SOHCAHTOA (Trigonometry)
SOHCAHTOA is a memory aid for the three trigonometric ratios in a right-angled triangle, which relate an angle to the lengths of two of the triangle's sides: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
Method
- Label the triangle's sides relative to the angle you are using: the Opposite (across from the angle), the Adjacent (next to the angle, not the hypotenuse), and the Hypotenuse (the longest side, opposite the right angle).
- Decide which two sides are involved, or which side and the angle, then use SOHCAHTOA to choose the correct ratio.
- To find a missing side, set up the equation using the ratio, then rearrange to make the unknown side the subject.
- To find a missing angle, set up the equation using the ratio, then use the inverse function (sin^-1, cos^-1 or tan^-1) on your calculator.
- Always check the angle is between 0 and 90 degrees and the side lengths make sense - the hypotenuse should be the longest side.
Worked example
In a right-angled triangle, the angle is 40 degrees and the hypotenuse is 12 cm. Find the length of the side opposite the angle, to 1 decimal place.
- The sides involved are Opposite and Hypotenuse, so use Sine (SOH): sin(angle) = Opposite/Hypotenuse.
- Substitute the known values: sin(40) = Opposite/12.
- Rearrange to make Opposite the subject: Opposite = 12 x sin(40).
- Calculate: 12 x sin(40) = 12 x 0.643 = 7.71...
- The opposite side is 7.7 cm, to 1 decimal place.
Practice questions
Try each question, then tap to reveal the answer.
Exam-style questions
Written in the style of a GCSE exam paper, with a full mark scheme.
A ladder leans against a vertical wall, making an angle of 68 degrees with the ground. The foot of the ladder is 1.5 m from the wall. Work out the length of the ladder, to 2 decimal places.
A right-angled triangle has a base of 7 cm and a height of 4 cm. Work out the angle between the base and the hypotenuse, to 1 decimal place.
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