NCERT Solutions for Class 12 Accountancy.NCERT Solutions for Class 12 Business Studies.NCERT Solutions for Class 12 Computer Science (C++).NCERT Solutions for Class 12 Computer Science (Python).RD Sharma Class 11 Solutions Free PDF Download.Using the unit circle, the values of any trigonometric function for many angles other than those labeled can be easily calculated by hand using the angle sum and difference formulas.

In fact, all six standard trigonometric functions – sine, cosine, tangent, cotangent, secant, and cosecant, as well as archaic functions like versine and exsecant – can be defined geometrically in terms of a unit circle, as shown at right.

However, when defined with the unit circle, these functions produce meaningful values for any real-valued angle measure – even those greater than 2 π. When working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than π / 2. A simple demonstration of the above can be seen in the equality sin( π / 4) = sin( 3π / 4) = 1 / √ 2. It may be inferred in a similar manner that tan(π − t) = −tan( t), since tan( t) = y 1 / x 1 and tan(π − t) = y 1 / − x 1. The conclusion is that, since (− x 1, y 1) is the same as (cos(π − t), sin(π − t)) and ( x 1, y 1) is the same as (cos( t),sin( t)), it is true that sin( t) = sin(π − t) and −cos( t) = cos(π − t). It can hence be seen that, because ∠ROQ = π − t, R is at (cos(π − t), sin(π − t)) in the same way that P is at (cos( t), sin( t)). The result is a right triangle △ORS with ∠SOR = t. Now consider a point S(− x 1,0) and line segments RS ⊥ OS. Having established these equivalences, take another radius OR from the origin to a point R(− x 1, y 1) on the circle such that the same angle t is formed with the negative arm of the x-axis. Because PQ has length y 1, OQ length x 1, and OP has length 1 as a radius on the unit circle, sin( t) = y 1 and cos( t) = x 1. The result is a right triangle △OPQ with ∠QOP = t. Now consider a point Q( x 1,0) and line segments PQ ⊥ OQ. First, construct a radius OP from the origin O to a point P( x 1, y 1) on the unit circle such that an angle t with 0 < t < π / 2 is formed with the positive arm of the x-axis. Triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. When broken into real and imaginary components z = x + i y, z=x+iy, this condition is | z | 2 = z z ¯ = x 2 + y 2 = 1. This is the set of complex numbers z such that | z | = 1. In the complex plane, numbers of unit magnitude are called the unit complex numbers. Main article: unit complex numbers Animation of the unit circle with angles

Trig unit circle cheat sheet