Trig Functions of Angles

a) Recall that the sine function is positive in Quadrants I and II, the cosine function is positive in Quadrants I and IV, and the tangent function is positive in Quadrants I and III.

b) In addition, the reference angle for $\theta$ is

1. $\theta$, if $0<\theta<\pi/2$.

2. $\pi-\theta$, if $\pi/2<\theta<\pi$.

3. $\theta-\pi$, if $\pi<\theta<3\pi/2$.

4. $2\pi-\theta$, if $3\pi/2<\theta<2\pi$.

c) If $\theta$ is an angle in standard position, and $(x,y)$ is a point on the terminal side of $\theta$ (other than the origin), then

$\cos\theta=\frac{x}{r}$ and $\sin\theta=\frac{y}{r}$

where $r=\sqrt{x^2+y^2}$

Ex 1 Find $\sin 11\pi/6$.

Sol The reference angle for $11\pi/6$ is $2\pi-11\pi/6=\pi/6$. Since the sine is negative in Quadrant IV, $\sin 11\pi/6=-\sin\pi/6=-1/2$.

Ex 2 Find $\cos 4\pi/3$.

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

Pr 1 Find $\sin 2\pi/3$.

Pr 2 Find $\cos 7\pi/6$.

Pr 3 Find $\tan 3\pi/4$.

Pr 4 Find $\cos 5\pi/3$.

Pr 5 Find $\sec 5\pi/4$.

Pr 6 Find $\cot 5\pi/6$.

Pr 7 Find $\tan 4\pi/3$.

Pr 8 Find $\csc 7\pi/4$.

Pr 9 If $\theta$ is an angle in standard position with the point $(6,-8)$ on its terminal side, find $\cos\theta$ and $\sin\theta$.

Pr 10 Find $\sin 29\pi/6$.

Pr 11 Find $\cos 35\pi/3$.

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