Find Trig Functions Using Identities csc(x)=5/3 , tan(x)=3/4

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Trigonometry

$\csc (x) = \frac{5}{3}$ , $\tan (x) = \frac{3}{4}$

To find the value of $\sin (x)$ , use the fact that $\frac{1}{\csc (x)}$ then substitute in the known values.

$\sin (x) = \frac{1}{\csc (x)} = \frac{1}{\frac{5}{3}}$

Simplify the result.

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Multiply the numerator by the reciprocal of the denominator.

$\sin (x) = \frac{1}{\csc (x)} = 1 (\frac{3}{5})$

Multiply $\frac{3}{5}$ by $1$ .

$\sin (x) = \frac{1}{\csc (x)} = \frac{3}{5}$

To find the value of $\cos (x)$ , use the fact that $\tan (x) = \frac{\sin (x)}{\cos (x)}$ so $\cos (x) = \frac{\sin (x)}{\tan (x)}$ then substitute in the known values.

$\cos (x) = \frac{\sin (x)}{\tan (x)} = \frac{\frac{3}{5}}{\frac{3}{4}}$

Simplify the result.

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Multiply the numerator by the reciprocal of the denominator.

$\cos (x) = \frac{\sin (x)}{\tan (x)} = \frac{3}{5} \cdot \frac{4}{3}$

Cancel the common factor of $3$ .

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Cancel the common factor.

$\cos (x) = \frac{\sin (x)}{\tan (x)} = \frac{3}{5} \cdot \frac{4}{3}$

Rewrite the expression.

$\cos (x) = \frac{\sin (x)}{\tan (x)} = \frac{1}{5} \cdot 4$

Combine $\frac{1}{5}$ and $4$ .

$\cos (x) = \frac{\sin (x)}{\tan (x)} = \frac{4}{5}$

To find the value of $\cot (x)$ , use the fact that $\frac{1}{\tan (x)}$ then substitute in the known values.

$\cot (x) = \frac{1}{\tan (x)} = \frac{1}{\frac{3}{4}}$

Simplify the result.

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Multiply the numerator by the reciprocal of the denominator.

$\cot (x) = \frac{1}{\tan (x)} = 1 (\frac{4}{3})$

Multiply $\frac{4}{3}$ by $1$ .

$\cot (x) = \frac{1}{\tan (x)} = \frac{4}{3}$

To find the value of $\sec (x)$ , use the fact that $\frac{1}{\cos (x)}$ then substitute in the known values.

$\sec (x) = \frac{1}{\cos (x)} = \frac{1}{\frac{4}{5}}$

Simplify the result.

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Multiply the numerator by the reciprocal of the denominator.

$\sec (x) = \frac{1}{\cos (x)} = 1 (\frac{5}{4})$

Multiply $\frac{5}{4}$ by $1$ .

$\sec (x) = \frac{1}{\cos (x)} = \frac{5}{4}$

The trig functions found are as follows:

$\sin (x) = \frac{3}{5}$

$\cos (x) = \frac{4}{5}$

$\tan (x) = \frac{3}{4}$

$\cot (x) = \frac{4}{3}$

$\sec (x) = \frac{5}{4}$

$\csc (x) = \frac{5}{3}$

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