In a right-angled triangle, name the sides relative to the angle θ you are using: the hypotenuse is opposite the right angle (the longest side), the opposite faces θ, and the adjacent is next to θ.
The three ratios
Each ratio links θ with two sides. Choose the one that contains the side you know and the side you want.
Key idea
sin θ = opp / hyp
cos θ = adj / hyp
tan θ = opp / adj
(Remember it as SOH-CAH-TOA.)
Finding a side
Write the correct ratio, put in the numbers, then rearrange. If the unknown is on top, multiply; if it is on the bottom, divide.
Worked example
A right-angled triangle has hypotenuse 10 cm and angle θ = 30°. Find the opposite side.
sin 30° = opp / 10
opp = 10 × sin 30° = 10 × 0.5 = 5 cm.
Another: if the adjacent is 8 cm and θ = 60°, the hypotenuse is adj ÷ cos 60° = 8 ÷ 0.5 = 16 cm. Here the unknown was on the bottom of the ratio, so we divided instead of multiplying.
Special angles give exact values you should know: sin 30° = cos 60° = 0.5, and tan 45° = 1. For other angles, use the sin, cos or tan button on your calculator, making sure it is set to degrees before you start.
Remember
- Label hyp, opp and adj before choosing a ratio.
- SOH-CAH-TOA picks the right formula.
- Unknown on top → multiply; unknown on the bottom → divide.
- The hypotenuse is always the longest side.