Exact values, graphs and trig equations
Exact values (Extended)
- Learn these without a calculator:
| $x$ | $0^{\circ}$ | $30^{\circ}$ | $45^{\circ}$ | $60^{\circ}$ | $90^{\circ}$ |
|---|---|---|---|---|---|
| $\sin x$ | $0$ | $\tfrac12$ | $\tfrac{\sqrt2}{2}$ | $\tfrac{\sqrt3}{2}$ | $1$ |
| $\cos x$ | $1$ | $\tfrac{\sqrt3}{2}$ | $\tfrac{\sqrt2}{2}$ | $\tfrac12$ | $0$ |
| $\tan x$ | $0$ | $\tfrac{1}{\sqrt3}$ | $1$ | $\sqrt3$ | — |
Practice
What is the exact value of sin 30°? (as a decimal)
sin 30° = 1/2 = 0.5.
Graphs and equations (Extended)
- For $0^{\circ}\leqslant x \leqslant 360^{\circ}$: $y=\sin x$ and $y=\cos x$ are waves between $-1$ and $1$; $y=\tan x$ repeats every $180^{\circ}$.
- A trig equation often has more than one answer in the range — use the wave's symmetry.
- Worked example: $\sin x = \tfrac{\sqrt3}{2}$ → $x = 60^{\circ}$ or $180 - 60 = 120^{\circ}$.
- Worked example: $2\cos x + 1 = 0$ → $\cos x = -\tfrac12$ → $x = 120^{\circ}$ or $240^{\circ}$.
Practice
Solve sin x = √3/2 for 0–360°. One answer is 60°. What is the other (degrees)?
The sine wave is symmetric about 90°: 180 − 60 = 120°.
Practice
Solve 2cos x + 1 = 0 for 0–360°. One answer is 120°. What is the other (degrees)?
cos x = −1/2 gives x = 120° or 240°.
You've got it
Key idea
- know the exact values: $\sin 30^{\circ}=\tfrac12$, $\cos 60^{\circ}=\tfrac12$, $\tan 45^{\circ}=1$
- sine and cosine waves run between $-1$ and $1$
- a trig equation usually has two answers in $0$–$360^{\circ}$