Trigonometry Examples Radian Measure and Circular Functions

Find Trig Functions Using Identities

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

Step 1

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.

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

Step 2

Simplify the result.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.2

Cancel the common factor of $4$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

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

Step 2.3

Combine $\frac{1}{5}$ and $3$ .

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

Step 3

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

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

Step 4

Simplify the result.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2

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

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

Step 5

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

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

Step 6

Simplify the result.

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Step 6.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.2

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

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

Step 7

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

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

Step 8

Simplify the result.

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Step 8.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.2

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

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

Step 9

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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