Trig Functions of Angles

Sol 1 The reference angle for $2\pi/3$ is $\pi-2\pi/3=\pi/3$. Since the sine is positive in Quadrant II, $\sin 2\pi/3=\sin\pi/3=\frac{\sqrt{3}}{2}$.

Sol 2 The reference angle for $7\pi/6$ is $7\pi/6-\pi=\pi/6$. Since the cosine is negative in Quadrant III, $\cos 7\pi/6=-\cos\pi/6=-\frac{\sqrt{3}}{2}$.

Sol 3 The reference angle for $3\pi/4$ is $\pi-3\pi/4=\pi/4$. Since the tangent is negative in Quadrant II, $\tan 3\pi/4=-\tan \pi/4=-1$.

Sol 4 The reference angle for $5\pi/3$ is $2\pi-5\pi/3=\pi/3$. Since the cosine is positive in Quadrant IV, $\cos 5\pi/3=\cos \pi/3= 1/2$.

Sol 5 The reference angle for $5\pi/4$ is $5\pi/4-\pi=\pi/4$. Since $\sec\theta=1/(\cos\theta)$ and the cosine is negative in Quadrant III, $\cos 5\pi/4=-\cos\pi/4=-1/\sqrt{2}$ and $\sec 5\pi/4=1/(\cos 5\pi/4)=-\sqrt{2}$.

Sol 6 The reference angle for $5\pi/6$ is $\pi-5\pi/6=\pi/6$. Since $\cot\theta=1/(\tan\theta)$ and the tangent is negative in Quadrant II, $\tan 5\pi/6=-\tan\pi/6=-1/\sqrt{3}$ and $\cot 5\pi/6=1/(\tan\pi/6)=-\sqrt{3}$.

Sol 7 The reference angle for $4\pi/3$ is $4\pi/3-\pi=\pi/3$. Since the tangent is positive in Quadrant III, $\tan 4\pi/3=\tan\pi/3=\sqrt{3}$.

Sol 8 The reference angle for $7\pi/4$ is $2\pi-7\pi/4=\pi/4$. Since $\csc\theta=1/(\sin\theta)$ and the sine is negative in Quadrant IV, $\sin 7\pi/4=-\sin\pi/4=-\sqrt{2}/2$ and $\csc 7\pi/4=1/(-\sqrt{2}/2)=-2/\sqrt{2}=-\sqrt{2}$.

Sol 9 Since $r=\sqrt{x^2+y^2}=\sqrt{6^2+(-8)^2}=\sqrt{100}=10$,

$\cos\theta=\frac{x}{r}=6/10=3/5$ and $\sin\theta=\frac{y}{r}=-8/10=-4/5$.

Sol 10 Since $2\pi=12\pi/6$, subtracting off a multiple of $2\pi$ gives $29\pi/6-2(2\pi)=29\pi/6-24\pi/6=5\pi/6$; so $\sin21\pi/6=\sin5\pi/6$. Since the reference angle for $5\pi/6$ is $\pi-5\pi/6=\pi/6$, and since the sine is positive in Quadrant II, $\sin5\pi/6=\sin\pi/6=1/2$.

Sol 11 Since $2\pi=6\pi/3$, subtracting off a multiple of $2\pi$ gives $35\pi/3-5(2\pi)=35\pi/3-30\pi/3=5\pi/3$; so $\cos35\pi/3=\cos5\pi/3$. Since the reference angle for $5\pi/3$ is $2\pi-5\pi/3=\pi/3$, and since the cosine is positive in Quadrant IV, $\cos 5\pi/3=\cos \pi/3= 1/2$.

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