Question

The terminal side of an angle θ intersects a circle with a radius of 3 at a point where the x-coordinate is x = -2 . Given that tan(θ) > 0 , determine the exact values of the six trigonometric functions (sin, cos, tan, cot, sec, csc ) for the angle θ.

Answer 1

The exact values of the six trigonometric functions for the angle θ in the second quadrant are: sin(θ) = (√5)/3, cos(θ) = -2/3, tan(θ) = (√5)/(-2), cot(θ) = (-2)/(√5), sec(θ) = -3/2, csc(θ) = 3/(√5).

Step-by-step explanation:

Given that the x-coordinate is -2 and the radius of the circle is 3, we can use the Pythagorean theorem to find the y-coordinate:

Using the Pythagorean theorem: x^2 + y^2 = r^2

-2^2 + y^2 = 3^2

4 + y^2 = 9

y^2 = 5

y = ±√5

Since tan(θ) > 0 and the x-coordinate is negative, the angle lies in the second quadrant.

Therefore, the exact values of the six trigonometric functions for the angle θ are:

sin(θ) = y/r = (√5)/3

cos(θ) = x/r = -2/3

tan(θ) = y/x = (√5)/(-2)

cot(θ) = 1/tan(θ) = (-2)/(√5)

sec(θ) = 1/cos(θ) = -3/2

csc(θ) = 1/sin(θ) = 3/(√5)