How do you find the sine, cosine, and tangent values on the unit circle? Provide an example.
The Correct Answer and Explanation is:
To find the sine, cosine, and tangent values on the unit circle, we start by understanding the definition of the unit circle. The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the Cartesian coordinate system. Each point on the unit circle can be described by its coordinates (x, y), where ( x ) represents the cosine of the angle ( \theta ) and ( y ) represents the sine of ( \theta ). This is based on the fundamental trigonometric relationships defined in a right triangle.
- Sine (sin): The sine of an angle is defined as the y-coordinate of the point on the unit circle corresponding to that angle. Thus, ( \sin(\theta) = y ).
- Cosine (cos): The cosine of an angle is defined as the x-coordinate of the point on the unit circle corresponding to that angle. Thus, ( \cos(\theta) = x ).
- Tangent (tan): The tangent of an angle is the ratio of the sine to the cosine. Hence, ( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{y}{x} ), provided that ( \cos(\theta) \neq 0 ).
Example:
Let’s consider the angle ( 45^\circ ) (or ( \frac{\pi}{4} ) radians).
- Locate the point: On the unit circle, the angle ( 45^\circ ) corresponds to the point ((\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})).
- Calculate sine and cosine:
- Sine: ( \sin(45^\circ) = \frac{\sqrt{2}}{2} )
- Cosine: ( \cos(45^\circ) = \frac{\sqrt{2}}{2} )
- Calculate tangent:
- Tangent: ( \tan(45^\circ) = \frac{\sin(45^\circ)}{\cos(45^\circ)} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 )
Summary:
Thus, for the angle ( 45^\circ ), we have:
- ( \sin(45^\circ) = \frac{\sqrt{2}}{2} )
- ( \cos(45^\circ) = \frac{\sqrt{2}}{2} )
- ( \tan(45^\circ) = 1 )
These relationships help visualize and calculate trigonometric values efficiently using the unit circle, which is fundamental in trigonometry.