If cosecant of theta equals 3 and cosine of theta is less than zero, find sine of theta, cosine of theta, tangent of theta, cotangent of theta and secant of theta

sin=1/3, cos is negative. Second quadrant.

draw the triangle in second quadrant. hyp=3, opposite=1, adjacent=1/sqrt8

check that.

To find the values of sine, cosine, tangent, cotangent, and secant of theta, given that cosecant of theta is 3 and cosine of theta is less than zero, we can use the following trigonometric identities:

1. Cosecant (csc) of theta is the reciprocal of sine (sin) of theta:
csc(theta) = 1/sin(theta)

2. Cosine (cos) of theta is negative in the second and third quadrants of the unit circle:
cos(theta) < 0

Given that csc(theta) = 3, we can find sin(theta) as follows:

1. csc(theta) = 3
Taking the reciprocal of both sides:
1/sin(theta) = 3
sin(theta) = 1/3

Now, we can determine the values of the other trigonometric functions:

1. Cosine (cos) of theta:
Given that cos(theta) < 0, and sin(theta) = 1/3, we can use the Pythagorean identity to find cos(theta):
cos(theta) = sqrt(1 - sin^2(theta))
= sqrt(1 - (1/3)^2)
= sqrt(1 - 1/9)
= sqrt(8/9)
= (-sqrt(8))/3 (negative because cos(theta) < 0)

2. Tangent (tan) of theta:
tan(theta) = sin(theta)/cos(theta)
= (1/3) / (-sqrt(8)/3)
= -1/sqrt(8)
= -sqrt(8)/8

3. Cotangent (cot) of theta:
cot(theta) = 1/tan(theta)
= 1 / (-1/sqrt(8))
= -sqrt(8)

4. Secant (sec) of theta:
sec(theta) = 1/cos(theta)
= 1 / (-sqrt(8)/3)
= -3/sqrt(8)
= -3sqrt(8)/8

Therefore, the values of the trigonometric functions for theta are:
sine of theta (sin(theta)) = 1/3
cosine of theta (cos(theta)) = -sqrt(8)/3
tangent of theta (tan(theta)) = -sqrt(8)/8
cotangent of theta (cot(theta)) = -sqrt(8)
secant of theta (sec(theta)) = -3sqrt(8)/8