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The calculator: ratios of any angle (degrees, minutes, seconds), and the angle from its ratio
Mathematics Prep 3 Term 1, Unit 3, Lesson 5
GoalUse a calculator in degree mode to find sin, cos, tan of any angle and to find the angle from a ratio.
New words
degree mode · the calculator setting where angles are in degrees (not radians)minute and second · 1° = 60′ and 1′ = 60″sin⁻¹, cos⁻¹, tan⁻¹ · the inverse keys: they return the angle from the ratioapproximately · ≈ means equal after rounding
Explanation
Most angles have no neat exact ratio, so we use a calculator. First check that it is in degree mode. Then enter the angle: sin 25° ≈ 0.4226, cos 40° ≈ 0.7660, tan 55° ≈ 1.4281. An angle can be given in degrees, minutes and seconds: 32°30′ means 32 + 30/60 = 32.5°, so sin 32°30′ ≈ 0.5373. To go the other way we use the inverse key: if sin θ = 0.8 then θ = sin⁻¹(0.8) ≈ 53.13°, and 0.13° × 60 ≈ 8′ gives 53°8′. Remember: sin⁻¹ does not mean 1 ÷ sin. Keys differ between calculators, so use your own calculator’s manual.
| Angle | sin | cos | tan |
|---|---|---|---|
| 25° | 0.4226 | 0.9063 | 0.4663 |
| 40° | 0.6428 | 0.7660 | 0.8391 |
| 55° | 0.8192 | 0.5736 | 1.4281 |
Worked examples
Find θ if tan θ = 1.2. θ = tan⁻¹(1.2) ≈ 50.19°. In degrees and minutes: 0.194° × 60 ≈ 11.7′, so θ ≈ 50°12′.
Convert 32°30′ to degrees: 30′ = 30 ÷ 60 = 0.5°, so 32°30′ = 32.5°. Then sin 32.5° ≈ 0.5373.
In a right triangle the hypotenuse is 20 cm and one acute angle is 40°. The side adjacent to it = 20 × cos 40° ≈ 20 × 0.7660 ≈ 15.32 cm.
Class activity
Check the mode. Type sin 30 on your calculator. If you see 0.5 it is in degree mode; if not, change the mode. Then find sin 60, cos 30 and compare.
Practice
1
Use a calculator for cos 25°, correct to 4 decimal places.
0.9063
2
Find θ if sin θ = 0.8, to the nearest degree.
θ ≈ 53°.
3
Write 18°15′ in degrees as a decimal.
15′ = 0.25°, so 18.25°.
4
A ramp is 12 m long and rises at 15° to the ground. How high does it rise?
12 × sin 15° ≈ 12 × 0.2588 ≈ 3.11 m.
5
Why is sin⁻¹(0.5) not the same as 1 ÷ sin(0.5)?
sin⁻¹ gives an angle (30°); 1 ÷ sin gives a reciprocal number.