The line with equation −a+3b=0 coincides with the terminal side of an angle θ in standard position and sinθ< 0 .
what is the value of cosθ ?

If is the variable of the horizontal axis, then you can solve for to get the equation of the line in slope-intercept form in the plane:



i.e. a line with slope  through the origin, which means it is contained in the first and third quadrants. Since the terminal side of has a negative sine, the angle must lie in the third quadrant.

Because the slope of the line is , you can choose any length along the line to make up the hypotenuse of a right triangle with reference angle . Any such right triangle will have , regardless of whether the angle is the first or third quadrant. But since is known to lie in the third quadrant, and so and are both negative, you have