Trigonometry (30-1)

Question 1

State the primary trigonometric ratios for an acute angle θ in a right triangle.

Answer 1

sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent

Question 2

Solve for θ if sin θ = 0.6, where 0° ≤ θ ≤ 90°.

Answer 2

θ = sin⁻¹(0.6) ≈ 36.87°

Question 3

Solve for θ if cos θ = 0.3, 0° ≤ θ ≤ 360°.

Answer 3

θ = cos⁻¹(0.3) ≈ 72.54° or θ ≈ 360° - 72.54° = 287.46°

Question 4

Solve for θ if tan θ = -1, 0° ≤ θ ≤ 360°.

Answer 4

θ ≈ 135° or θ ≈ 315° (quadrants II and IV where tan is negative)

Question 5

Write the exact values of sin 30°, cos 45°, and tan 60°.

Answer 5

sin 30° = 1/2, cos 45° = √2/2, tan 60° = √3

Question 6

State the CAST rule.

Answer 6

Quadrant I: all trig ratios positive, Quadrant II: sin positive, Quadrant III: tan positive, Quadrant IV: cos positive

Question 7

Determine the amplitude, period, phase shift, and vertical displacement for y = 3sin(2x - 90°) + 4.

Answer 7

Amplitude = 3, Period = 360°/2 = 180°, Phase shift = 90°/2 = 45° right, Vertical displacement = 4 up

Question 8

Graph y = cos x for 0° ≤ x ≤ 360°. Identify key points.

Answer 8

Key points: (0°, 1), (90°, 0), (180°, -1), (270°, 0), (360°, 1)

Question 9

Determine the general solution for sin θ = 0.5.

Answer 9

θ = 30° + 360°n or θ = 150° + 360°n, n ∈ ℤ

Question 10

Determine the general solution for cos θ = -√3/2.

Answer 10

θ = 150° + 360°n or θ = 210° + 360°n, n ∈ ℤ

Question 11

Determine the general solution for tan θ = √3.

Answer 11

θ = 60° + 180°n, n ∈ ℤ

Question 12

Solve 2sin²x - 1 = 0 for 0° ≤ x ≤ 360°.

Answer 12

sin²x = 1/2 → sin x = ±√(1/2) = ±√2/2 → x = 45°, 135°, 225°, 315°

Question 13

Prove the identity: (1 + tan²x) = sec²x.

Answer 13

Divide both sides of sin²x + cos²x = 1 by cos²x: tan²x + 1 = sec²x ✔

Question 14

Prove the identity: sin x cos x = (1/2)sin 2x.

Answer 14

Double-angle identity: sin 2x = 2 sin x cos x → sin x cos x = (1/2) sin 2x ✔

Question 15

Solve cos 2x = 0.5 for 0° ≤ x ≤ 360°.

Answer 15

2x = 60° or 300° → x = 30° or 150°

Question 16

Write the double-angle identity for cos 2x in three forms.

Answer 16

cos 2x = cos²x - sin²x = 2cos²x - 1 = 1 - 2sin²x

Question 17

Solve 2sin x + √3 = 0 for 0° ≤ x ≤ 360°.

Answer 17

sin x = -√3/2 → x = 240°, 300° (quadrants III and IV where sin is negative)

Question 18

Sketch y = 2cos(3x) on 0° ≤ x ≤ 360°.

Answer 18

Amplitude = 2, Period = 360°/3 = 120°, 3 cycles occur in 0°–360°

Question 19

Determine the exact solution for sin 3x = 0 on 0° ≤ x ≤ 360°.

Answer 19

3x = 0°, 180°, 360° → x = 0°, 60°, 120°

Question 20

Use the sine law to solve for ∠C given a = 10, b = 7, ∠A = 40°.

Answer 20

sin C / c = sin A / a → sin C = c × sin A / a → calculate numeric solution as needed

Question 21

Use the cosine law to solve for side c given a = 8, b = 6, ∠C = 120°.

Answer 21

c² = a² + b² - 2ab cos C → c² = 64 + 36 - 2(8)(6)(cos 120°) = 100 - (-48) = 148 → c ≈ 12.17

Question 22

Determine all solutions for cos x = 0 in the interval 0° ≤ x ≤ 360°.

