Sure thing, Erin! Let's find the trigonometric values for the given cosine value of -3/5.
First, let's find the sine value. Since the cosine is negative and theta is in the second quadrant (pi/2 < theta < pi), we know that the sine will be positive. We can use the Pythagorean identity to find the sine:
sin^2(theta) + cos^2(theta) = 1
Substituting the given value of cosine:
sin^2(theta) + (-3/5)^2 = 1
Simplifying:
sin^2(theta) + 9/25 = 1
sin^2(theta) = 1 - 9/25
sin^2(theta) = 16/25
Taking the square root of both sides:
sin(theta) = ±4/5
Since theta is in the second quadrant, the sine will be positive. So, the sine value is 4/5.
Next, let's find the tangent value. We know that tangent is equal to sine divided by cosine:
tan(theta) = sin(theta) / cos(theta)
Substituting the values we found:
tan(theta) = (4/5) / (-3/5)
Simplifying:
tan(theta) = -4/3
For the cosecant value, we know that it is the reciprocal of the sine:
csc(theta) = 1 / sin(theta)
Substituting the sine value we found:
csc(theta) = 1 / (4/5)
Simplifying:
csc(theta) = 5/4
For the secant value, we know that it is the reciprocal of the cosine:
sec(theta) = 1 / cos(theta)
Substituting the given cosine value:
sec(theta) = 1 / (-3/5)
Simplifying:
sec(theta) = -5/3
Finally, for the cotangent value, we know that it is the reciprocal of the tangent:
cot(theta) = 1 / tan(theta)
Substituting the tangent value we found:
cot(theta) = 1 / (-4/3)
Simplifying:
cot(theta) = -3/4
So, the trigonometric values for the given cosine value of -3/5, where pi/2 < theta < pi, are:
sin(theta) = 4/5
tan(theta) = -4/3
csc(theta) = 5/4