The Amazing Unit Circle
Reference Angles in Quadrant II |

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 |

For an angle θ in the second quadrant the reference angle φ is the remaining angle needed to complete a straight angle, that is, π radians or 180°. Thus θ + φ = π or θ + φ = 180°, and so φ = π - θ or φ = 180° - θ. 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 the reflection of the point (cos φ, sin φ) in the y-axis. The x-coordinates of these two points have opposite signs, while the y-coordinates are equal. Thus, for θ in Quadrant II: cos θ = - cos φ and sin θ = sin φ. Conclusion: to compute the value of cosine and sine of an angle θ in the second quadrant, find the value of the function at the reference angle φ and then attach the correct sign (- for cosine and + for sine in Quadrant II). The method also works for the other trigonometric functions. For example, tan θ = - tan φ in Quadrant II. Restore initial diagram |

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