Basic Trigonometric Identities

1.

$$\tan\theta=\frac{\sin\theta}{\cos\theta}$$

2.

$$\cot\theta=\frac{\cos\theta}{\sin\theta}=\frac{1}{\tan\theta}$$

3.

$$\sec\theta=\frac{1}{\cos\theta}$$

4.

$$\csc\theta=\frac{1}{\sin\theta}$$

Pythagorean Identities

1.

$$\sin^2\theta+\cos^2\theta=1$$

2.

$$\tan^2\theta+1=\sec^2\theta$$

3.

$$\cot^2\theta+1=\csc^2\theta$$

Ex 1 Find $\tan\theta$ if $\sin\theta=24/25$ and $\theta$ is in Quadrant II.

Sol $\cos^2\theta=1-\sin^2\theta=1-(24/25)^2=1-576/625=49/625$, so $\cos\theta=-7/25$ since $\cos\theta<0$ if $\theta$ is in Quadrant II. Then

$$\tan\theta=\frac{\sin\theta}{\cos\theta}=(24/25)/(-7/25)=-24/7$$

.

Ex 2 Find $\csc\theta$ if $\tan\theta=5/12$ and $\theta$ is in Quadrant III.

Sol Since $\tan\theta=5/12$, $\cot\theta=\frac{1}{\tan\theta}=12/5$ and $\csc^2\theta=\cot^2\theta+1=(12/5)^2+1=144/25+1=169/25$ and therefore $\csc\theta=-13/5$ since $\csc\theta<0$ for $\theta$ in Quadrant III.

Pr A Find $\sin\theta$ if $\cos\theta=12/13$ and $3\pi/2<\theta<2\pi$.

Pr B Find $\sec\theta$ if $\tan\theta=3/4$ and $\pi<\theta<3\pi/2$.

Pr C Find $\cot\theta$ if $\sec\theta=25/7$ and $3\pi/2<\theta<2\pi$.

Pr 1 Find $\cos\theta$ if $\tan\theta=-3$ and $3\pi/2<\theta<2\pi$.

Pr 2 Find $\sin\theta$ if $\tan\theta=4/3$ and $\pi<\theta<3\pi/2$.

Pr 3 Find $\tan\theta$ if $\cos\theta=-3/5$ and $\pi<\theta<3\pi/2$.

Pr 4 Simplify the expression $(\sin\theta+\cos\theta)^2$.

Pr 5 Simplify the expression $(\sec 4\theta-1)(\sec 4\theta+1)$.

Pr 6 Simplify the expression $(\csc\theta-\cot\theta)(\csc\theta+\cot\theta)$.

Pr 7 Simplify the expression $\sqrt{9-x^2}$ for $x=3\sin\theta$ with $-\pi/2\le\theta\le\pi/2$.

Pr 8 Simplify the expression $\sqrt{x^2+4}$ for $x=2\tan\theta$ with $-\pi/2<\theta<\pi/2$.

Pr 9 Use trig identities to change $\frac{\cos^2\theta}{\sin\theta}$ to $\csc\theta-\sin\theta$.

Pr 10 Use trig identities to change $\sin^4\theta\cos^3\theta$ to $(\sin^4\theta-\sin^6\theta)\cos\theta$.

Pr 11 Use trig identities to change $\sin^5\theta$ to $(1-2\cos^2\theta+\cos^4\theta)\sin\theta$.

Pr 12 Use trig identities to change $\sec^6\theta$ to $(1+2\tan^2\theta+\tan^4\theta)\sec^2\theta$.

Pr 13 Use trig identities to change $\tan\theta+\cot\theta$ to $\sec\theta\csc\theta$.

Pr 14 Use trig identities to change $\frac{\sec^3\theta}{\tan\theta}$ to $\sec\theta\tan\theta+\csc\theta$.

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