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Section 2.2

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1. (a) \displaystyle{\sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2}}, \displaystyle{\cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2}},
\displaystyle{\tan\left(\frac{4\pi}{3}\right) = \sqrt{3}}, \displaystyle{\sec\left(\frac{4\pi}{3}\right) = -2}

(c) \displaystyle{\sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2}}, \displaystyle{\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}},
\displaystyle{\tan\left(\frac{4\pi}{3}\right) = -\sqrt{3}}, \displaystyle{\sec\left(\frac{4\pi}{3}\right) = -2}

(e) \displaystyle{\sin\left(-\frac{2\pi}{3}\right) = -\frac{\sqrt{3}}{2}}, \displaystyle{\cos\left(-\frac{2\pi}{3}\right) = -\frac{1}{2}},
\displaystyle{\tan\left(-\frac{2\pi}{3}\right) = \sqrt{3}}, \displaystyle{\sec\left(-\frac{2\pi}{3}\right) = -2}

(g) \displaystyle{\sin\left(\frac{11\pi}{6}\right) = -\frac{1}{2}}, \displaystyle{\cos\left(\frac{11\pi}{6}\right) = \frac{\sqrt{3}}{2}},
\displaystyle{\tan\left(\frac{11\pi}{6}\right) = -\frac{1}{\sqrt{3}}}, \displaystyle{\sec\left(\frac{11\pi}{6}\right) = \frac{2}{\sqrt{3}}}

2. (a) Amplitude = 1 , period = \dfrac{2\pi}{3} , phase angle = 0

(c) Amplitude = 1 , period = 2 , phase angle = \pi

(e) Amplitude = 4 , period = 2 , phase angle = 0

(e) Amplitude = 5 , period = = \pi , phase angle = -\dfrac{\pi}{2}

4. (a) \sin(2x) = \sin(x + x) = \sin(x)\cos(x) + \cos(x)\sin(x)
= 2\sin(x)\cos(x)

5. (a) This follows from:
\cos(2x) = \cos^2(x) - \sin^2(x) = \cos^2(x) - (1 - \cos^2(x)) = 2\cos^2(x) - 1.

6. (a) \displaystyle{\sin\left(x-\frac{\pi}{2}\right) = \sin(x)\cos\left(-\frac{\pi}{2}\right) + \cos(x)\sin\left(-\frac{\pi}{2}\right) = -\cos(x)}

(c) \displaystyle{\sin\left(x+\frac{\pi}{2}\right) = \sin(x)\cos\left(\frac{\pi}{2}\right) + \cos(x)\sin\left(\frac{\pi}{2}\right) = \cos(x)}

8. \displaystyle{\tan(t + \pi) = \frac{\sin(t + \pi)}{\cos(t + \pi)} = \frac{\sin(t)\cos(\pi) + \cos(t)\sin(\pi)}{\cos(t)\cos(\pi) - \sin(t)\sin(\pi)}}
\displaystyle{= \frac{-\sin(t)}{-\cos(t)} = \tan(t)}

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