Trigonometry Examples

$\sin (θ) = \frac{3}{5}$ , $\cos (θ) = - \frac{4}{5}$

Step 1

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

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

Step 2

Simplify the result.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.2

Cancel the common factor of $5$ .

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Step 2.2.1

Move the leading negative in $- \frac{5}{4}$ into the numerator.

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

Step 2.2.2

Factor $5$ out of $- 5$ .

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

Step 2.2.3

Cancel the common factor.

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

Step 2.2.4

Rewrite the expression.

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = 3 (\frac{- 1}{4})$

Step 2.3

Combine $3$ and $\frac{- 1}{4}$ .

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{3 \cdot - 1}{4}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Multiply $3$ by $- 1$ .

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{- 3}{4}$

Step 2.4.2

Move the negative in front of the fraction.

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = - \frac{3}{4}$

Step 3

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

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

Step 4

Simplify the result.

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Step 4.1

Cancel the common factor of $1$ and $- 1$ .

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Step 4.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{- 1 \cdot - 1}{- \frac{3}{4}}$

Step 4.1.2

Move the negative in front of the fraction.

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

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3

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

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

Step 5

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

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

Step 6

Simplify the result.

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Step 6.1

Cancel the common factor of $1$ and $- 1$ .

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Step 6.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{- 1 \cdot - 1}{- \frac{4}{5}}$

Step 6.1.2

Move the negative in front of the fraction.

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

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.3

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

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

Step 7

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

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

Step 8

Simplify the result.

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Step 8.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.2

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

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

Step 9

The trig functions found are as follows:

$\sin (θ) = \frac{3}{5}$

$\cos (θ) = - \frac{4}{5}$

$\tan (θ) = - \frac{3}{4}$

$\cot (θ) = - \frac{4}{3}$

$\sec (θ) = - \frac{5}{4}$

$\csc (θ) = \frac{5}{3}$