The sine, cosine and tangent of 30°, 45° and 60° have exact values that can be found from two right triangles. You do not need a decimal approximation or a calculator to obtain them. The table below collects these values; the following constructions explain where they come from.
Sine, cosine and tangent table
Read across the row for the angle you need. The entries are ratios, so they have no length unit. Fractions and square roots keep the values exact.
| α | sin α | cos α | tan α |
|---|---|---|---|
| 30° | |||
| 45° | |||
| 60° |
Why the 30° and 60° values contain a square root of three
Start with an equilateral triangle whose sides have length 2. Its altitude splits it into two congruent right triangles. Each has a hypotenuse of length 2, a shorter leg of length 1, and angles of 30° and 60°. Pythagoras gives the remaining leg h:
For 30°, the opposite leg is 1 and the hypotenuse is 2, so the sine is one half. The adjacent leg is h, giving the cosine shown in the table. For 60°, the opposite and adjacent legs swap roles. That is why the sine of 30° equals the cosine of 60°, and vice versa.
Deriving the 45° values from a square
Draw a diagonal in a square with sides of length 1. It divides the square into two right triangles with equal legs. Each acute angle is 45°, and the diagonal is the hypotenuse.
Because the two legs are equal, sine and cosine must be equal. Dividing a leg by the hypotenuse and rationalising the denominator gives the exact value in the 45° row. The tangent is 1 because it compares two equal legs.
Calculating tangent from sine and cosine
Tangent is the opposite leg divided by the adjacent leg. It can also be obtained from the other two ratios:
Finding an opposite leg from a 60° angle
In a right triangle, the leg adjacent to a 60° angle measures 4 cm. Find the opposite leg h. The given length is a leg, not the hypotenuse.
Use tangent because it relates the opposite and adjacent legs. Put h in the numerator and 4 in the denominator.
Multiply both sides by 4 to isolate h. This step converts the ratio into the required length.
Replace the tangent by its exact value from the table, then calculate a decimal approximation only at the end.
The opposite leg is exactly four times the square root of three centimetres, approximately 6.93 cm. A calculator check must use degree mode.
At 30°, this produces the reciprocal of the square root of three. Multiplying numerator and denominator by that square root gives the equivalent rationalised form in the table. At 60°, the tangent is the square root of three.
Rebuilding the table without guessing
For sine at 30°, 45° and 60°, take the square roots of 1, 2 and 3 and divide each by 2. Cosine uses the same sequence in reverse. Then calculate tangent from their quotient. This is a memory aid backed by the triangles, not a rule for arbitrary angles.
If you use a calculator to check a decimal value, select degree mode. Radian mode interprets the input differently. These are exact values for the stated angles, while a rounded decimal is only an approximation. For the geometric meaning of the angle measures, see degrees and radians.
