If vac theta =-2 and theta lies in quadrant 3, find tan theta

Answer 1

To find the value of tan(theta), we need to determine the value of sin(theta) and cos(theta) first because tan(theta) is defined as sin(theta)/cos(theta).

Given that sec(theta)=-2 and theta lies in quadrant 3, we can use the following definitions:
sec(theta) = 1/cos(theta)
and
sec(theta) = -2

So, we have:
1/cos(theta) = -2

To solve for cos(theta), we can multiply both sides of the equation by cos(theta):
1 = -2cos(theta)

Then, divide both sides of the equation by -2:
cos(theta) = -1/2

Since theta lies in quadrant 3, where cos(theta) is negative and sin(theta) is positive, we can determine that sin(theta) = √(1 - cos^2(theta)):
sin(theta) = √(1 - (-1/2)^2) = √(1 - 1/4) = √(3/4) = √3/2

Finally, we can find tan(theta) by dividing sin(theta) by cos(theta):
tan(theta) = sin(theta)/cos(theta) = (√3/2) / (-1/2) = -(√3/2) / (1/2) = -√3

Therefore, tan(theta) = -√3.