Find the exact value of the following trigonometric functions:
sin(13pi), cos(14pi), tan(15pi)
follow the steps that I showed you in previous post
To find the exact value of trigonometric functions at specific angles, we can use the unit circle. The unit circle is a circle with a radius of 1 and is centered at the origin of a coordinate plane.
1. sin(13π):
In the unit circle, angles are measured in radians. To find the value of sin(13π), we need to locate the angle on the unit circle. Since 13π is equivalent to 6 full rotations (2π) plus an additional angle of π, we can disregard the rotations and only focus on the additional angle.
If we move counterclockwise from the positive x-axis by π (180 degrees), we end up on the negative side of the y-axis. Therefore, sin(13π) = sin(π) = 0.
2. cos(14π):
Similarly, for cos(14π), we need to determine where the angle falls on the unit circle. Since 14π is equivalent to 7 full rotations (2π), the angle returns to its starting position on the positive x-axis. Therefore, cos(14π) = cos(0) = 1.
3. tan(15π):
To find tan(15π), we can divide sin(15π) by cos(15π) because tan(theta) = sin(theta) / cos(theta).
We have already determined that sin(15π) = 0 (similar to sin(13π)) and cos(15π) = -1 (since cos(theta) = -1 when theta is π and multiples of π).
Dividing sin(15π) by cos(15π) gives us 0 / -1 = 0.
Therefore, sin(13π) = 0, cos(14π) = 1, and tan(15π) = 0.