The various trigonometric identities and their principal angle values can be calculated through the use of a unit circle. In the unit circle, we have cosine as the x-coordinate and sine as the y-coordinate. Let us now find their respective values for θ = 0°, and θ = 90º.
For θ = 0°, the x-coordinate is 1 and the y-coordinate is 0. Therefore, we have cos0º = 1, and sin0º = 0. Let us look at another angle of 90º. Here the value of cos90º = 1, and sin90º = 1. Further, let us use this unit circle and find the important trigonometric function values of θ such as 30º, 45º, 60º. Also, we can also measure these θ values in radians. We know that 360° = 2π radians. We can now convert the angular measures to radian measures and express them in terms of the radians.
Unit Circle Table:
The unit circle table is used to list the coordinates of the points on the unit circle that corresond to common angles with the help of trigonometric ratios.
| Angle θ | Radians | Sinθ | Cosθ | Tanθ = Sinθ/Cosθ | Coordinates |
|---|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 | (1, 0) |
| 30° | π/6 | 1/2 | √3/2 | 1/√3 | (√3/2, 1/2) |
| 45° | π/4 | 1/√2 | 1/√2 | 1 | (1/√2, 1/√2) |
| 60° | π/3 | √3/2 | 1/2 | √3 | (1/2, √3/2) |
| 90° | π/2 | 1 | 0 | undefined | (0,1) |
We can find the secant, cosecant, and cotangent functions also using these formulas:
- secθ = 1/cosθ
- cosecθ = 1/sinθ
- cotθ = 1/tanθ
We have discussed the unit circle for the first quadrant. Similarly, we can extend and find the radians for all the unit circle quadrants. The numbers 1/2, 1/√2, √3/2, 0, 1 repeat along with the sign in all 4 quadrants.
Unit Circle in Complex Plane
A unit circle consists of all complex numbers of absolute value as 1. Therefore, it has the equation of |z| = 1. Any complex number z = x + iiy will lie on the unit circle with equation given as x2 + y2 = 1.
The unit circle can be considered as unit complex numbers in a complex plane, i.e., the set of complex numbers z given by the form,
z = eiit = cos t + ii sin t = cis(t)
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