Any point on the unit circle has coordinates(x, y), which are equal to the trigonometric identities of (cosθ, sinθ). For any values of θ made by the radius line with the positive x-axis, the coordinates of the endpoint of the radius represent the cosine and the sine of the θ values. Here we have cosθ = x, and sinθ = y, and these values are helpful to compute the other trigonometric ratio values. Applying this further we have tanθ = sinθ/cosθ or tanθ = y/x.
Another important point to be understood is that the sinθ and cosθ values always lie between 1 and -1, and the radius value is 1, and it has a value of -1 on the negative x-axis. The entire circle represents a complete angle of 360º and the four quadrant lines of the circle make angles of 90º, 180º, 270º, 360º(0º). At 90º and at 270º the cosθ value is equal to 0 and hence the tan values at these angles are undefined.
Example: Find the value of tan 45º using sin and cos values from the unit circle.
Solution:
We know that, tan 45° = sin 45°/cos 45°
Using the unit circle chart:
sin 45° = 1/√2
cos 45° = 1/√2
Therefore, tan 45° = sin 45°/cos 45°
= (1/√2)/(1/√2)
= 1
Answer: Therefore, tan 45° = 1
Unit Circle Chart in Radians
The unit circle represents a complete angle of 2π radians. And the unit circle is divided into four quadrants at angles of π/2, π. 3π/2, and 2π respectively. Further within the first quadrant at the angles of 0, π/6, π/4, π/3, π/2 are the standard values, which are applicable to the trigonometric ratios. The points on the unit circle for these angles represent the standard angle values of the cosine and sine ratios. On close observation of the below figure the values are repeated across the four quadrants, but with a change in sign. This change in sign is because of the reference x-axis and y-axis, which are positive on one side and negative on the other side of the origin. Now with the help of this, we can easily find the trigonometric ratio values of standard angles, across the four quadrants of the unit circle.
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