Answer 22

x = 90°, 270°

Question 23

Solve the equation tan²x - 3 = 0, 0° ≤ x ≤ 360°.

Answer 23

tan²x = 3 → tan x = ±√3 → x = 60°, 120°, 240°, 300°

Question 24

Simplify sin x / cos x.

Answer 24

tan x

Question 25

State the domain and range of y = sin x.

Answer 25

Domain: x ∈ ℝ, Range: -1 ≤ y ≤ 1

Question 26

State the period of y = sin(bx).

Answer 26

Period = 360° / b (or 2π / b in radians)

Question 27

Determine the amplitude and vertical shift of y = -4cosx + 2.

Answer 27

Amplitude = 4, Reflection over x-axis, Vertical shift 2 units up

Question 28

State the coordinates of the terminal point on the unit circle for θ = 0°.

Answer 28

(1, 0)

Question 29

State the coordinates of the terminal point on the unit circle for θ = 90°.

Answer 29

(0, 1)

Question 30

State the coordinates of the terminal point on the unit circle for θ = 180°.

Answer 30

(-1, 0)

Question 31

State the coordinates of the terminal point on the unit circle for θ = 270°.

Answer 31

(0, -1)

Question 32

State the coordinates of the terminal point on the unit circle for θ = 30°.

Answer 32

(√3/2, 1/2)

Question 33

State the coordinates of the terminal point on the unit circle for θ = 45°.

Answer 33

(√2/2, √2/2)

Question 34

State the coordinates of the terminal point on the unit circle for θ = 60°.

Answer 34

(1/2, √3/2)

Question 35

Determine exact values for sin 120°.

Answer 35

√3/2

Question 36

Determine exact values for cos 225°.

Answer 36

-√2/2

Question 37

Determine exact values for tan 300°.

Answer 37

-√3

Question 38

Explain why cos²θ + sin²θ = 1 using the unit circle.

Answer 38

The terminal point (cos θ, sin θ) lies on a circle of radius 1 → x² + y² = 1² → cos²θ + sin²θ = 1

Question 39

Identify which quadrants sine is positive on the unit circle.

Answer 39

Quadrants I and II

Question 40

Identify which quadrants tangent is negative on the unit circle.

Answer 40

Quadrants II and IV

Question 41

Prove the identity: tan θ = sin θ / cos θ using the unit circle.

Answer 41

tan θ = (y/x) from (x, y) point → y/x = (sin θ)/(cos θ)

Question 42

Prove the identity: cos(90° - θ) = sin θ using the unit circle.

Answer 42

The x-coordinate of 90° - θ is equal to the y-coordinate of θ

Question 43

Prove the identity: sin(180° - θ) = sin θ using the unit circle.

Answer 43

Terminal point for 180° - θ is a reflection across the y-axis → same y-value

Question 44

Prove the identity: cos(180° - θ) = -cos θ using the unit circle.

Answer 44

Terminal point for 180° - θ has opposite x-value

Question 45

Prove the identity: 1 + tan²θ = sec²θ using the Pythagorean identity.

Answer 45

Divide both sides of sin²θ + cos²θ = 1 by cos²θ → tan²θ + 1 = sec²θ

Question 46

Prove the identity: (1 - cos 2x)/2 = sin²x.

Answer 46

cos 2x = 1 - 2sin²x → rearrange: 1 - cos 2x = 2sin²x → divide by 2

Question 47

Prove the identity: (1 + cos 2x)/2 = cos²x.

Answer 47

cos 2x = 2cos²x - 1 → rearrange: cos 2x + 1 = 2cos²x → divide by 2

Question 48

Verify the identity: (1 - cos x)(1 + cos x) = sin²x.

Answer 48

Left side = 1 - cos²x = sin²x

Question 49

Verify the identity: (tan x)(cot x) = 1.

Answer 49

tan x = sin x / cos x, cot x = cos x / sin x → product = 1

Question 50

Solve for all θ such that cos θ = -√3/2 using the unit circle.

Answer 50

θ = 150°, 210° (Quadrants II and III where cos is negative)

Question 51

Solve for all θ such that sin θ = -1/2 using the unit circle.

Answer 51

θ = 210°, 330° (Quadrants III and IV where sin is negative)