Trigonometry
Grade 5-7Right-Angled Trigonometry (SOHCAHTOA)
Every right-angled trigonometry question comes down to the same three steps: label the sides relative to the angle, pick the correct ratio using SOHCAHTOA, then rearrange to find whatever is missing. This lesson covers labelling the hypotenuse, opposite and adjacent sides, choosing the correct trig ratio, finding a missing side, and finding a missing angle.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Labelling the sides of a right-angled triangle
The hypotenuse is always the longest side, opposite the right angle. Once a specific angle (other than the right angle) is chosen, the side directly across from it is the opposite, and the remaining side next to that angle is the adjacent.
In a right-angled triangle with a marked angle at the bottom right, the side going straight up from the right angle is usually the opposite, the side along the bottom is the adjacent, and the sloped side is the hypotenuse.
Choosing the correct ratio with SOHCAHTOA
SOHCAHTOA is a memory aid for the three trigonometric ratios: Sine = Opposite ÷ Hypotenuse, Cosine = Adjacent ÷ Hypotenuse, Tangent = Opposite ÷ Adjacent. Identify which two sides are involved in the question, known or unknown, and pick the ratio that connects them.
If a question gives the hypotenuse and asks for the opposite side, use sine, since sine is the ratio that connects opposite and hypotenuse.
Finding a missing side
Set up the equation using the chosen ratio, substituting the known angle and known side, then rearrange to make the missing side the subject.
To find x, opposite a 40° angle, with a hypotenuse of 12 cm: sin 40° = x ÷ 12, so x = 12 × sin 40° = 7.71 cm (3 s.f.).
Finding a missing angle
When two sides are known but the angle is missing, set up the ratio using the two known sides, then use the inverse trig function, sin⁻¹, cos⁻¹ or tan⁻¹, to find the angle.
With an opposite side of 7 cm and a hypotenuse of 15 cm: sin θ = 7 ÷ 15, so θ = sin⁻¹(7 ÷ 15) = 27.8° (1 d.p.).
Worked Examples
Three exam-style questions, fully solved.
Find the length of side x, opposite a 40° angle, with a hypotenuse of 12 cm. Give your answer correct to 3 significant figures.
Easy- 1.Identify the sides involved: opposite (x) and hypotenuse (12), so use sine
- 2.Set up the equation: sin 40° = x ÷ 12
- 3.Rearrange and calculate: x = 12 × sin 40°
Answer: 7.71 cm
Find the length of side x, opposite a 34° angle, with an adjacent side of 9 cm. Give your answer correct to 3 significant figures.
Medium- 1.Identify the sides involved: opposite (x) and adjacent (9), so use tangent
- 2.Set up the equation: tan 34° = x ÷ 9
- 3.Rearrange and calculate: x = 9 × tan 34°
Answer: 6.07 cm
A ladder of length 6.5 m leans against a vertical wall. The base of the ladder is 2.1 m from the wall. Find the angle the ladder makes with the ground, correct to 1 decimal place.
Hard- 1.Identify the sides involved relative to the angle at the ground: adjacent (2.1 m) and hypotenuse (6.5 m), so use cosine
- 2.Set up the equation: cos θ = 2.1 ÷ 6.5
- 3.Use the inverse cosine function: θ = cos⁻¹(2.1 ÷ 6.5)
Answer: 71.2°
Avoid These
The most common mistakes students make.
Mislabelling which side is opposite and which is adjacent relative to the given angle, especially when the triangle is drawn in an unfamiliar orientation.
Choosing the wrong trig ratio for the two sides involved, for example using sine when the sides given are adjacent and hypotenuse, which needs cosine.
Forgetting to use the inverse trig function (sin⁻¹, cos⁻¹ or tan⁻¹) when finding a missing angle, and instead dividing the two sides directly.
Rounding too early during the calculation, which makes the final answer less accurate than the required degree of precision.
Leaving the calculator in the wrong angle mode (radians instead of degrees), which gives a completely incorrect answer.
FAQ
Questions parents and students ask.
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