Review: directed angles, radians, the six trigonometric functions, related angles, graphs and finding an angle
Mathematics Sec 1 Term 1, Units 1–4, Review
GoalRevise Unit 3 with new mixed problems.
New words
equivalent angles · angles that differ by whole turnsamplitude · the greatest distance from the middle linereference angle · the acute angle used to find a ratio
Explanation
Checklist for Unit 3. (1) π radians = 180°; ℓ = θ × r. (2) Equivalent angles differ by 360°; find the smallest positive one by adding or subtracting turns. (3) On a point (x, y) with r = √(x² + y²): sin = y/r, cos = x/r, tan = y/x. (4) Signs: All, Sin, Tan, Cos. (5) Special values for 30°, 45°, 60°. (6) Related angles: 180° ± θ, 360° − θ keep the function; 90° ± θ and 270° ± θ change it to the co-function; take the sign from the quadrant. (7) sin α = cos β with acute angles means α + β = 90°. (8) y = a sin θ and y = a cos θ: amplitude |a|, period 2π. (9) To find an angle from a ratio, use the reference angle and the two quadrants.
Quantity
Value
sin 330°
−1/2
cos 225°
−√2/2
tan 120°
−√3
sin(−60°)
−√3/2
Worked examples
Change 150° to radians (5π/6) and 7π/4 to degrees (315°). On a circle of radius 8 m, an angle of 3π/4 cuts an arc ℓ = 3π/4 × 8 = 6π ≈ 18.85 m.
The terminal side passes through (−8, −15). r = 17, sin θ = −15/17, cos θ = −8/17, tan θ = 15/8 (third quadrant: only tan is positive ✓). Also sin θ = −4/5 with cos θ > 0 gives cos θ = 3/5 and tan θ = −4/3.
Evaluate: cos 225° = −cos 45° = −√2/2; sin 330° = −sin 30° = −½; tan 120° = −tan 60° = −√3; sin(−60°) = −√3/2. The smallest positive measure of −1150° is −1150° + 1440° = 290°.
Solve sin 4x = cos(x + 20°) with acute angles: 4x + x + 20° = 90°, x = 14°. Check: sin 56° = cos 34° ✓. For 0° ≤ θ < 360°, cos θ = 0.5 gives θ = 60° or 300°.
y = −3 cos θ has amplitude 3, range [−3, 3] and period 2π. A ramp 6 m long rises 0.9 m: sin θ = 0.15, so θ ≈ 8.63°.
Class activity
Quick-fire round. In pairs, one student says an angle (such as 2π/3, 225°, −90°, 405°) and the other gives its quadrant, its smallest positive measure and its sine. Swap after five angles.
Practice
1
Change 5π/3 to degrees and 135° to radians.
300° and 3π/4.
2
The terminal side passes through (5, −12). Find sin θ, cos θ, tan θ.
r = 13: sin −12/13, cos 5/13, tan −12/5.
3
Find sin 210° + cos 300°.
−½ + ½ = 0.
4
Solve tan θ = −√3 for 0° ≤ θ < 360°.
Reference 60°, quadrants 2 and 4: 120° or 300°.
5
Find the range of y = 2.5 sin θ and sin(2π + π/6).
