Choosing the right ratio
Introduced| English | Meaning |
|---|---|
| sine rule/saɪn ruːl/ | The rule that each side of a triangle divided by the sine of its opposite angle has the same value. |
| cosine rule/ˈkəʊsaɪn ruːl/ | The rule relating a triangle's sides and an angle: one side squared equals the sum of the other two sides squared minus twice their product times the cosine of the included angle. |
| unit circle/ˈjuːnɪt ˈsɜːkl/ | A circle with radius one, usually centred at the origin. |
| periodic/ˌpɪərɪˈɒdɪk/ | Repeating the same pattern after a fixed interval of the independent variable. |
| radians/ˈreɪdɪənz/ | Units of angle defined by arc length divided by radius; one full turn equals two pi of these units. |
| bearing/ˈbeərɪŋ/ | A direction measured as an angle clockwise from north, usually written with three digits. |
| angle of elevation/ˈæŋɡl ɒv ˌelɪˈveɪʃn/ | The angle measured upwards from a horizontal line to the line of sight to an object. |
| angle of depression/ˈæŋɡl ɒv dɪˈpreʃn/ | The angle measured downwards from a horizontal line to the line of sight to an object. |
Match an angle to its sides
- Right-triangle ratios relate a chosen acute angle to opposite, adjacent and hypotenuse lengths. Opposite and adjacent change when the chosen angle changes.
- Sine is opposite over hypotenuse, cosine adjacent over hypotenuse, and tangent opposite over adjacent. Identify the known and requested lengths before choosing a ratio.
Select the equation before calculating
- Draw and label the right triangle. The longest side opposite the right angle is the hypotenuse.
- With opposite and adjacent, use $\tan\theta=o/a$. Rearrange for the unknown; use inverse tangent if the angle is unknown, and check the angle mode.
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?
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Tangent is opposite over adjacent — the only ratio that involves neither hypotenuse.
Extend the method beyond right triangles
- The sine rule 正弦定理 pairs sides with opposite angles: $a/\sin A=b/\sin B$. The cosine rule 余弦定理 uses $c^2=a^2+b^2-2ab\cos C$, where C is the included angle.
- The unit circle 单位圆 extends sine and cosine beyond acute angles. They are periodic 周期的, repeating every 360 degrees or $2\pi$ radians; tangent has a different period and undefined values.
From 40 m away, the angle of elevation of a tower top is 32°. Find the height in metres, to 1 decimal place.
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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.
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The ratio line is the marked step. A number with no ratio line above it earns nothing if it is wrong.
Keep the reference direction explicit
- A bearing 方位角 is clockwise from north and uses three digits in degrees. An angle of elevation 仰角 is upward from the horizontal, and an angle of depression 俯角 downward.
- Radians 弧度 measure angles with $180^\circ=\pi$ radians. Convert units or set the calculator correctly before substituting.
You know two sides of a triangle and the angle between them. Which rule finds the third side?
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Two sides and the included angle is exactly what a² = b² + c² − 2bc cos A needs. Pythagoras needs a right angle.
Two observations share one height. A tower seen at 35 degrees, then 50 degrees after moving 10 m closer, gives $h=d\tan35^\circ=(d-10)\tan50^\circ$. The resulting initial distance is about 24.245 m. Add an instrument height only when converting h to tower height above ground.
Write a bearing of seventy-five degrees in the correct three-figure form.
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Bearings always take three figures, so a leading zero is required.
How many degrees is π radians?
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Half a full turn. A full turn is 2π radians, or 360°.
Explain which data select the rule. Two sides and their included angle fit the cosine rule. A known side-angle opposite pair supports the sine rule. Sheet 1.5 combines triangle lengths, bearings and two observations.
With side-side-angle data, inverse sine may produce an acute angle while its supplement gives another valid triangle. Check the remaining angle sum before discarding that possibility. Do not assume a right angle that the question does not state.