Trigonometry Examples

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

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

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

Simplify the result.

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Cancel the common factor of $3$ .

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Factor out the greatest common factor $3$ .

$\sin (x) = \tan (x) \cdot \cos (x) = \frac{4}{3 \cdot 1} \cdot \frac{3 \cdot 1}{5}$

Cancel the common factor.

$\sin (x) = \tan (x) \cdot \cos (x) = \frac{4}{3 \cdot 1} \cdot \frac{3 \cdot 1}{5}$

Rewrite the expression.

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

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

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

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

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

Simplify the result.

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Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

$\sec (x) = \frac{1}{\cos (x)} = \frac{1}{\frac{3}{5}}$

Simplify the result.

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Multiply the numerator by the reciprocal of the denominator.

$\sec (x) = \frac{1}{\cos (x)} = 1 (\frac{5}{3})$

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

$\sec (x) = \frac{1}{\cos (x)} = \frac{5}{3}$

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

$\csc (x) = \frac{1}{\sin (x)} = \frac{1}{\frac{4}{5}}$

Simplify the result.

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Multiply the numerator by the reciprocal of the denominator.

$\csc (x) = \frac{1}{\sin (x)} = 1 (\frac{5}{4})$

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

$\csc (x) = \frac{1}{\sin (x)} = \frac{5}{4}$

The trig functions found are as follows:

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

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

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

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

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

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

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