The Amazing Unit Circle
Reference Angles in Quadrant IV |

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 fourth quadrant the reference angle φ is the remaining angle needed to complete a full circle angle, that is, 2π radians or 360°. Thus θ + φ = 2π or θ + φ = 360°, and so φ = 2π - θ or φ = 360° - 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 x-axis. The x-coordinates of these two points are equal, while the y-coordinates have opposite signs. Thus, for θ in Quadrant IV: cos θ = cos φ and sin θ = - sin φ. Conclusion: to compute the value of cosine and sine of an angle θ in the fourth 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 IV). The method also works for the other trigonometric functions. For example, tan θ = - tan φ in Quadrant IV. Restore initial diagram |

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