Choosing the right ratio
| English | Chinese | Pinyin |
|---|---|---|
| angle of elevation | 仰角 | yǎng jiǎo |
| sine rule | 正弦定理 | zhèng xián dìng lǐ |
| cosine rule | 余弦定理 | yú xián dìng lǐ |
| unit circle | 单位圆 | dān wèi yuán |
| periodic | 周期的 | zhōu qī de |
| bearing | 方位角 | fāng wèi jiǎo |
| angle of depression | 俯角 | fǔ jiǎo |
| radians | 弧度 | hú dù |
One angle, three ratios
- A right-angled triangle with a known angle has fixed side ratios. That is the whole of basic trigonometry.
- $\sin\theta = \frac{\text{opp}}{\text{hyp}}$, $\cos\theta = \frac{\text{adj}}{\text{hyp}}$, $\tan\theta = \frac{\text{opp}}{\text{adj}}$.
- The marked skill is choosing the ratio. The arithmetic after that is one button.
Which ratio, and why
- Label the sides relative to the angle you know: opposite, adjacent, hypotenuse.
- Cross out the side you neither have nor want. The two that remain name the ratio.
- If you have opposite and adjacent, it is tangent, every time.
Watch the ratios as the angle changes
See why tangent grows without limit while sine and cosine never leave [−1, 1].
You know the adjacent side and want the opposite side. Which ratio do you use?
Tangent is opposite over adjacent — the only ratio that involves neither hypotenuse.
From 40 m away, the angle of elevation of a tower's top is $32°$. How tall is it?
You have the adjacent (40 m) and you want the opposite (the height), so use tangent.
$\tan 32° = \dfrac{h}{40}$, so $h = 40\tan 32° = 25.0$ m.
Write the ratio line before the number line. That line is the method mark, and it survives an arithmetic slip.
From 40 m away, the angle of elevation of a tower top is 32°. Find the height in metres, to 1 decimal place.
h = 40 tan 32° = 25.0 m. Make sure the calculator is in degree mode.
Put a trigonometry solution in the order that earns the method marks.
The ratio line is the marked step. A number with no ratio line above it earns nothing if it is wrong.
Beyond the right angle
- For any triangle, the sine rule 正弦定理 is $\dfrac{a}{\sin A} = \dfrac{b}{\sin B}$.
- The cosine rule 余弦定理 is $a^2 = b^2 + c^2 - 2bc\cos A$ — use it when you know two sides and the angle between them.
- The unit circle 单位圆 extends the ratios past $90°$ and shows why they are periodic 周期的.
You know two sides of a triangle and the angle between them. Which rule finds the third side?
Two sides and the included angle is exactly what a² = b² + c² − 2bc cos A needs. Pythagoras needs a right angle.
Bearings and angles of elevation
- A bearing 方位角 is measured clockwise from north and always written with three figures: $075°$.
- An angle of elevation 仰角 is measured up from the horizontal; an angle of depression 俯角 down from it.
- Radians 弧度 are the other angle unit: $\pi$ radians is $180°$.
Write a bearing of seventy-five degrees in the correct three-figure form.
Bearings always take three figures, so a leading zero is required.
How many degrees is π radians?
Half a full turn. A full turn is 2π radians, or 360°.
Check the calculator's angle mode before every trigonometry question. A correct method in radian mode gives a wrong number and looks like a misunderstanding. It is the cheapest mark in the module to protect.
The angle of depression from a cliff to a boat equals the angle of elevation from the boat to the cliff — they are alternate angles between parallel horizontals. Students often draw the depression angle inside the triangle at the wrong vertex, which changes the ratio and the answer.