=2π/3
6. B — Phase shift = (π/2)/2 = π/4 right
7. A — Pythag: cos θ=12/13 (QI positive)
8. C — sin θ·csc θ=1, then 1−cos²θ=sin²θ
9. B — sin θ=4/5; sin2θ=2(4/5)(3/5)=24/25
10. A — cos(45°+30°)=cos45°cos30°−sin45°sin30°=(√6−√2)/4
11. A — arcsin range [−π/2,π/2]: arcsin(−√2/2)=−π/4
12. C — Factors to (2cosx+1)(cosx−1)=0: x=0, 2π/3, 4π/3
13. A — Law of Sines: b=16·sin45°=8√2
14. A — Law of Cosines: c²=74−35=39, c=√39
15. B — cos²θ/cosθ=cosθ
16. C — Period of tan(πx/2)=π÷(π/2)=2
17. C — Cofunction: cos(π/2−θ)=sinθ
18. B — Area=(1/2)(6)(10)sin30°=15
19. B — cos2θ=1−2sin²θ
20. A — A=2, T=2π/π=2, shift=(π/2)/π=1/2 right

Trigonometry20 Essential Questions

Key Concepts to Memorize

Critical Values to Know

Sine & Cosine — General Form

Must-Know Identities

Inverse Function Domains & Ranges

Triangle Solving

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Answer Key & Full Solutions

Trigonometry
20 Essential Questions