An angle θ, in standard position, has its terminal side in Quadrant IV on the
line 12x + 5y = 0. Find the exact values of sin θ and cos θ.
sin=-12/sqrt(144+25)
cos=5/13
To find the exact values of sin θ and cos θ for an angle in standard position, we need to determine the coordinates of the point where the terminal side of the angle intersects the unit circle.
Let's start by finding the intersection point of the line 12x + 5y = 0 with the x and y-axes.
1. Intersection with the x-axis:
When y = 0, we can solve the equation to find the x-coordinate:
12x + 5(0) = 0
12x = 0
x = 0
So, the line intersects the x-axis at the point (0, 0).
2. Intersection with the y-axis:
When x = 0, we can solve the equation to find the y-coordinate:
12(0) + 5y = 0
5y = 0
y = 0
So, the line intersects the y-axis at the point (0, 0).
Since the terminal side of the angle is in Quadrant IV (bottom right quadrant) and the line intersects the x and y-axes at (0, 0), the terminal side of the angle will pass through the point (0, 0) in Quadrant IV.
On the unit circle, the x-coordinate represents the cosine (cos) value of the angle, and the y-coordinate represents the sine (sin) value of the angle.
Since the point (0, 0) is on the unit circle, the cos θ and sin θ values in this case will both be 0.
Therefore, the exact values of sin θ and cos θ for this angle are both 0.