Standard Angles

Ratios of some standard angles

ratio\angle 30° 45° 60° 90°
sine 0 1/2 1/√2 √3/2 1
cos 1 √3/2 1/√2 1/2 0
tan 0 1/√3 1 √3
cot √3 1 1/√3 0
sec 1 2/√3 √2 2
cosec 2 √2 2/√3 1

Angle of 45°


Let the angle AOP traced out be 45°.
Then, since the three angles of a triangle are together equal to two right angles,
∠OPM = 180° - ∠POM - ∠PMO
=180° - 45° - 90° = 45° = ∠POM.
∴OM = MP.

If OP = 2a, we have
4a² = OP² = OM² + MP² = 2.OM²,
so that OM = a√2;
∴ sin 45° = MP/OP = a√2/2a = 1/√2,
cos 45° = OM/OP = a√2/2a = 1/√2
and tan 45° = 1

Angle of 30°

angle of 30 degree
Let the angle AOP traced out be 30°.
Produce PM to P' making MP' equal to PM.

The two triangles OMP and OMP' have their sides OM and MP equal to OM and MP' and also the contained angle equal.

Therefore OP' = OP, and ∠OP'P = ∠OPP' = 60°, so that the triangle P'OP is equilateral.

Hence, if OP = 2a, we have
MP = ½P'P = ½OP = a.

Also OM = √(OP²-MP²) = √(4a² - a²) = a√3.
∴sin 30° = MP/OP = 1/2
cos 30° = OM/OP = a√3/2a = √3/2.
and tan 30° = sin 30°/cos 30° = 1/√3

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