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Unformatted text preview: 2 θ= 5π
6 x x 1 1 π
In the above scenario, the angle π is called the reference angle for the angle 56 . In general, for
a non-quadrantal angle θ, the reference angle for θ (usually denoted α) is the acute angle made
between the terminal side of θ and the x-axis. If θ is a Quadrant I or IV angle, α is the angle
between the terminal side of θ and the positive x-axis; if θ is a Quadrant II or III angle, α is
the angle between the terminal side of θ and the negative x-axis. If we let P denote the point
(cos(θ), sin(θ)), then P lies on the Unit Circle. Since the Unit Circle possesses symmetry with
respect to the x-axis, y -axis and origin, regardless of where the terminal side of θ lies, there is a
point Q symmetric with P which determines θ’s reference angle, α as seen below.
y y 1 1 P =Q P
1 x Reference angle α for a Quadrant I angle Q
1 x Reference angle α for a Quadrant II angle 10.2 The Unit Circle: Cosine and Sine 617 y y 1 1 Q Q α α
1 α x 1 α P x P Reference angle α for a Quadrant III angle Reference angle α for a Quadrant IV angle We have just outlined the proof of the following theorem.
Theorem 10.2. Reference Angle Theorem. Suppose α is the reference angle for θ. Then
cos(θ) = ± cos(α) and sin(θ) = ± sin(α), where the choice of the (±) depends on the quadrant in
which the terminal side of θ lies.
In light of Theorem 10.2, it pays to know the cosine and sine values for certain common an...
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