When two sides of a right-angled triangle are known, you can find the angle by using the inverse trigonometric ratios sin⁻¹, cos⁻¹ and tan⁻¹.
From a ratio to an angle
Work out the ratio of the two sides, then apply the matching inverse function.
Key idea
θ = sin⁻¹(opp/hyp)
θ = cos⁻¹(adj/hyp)
θ = tan⁻¹(opp/adj)
Worked example
A right-angled triangle has opposite 5 cm and hypotenuse 10 cm. Find θ.
sin θ = 5/10 = 0.5
θ = sin⁻¹(0.5) = 30°.
Elevation and depression
The angle of elevation is measured upward from the horizontal to an object above you; the angle of depression is measured downward from the horizontal to an object below. For example, if a tower is 30 m tall and you stand 30 m from its base, the elevation of the top is tan⁻¹(30/30) = tan⁻¹(1) = 45°.
To solve a real problem, first draw the right-angled triangle and label the horizontal distance, the height and the line of sight. Then decide which two sides you know: the height is the opposite side, the horizontal distance is the adjacent side, and the line of sight is the hypotenuse. Choosing the ratio that uses your two known sides gives the quickest route to the angle. A neat sketch prevents you from mixing up elevation and depression.
Remember
- Two sides known → use the inverse ratio to get the angle.
- sin⁻¹, cos⁻¹, tan⁻¹ undo sin, cos, tan.
- Elevation looks up; depression looks down — both from the horizontal.
- Set the calculator to degrees.