lesson
Find a missing side
1 · Learn
Start by marking the right angle and the chosen acute angle. Label opposite, adjacent and hypotenuse relative to that acute angle. Choose the ratio containing the known side and the unknown side: sin(theta) = opposite/hypotenuse; cos(theta) = adjacent/hypotenuse; tan(theta) = opposite/adjacent.
For a right triangle with hypotenuse 10 cm and angle 30°, the opposite side x satisfies sin(30°) = x/10. Multiply both sides by 10: x = 10 sin(30°) = 5 cm. A unit triangle with hypotenuse 1 has opposite side 0.5; scaling every length by 10 gives the same answer.
The unknown may be underneath the fraction. If the adjacent side is 9 cm and the angle is 50°, cos(50°) = 9/h. Multiply by h, then divide by cos(50°): h = 9/cos(50°), approximately 14.00 cm. Multiplying 9 by the cosine instead would not isolate h.
Use DEG mode for degree angles. Do not round a sine, cosine or tangent before the last calculation. Check that the hypotenuse is the longest side. Write the length unit and distinguish an approximate answer from an exact one.
2 · Worked example
A diagram of a straight ramp is modelled as a right triangle. Its horizontal run is 8 m and its angle to the horizontal is 35°. Height x is opposite and run is adjacent: tan(35°) = x/8. Thus x = 8 tan(35°), approximately 5.60 m. This is a paper calculation, not an instruction to use the ramp.
3 · Your turn
In one right triangle, the hypotenuse is 12 cm and the chosen angle is 40°. Find the adjacent side. In another, the adjacent side is 9 cm and the angle is 50°: find its hypotenuse. Round each final answer to two decimal places.
Check your answer
Adjacent = 12 cos(40°), approximately 9.19 cm. Hypotenuse = 9/cos(50°), approximately 14.00 cm. In each triangle the hypotenuse exceeds either shorter side.
4 · Apply your learning
For the 10 cm, 30° worked example, calculate the adjacent side using cosine and check the two shorter sides with Pythagoras. Keep the unrounded cosine result during the check; rounding first can produce a small discrepancy.
Adult guidance and safety
Check similarity, side naming, calculator degree mode and equation rearrangement before moving on. Use supplied paper models, not climbing or surveying unsafe structures. Keep full calculator precision until the requested final rounding.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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