lesson
Choose sides relative to the angle
1 · Learn
In a right triangle, opposite and adjacent depend on the chosen acute angle, while the hypotenuse remains opposite the right angle. Sine is opposite/hypotenuse, cosine adjacent/hypotenuse and tangent opposite/adjacent.
Use the correct angle mode when calculating degrees. These right-triangle ratios do not apply unchanged to an arbitrary non-right triangle.
Why do the ratios depend on the angle, not the size? Similar right triangles scale all sides by the same factor, which cancels in a ratio. A 3-4-5 triangle and a 6-8-10 triangle have the same opposite/hypotenuse ratio: 3/5 = 6/10 = 0.6.
The diagram uses a quarter of a unit circle: its radius is 1. A point P on the circle makes a right triangle with horizontal length 0.8 and vertical length 0.6. Its hypotenuse is 1, so cos(theta) = 0.8 and sin(theta) = 0.6. Moving P changes the angle and these two lengths, but the radius stays 1. Scaling this triangle to hypotenuse 5 gives sides 4 and 3. Tangent compares vertical to horizontal: 0.6/0.8 = 3/4.
2 · Worked example
For an acute angle with opposite 3 and hypotenuse 5, sine is 3/5 = 0.6. The other shorter side is 4, so cosine is 4/5.
3 · Your turn
Relative to an acute angle in a right triangle, opposite is 5 and adjacent is 12. What is its tangent?
Check your answer
5/12.
4 · Apply your learning
Label the three sides from each acute angle in the same triangle and explain which names change.
Adult guidance and safety
Require reasoning and a check. Provide accurate graphs, construction instruments and varied practice. Annual KS3 allocation is proposed; do not treat these short models as a complete assessed programme.
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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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