In g7 SOHCAHTOA you used sine, cosine and tangent to solve right-angled triangles, and in g15 Sine & Cosine Rules you extended them to any triangle — but both of those only ever used angles between 0° and 180°. Set sine or cosine free from a triangle altogether and something new appears: a smooth, endlessly repeating wave. This chapter is about that wave — the graphs of sin(x), cos(x) and tan(x), and what they tell you about angles you can't just look up in a triangle.
1. Degrees and radians
Angles can be measured in degrees (a full turn = 360°) or in radians (a full turn = 2π radians). Radians measure an angle by the length of arc it cuts out on a unit circle, which is why calculus and physics almost always use them. To convert: multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees.
| Degrees | Radians |
|---|---|
| 360° | 2π |
| 180° | π |
| 90° | π/2 |
| 60° | π/3 |
| 45° | π/4 |
| 30° | π/6 |
2. The unit circle: sine and cosine beyond the triangle
Picture a point going anticlockwise around a circle of radius 1, centred on the origin, starting on the positive x-axis. As the point turns through angle x, its y-coordinate is defined to be sin(x) and its x-coordinate is defined to be cos(x). For 0° to 90° this matches SOHCAHTOA exactly (opposite/hypotenuse and adjacent/hypotenuse on a radius-1 triangle) — but the unit circle keeps working past 90°, giving sine and cosine values for any angle, including negative angles and angles over 360°.
3. The graphs of sin(x), cos(x) and tan(x)
- y = sin(x) starts at 0, rises to a peak of 1 at 90°, falls through 0 at 180°, drops to a trough of −1 at 270°, and returns to 0 at 360°.
- y = cos(x) has the exact same wave shape, just shifted — it starts at its peak, 1, at 0°, crosses 0 at 90°, hits −1 at 180°, and returns to 1 at 360°.
- Both sin(x) and cos(x) have amplitude 1 (the wave reaches from −1 to +1) and period 360° (the shape repeats exactly every 360°).
- y = tan(x) looks completely different: it climbs from 0, shoots up to infinity as x approaches 90°, then reappears from below at just past 90° and climbs again. It is undefined at 90° and 270° (because tan = sin/cos, and cos = 0 there — dividing by zero), and its period is only 180°, half that of sine and cosine.
4. Solving a trig equation: reading multiple solutions off the graph
Because the sine curve is symmetric, an equation like sin(x) = 0.5 almost always has more than one solution in a given range. Your calculator (inverse sine) only ever gives you the principal value — the first one. To find the rest, use the graph's symmetry:
- For sine, the second solution in 0°–360° is 180° − (principal value), because the curve is symmetric about x = 90°.
- For cosine, the second solution is 360° − (principal value), because the curve is symmetric about x = 180°.
Calculator: sin-1(0.5) = 30°. This is the principal value.
Sine is also positive in the second "hump" of the graph (90° to 180°), symmetric about x = 90°, giving a second solution of 180° − 30° = 150°.
Check on the graph: at x = 30° and x = 150°, the curve crosses the horizontal line y = 0.5 at exactly two points between 0° and 360° — sine is negative for the rest of the range, so there are no more.
x = 30° or x = 150°.
Calculator: cos-1(0.6) = 53.1° (principal value).
Cosine is symmetric about x = 180°, so the second solution is 360° − 53.1° = 306.9°.
x = 53.1° or x = 306.9° (1 d.p.).
Exam technique: always sketch a quick, rough sine or cosine curve first and mark the target y-value as a horizontal line across it. Wherever that line crosses the curve within the given range is a valid solution — count the crossings before you start calculating so you know how many answers to expect.