Trigonometry (2-D)
sin, cos and tan as functions of any angle and their graphs, solving simple trigonometric equations, and (NF) the area formula ½ab sin C, the sine and cosine formulae and Heron's formula. The 3-D part of the unit follows in F5.
What you will learn
- 14.1 sin, cos and tan as functions: graphs, maximum and minimum values, period. Simplify expressions with θ, 90° ± θ, 180° ± θ, 360° − θ and so on.
- 14.2 Solve a sin θ = b, a cos θ = b, a tan θ = b for 0° ≤ θ ≤ 360°. NF: other trigonometric equations; those that become quadratics are in F5 more about equations.
- NF 14.3 Area = ½ab sin C.
- NF 14.4 Sine formula and cosine formula.
- NF 14.5 Heron's formula.
You need first: Trigonometry, Functions and graphs
Key terms
| English | 中文 |
|---|---|
| trigonometric function | 三角函數 |
| period, periodic | 週期，週期性 |
| quadrant | 象限 |
| trigonometric equation | 三角方程 |
| sine formula / cosine formula | 正弦公式／餘弦公式 |
| Heron's formula | 希羅公式 |
| included angle | 夾角 |
Standard steps
Signs in each quadrant: "ASTC"
| Quadrant | Angle | Positive |
|---|---|---|
| I | 0°–90° | All |
| II | 90°–180° | Sin |
| III | 180°–270° | Tan |
| IV | 270°–360° | Cos |
Simplify sin(180° − θ) and friends
- Pretend θ is small (acute). Find which quadrant the angle is in.
- Use ASTC to decide the sign.
- 180° ± θ and 360° − θ keep the same function (sin stays sin); 90° ± θ and 270° ± θ swap sin ↔ cos (tan → 1/tan).
For example, sin(180° − θ) = sin θ, cos(180° + θ) = −cos θ, sin(90° + θ) = cos θ.
Solve a sin θ = b (0° ≤ θ ≤ 360°)
- Make sin θ the subject.
- Find the reference angle α with the calculator, using the positive value.
- Use ASTC to choose the quadrants where the sign is correct.
- Write each answer: Quadrant II → 180° − α; III → 180° + α; IV → 360° − α.
Which triangle formula (NF)
| You know | Use |
|---|---|
| 2 sides and the angle between them; want the third side | cosine formula: c² = a² + b² − 2ab cos C |
| 3 sides; want an angle | cosine formula: cos C = (a² + b² − c²)/(2ab) |
| 2 angles and a side, or a side and its opposite angle plus one more | sine formula: a/sin A = b/sin B = c/sin C |
| 2 sides and the included angle; want the area | ½ab sin C |
| 3 sides; want the area | Heron: √[s(s − a)(s − b)(s − c)], s = (a + b + c)/2 |
- List what you know: which sides, which angles, and whether the known angle is between the two known sides.
- Match it to a row of the table and write the formula before substituting.
- After each new value, check the table again for the next step. An angle found with the sine formula may also have an obtuse answer (180° − θ).
Worked examples
Example 1: Solve 5 sin θ + 3 = 0 for 0° ≤ θ ≤ 360°.
sin θ = −0.6
reference angle α = sin⁻¹(0.6) ≈ 36.87°
sin is negative in quadrants III and IV
θ ≈ 180° + 36.87° = 216.9° or θ ≈ 360° − 36.87° = 323.1°Example 2: NF: in △ABC, a = 8, b = 5 and C = 60°. Find the area and c.
area = ½(8)(5) sin 60° = 10√3 ≈ 17.3
c² = 8² + 5² − 2(8)(5) cos 60° = 64 + 25 − 40 = 49
c = 7Example 3: NF: A = 40°, B = 65°, a = 10. Find b.
b / sin 65° = 10 / sin 40°
b = 10 sin 65° / sin 40° ≈ 14.1Example 4: NF: area of a triangle with sides 13, 14, 15.
s = 21
area = √[21(8)(7)(6)] = √7056 = 84Example 5: NF: which triangle formula? A triangle has sides 7, 8 and 9. Find its largest angle.
3 sides, want an angle → cosine formula
the largest angle C is opposite the longest side, 9
cos C = (7² + 8² − 9²)/(2 × 7 × 8) = 32/112
C ≈ 73.4°Common mistakes
- Solving sin θ = −0.6 by typing sin⁻¹(−0.6) and giving −36.9°, which is outside 0°–360°.
- Giving only one answer when there are two.
- Cosine formula: forgetting the "2ab cos C" or using the wrong angle (C must be opposite c).
- Sine formula when the angle isn't opposite the known side.
- Calculator in RAD mode.
How the DSE tests it
- Paper 2 Section A: simplify expressions with 180° ± θ; read features from a graph of y = a sin bx; solve simple equations.
- Section B (NF): sine and cosine formulas, often as a step in a 3-D question.
- Paper 1: area by ½ab sin C or Heron within longer questions.
Questions you might not ask in class
Why are there two answers?
Sine is a wave. A horizontal line at height −0.6 crosses one full wave twice between 0° and 360°.
How do I remember which angles swap sin and cos?
90° and 270° are on the vertical axis: they swap. 180° and 360° are on the horizontal axis: they keep.
Can I always use the cosine formula instead of the sine formula?
Often, but not always. Pick the formula from what you know, using the table.