Exact Values of Sine, Cosine and Tangent
Know and use the exact values of sin, cos and tan for 0, 30, 45, 60 and 90 degrees, taken from a halved equilateral triangle and a half square.
What a learner can do afterwards
- Recall sin 30, cos 60 and tan 45 without a calculator
- Derive the 30 and 60 degree values from an equilateral triangle cut in half
- Leave an answer in surd form instead of rounding it
1 · Read
Some angles appear so often that you must know their sine, cosine, and tangent with no calculator. Your must-know trio is sin 30 = 1/2, cos 60 = 1/2, and tan 45 = 1. A close friend is sin 60 = root 3 over 2, which you leave in surd form.
Each value comes from a special triangle, so learn the source and you will never mix them up. Cut an equilateral triangle in half and you get a 30-60-90 triangle: the hypotenuse is the old side, the shortest leg is half of it, and that half-over-whole is exactly sin 30 = 1/2. Cut a square in half instead and you get a 45-45-90 triangle with equal legs, so opposite over adjacent gives tan 45 = 1.
Practise by covering the table, writing what you remember, then checking your gaps. Leave an answer in surd form whenever it holds a root that never comes out neat, like root 3 over 2. And use the symmetries to check yourself: sin 30 matches cos 60 because 30 and 60 are co-angles.
Halve an equilateral triangle for 30 and 60, halve a square for 45, and keep roots in surd form.
2 · Watch
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Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.