Finding angles, elevation and bearings

Undo the ratio

To find an angle, work out the ratio from the two sides, then use the inverse: sin⁻¹, cos⁻¹ or tan⁻¹ (SHIFT then sin on most calculators).

1. opposite 7 cm, hypotenuse 12 cm
2. sin θ = 7/12
3. θ = sin⁻¹(7/12) = 35.68... = 35.7° (1 d.p.)

Keep the whole fraction in the calculator: rounding 7/12 to 0.58 first gives 35.5°, which is wrong to 1 d.p.

Pick the inverse from the two sides you know, just as for a side: opposite and adjacent give tan⁻¹(opposite/adjacent), adjacent and hypotenuse give cos⁻¹. A road rising 20 m over 250 m across makes tan⁻¹(20/250) = 4.6° with the horizontal; a slope 300 m long dropping 60 m makes sin⁻¹(60/300) = 11.5°.

For a ladder, the angle with the ground has the foot distance next to it (adjacent) and the ladder as hypotenuse: a 5 m ladder 1.2 m out makes cos⁻¹(1.2/5) = 76.1° with the ground. Compare it with any limits you are given.

For an isosceles triangle, cut it in half to make right-angled triangles. Equal sides 13 cm and base 10 cm: the half-base is 5, so a base angle is cos⁻¹(5/13) = 67.4°, and the top angle is 180 − 2 × 67.4 = 45.2°.

If an answer looks far too big or small (such as 29 × cos 24° = 12.3 instead of 26.5), check the calculator is in degrees, not radians.