The Amazing Unit Circle
Reference Angles in Quadrant III

Reference angles are used to determine the values of the trigonometric functions in the second, third and fourth quadrants, in particular, for the "nice" angles. The reference angle for an angle θ is the smallest angle φ from the (positive or negative) x-axis to the terminal ray of the angle θ.

signs by quadrant

For an angle θ in the third quadrant the reference angle φ is the angle that must be subtracted from θ to leave a straight angle, that is, π radians or 180°. Thus θ - φ = π or θ - φ = 180°, and so

φ = θ - π or φ = θ - 180°.

Next plot the reference angle φ in the first quadrant, that is, in standard position. We see that the point (cos θ, sin θ) is on the opposite side of the unit circle from the point (cos φ, sin φ). The x- and y-coordinates of these two points have opposite signs. Thus, for θ in Quadrant III:

cos θ = - cos φ and sin θ = - sin φ.

Conclusion: to compute the value of cosine and sine of an angle θ in the third quadrant, find the value of the function at the reference angle φ and then attach the correct sign (- for both cosine and sine in Quadrant III). The method also works for the other trigonometric functions. For example, tan θ = sin θ/cos θ = (- sin φ)/(- cos φ) = sin θ/cos θ = tan φ in Quadrant III.

Restore initial diagram

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