lesson
Use exact values before decimals
1 · Learn
For special angles, trigonometric ratios have exact values. sin 0°=0, sin 30°=1/2, sin 45°=√2/2, sin 60°=√3/2 and sin 90°=1. Cosine uses the same values in reverse order.
tan θ=sin θ/cos θ where cosine is non-zero: tan 0°=0, tan 30°=√3/3, tan 45°=1 and tan 60°=√3. tan 90° is undefined.
Exact values support exact calculation and checking. Keep DEG mode selected; a decimal approximation is not a replacement when an exact form is requested.
2 · Worked example
In a 45°-45°-90° triangle with legs 1, the hypotenuse is √2, so sin45°=1/√2=√2/2 and tan45°=1.
3 · Your turn
State sin30°, cos60° and tan45° exactly. Then calculate 8sin30°.
Check your answer
1/2, 1/2 and 1; 8sin30°=4.
4 · Apply your learning
Draw the 45° and 30°-60° right triangles, label exact side ratios, and derive rather than merely copy the values.
Adult guidance and safety
This is England national-curriculum core teaching, not an exam-board specification. Check prerequisite understanding, require clear working and use additional varied practice before judging fluency.
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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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