The general equation of a circle is (x – a)2 + (y – b)2 = r2, which represents a circle having the center (a, b) and the radius r. This equation of a circle is simplified to represent the equation of a unit circle. A unit circle is formed with its center at the point(0, 0), which is the origin of the coordinate axes. and a radius of 1 unit. Hence the equation of the unit circle is (x – 0)2 + (y – 0)2 = 12. This is simplified to obtain the equation of a unit circle.
Equation of a Unit Circle: x2 + y2 = 1
Here for the unit circle, the center lies at (0,0) and the radius is 1 unit. The above equation satisfies all the points lying on the circle across the four quadrants.
Finding Trigonometric Functions Using a Unit Circle
We can calculate the trigonometric functions of sine, cosine, and tangent using a unit circle. Let us apply the Pythagoras theorem in a unit circle to understand the trigonometric functions. Consider a right triangle placed in a unit circle in the cartesian coordinate plane. The radius of the circle represents the hypotenuse of the right triangle. The radius vector makes an angle θ with the positive x-axis and the coordinates of the endpoint of the radius vector is (x, y). Here the values of x and y are the lengths of the base and the altitude of the right triangle. Now we have a right angle triangle with the sides 1, x, y. Applying this in trigonometry, we can find the values of the trigonometric ratio, as follows:
- sinθ = Altitude/Hypoteuse = y/1
- cosθ = Base/Hypotenuse = x/1
We now have sinθ = y, cosθ = x, and using this we now have tanθ = y/x. Similarly, we can obtain the values of the other trigonometric ratios using the right-angled triangle within the unit circle. Also by changing the θ values we can obtain the principal values of these trigonometric ratios.
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