In a right-angled triangle, sine, cosine and tangent connect one angle to two of its sides. The whole topic comes down to a two-step routine: fix your angle, then label the three sides relative to it — opposite, adjacent, hypotenuse — and pick the ratio that uses the two sides you care about.
The hypotenuse is always opposite the right angle and is the longest side. The opposite is across from the angle you are working with; the adjacent is the remaining side next to it. Re-label every time you change angle.
To find a side, rearrange the ratio and evaluate directly. To find an angle, use the inverse function: sin⁻¹, cos⁻¹ or tan⁻¹ on your calculator.
List which two sides are opposite / adjacent / hypotenuse, then read off SOH-CAH-TOA. Two sides given → you can always find the third side or the angle.
A right-angled triangle has hypotenuse 9 cm and an angle of 34°. Find the length x opposite that angle.
In a right-angled triangle the side opposite angle θ is 7 cm and the adjacent side is 5 cm. Find θ.
In a right-angled triangle the side opposite an angle is 5 cm and the hypotenuse is 13 cm. Find the angle.
A right-angled triangle has an angle of 28° and hypotenuse 20 cm. Find the side opposite the 28° angle.
A ladder of length 6.2 m rests against a vertical wall. The foot of the ladder is 1.8 m from the base of the wall. Calculate the angle the ladder makes with the ground.
An isosceles triangle has a base of 10 cm and two equal sides of 13 cm. Calculate the size of each base angle.
A rectangle measures 24 cm by 10 cm. A diagonal is drawn. Calculate the acute angle between the diagonal and the longer side of the rectangle.