Trigonometry Examples

$\cos (θ) = \sin (θ)$

Step 1

Divide each term in the equation by $\cos (θ)$ .

$\frac{\cos (θ)}{\cos (θ)} = \frac{\sin (θ)}{\cos (θ)}$

Step 2

Cancel the common factor of $\cos (θ)$ .

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Step 2.1

Cancel the common factor.

$\frac{\cos (θ)}{\cos (θ)} = \frac{\sin (θ)}{\cos (θ)}$

Step 2.2

Rewrite the expression.

$1 = \frac{\sin (θ)}{\cos (θ)}$

Step 3

Convert from $\frac{\sin (θ)}{\cos (θ)}$ to $\tan (θ)$ .

$1 = \tan (θ)$

Step 4

Rewrite the equation as $\tan (θ) = 1$ .

$\tan (θ) = 1$

Step 5

Take the inverse tangent of both sides of the equation to extract $θ$ from inside the tangent.

$θ = \arctan (1)$

Step 6

Simplify the right side.

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Step 6.1

The exact value of $\arctan (1)$ is $45$ .

$θ = 45$

Step 7

The tangent function is positive in the first and third quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the fourth quadrant.

$θ = 180 + 45$

Step 8

Add $180$ and $45$ .

$θ = 225$

Step 9

Find the period of $\tan (θ)$ .

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Step 9.1

The period of the function can be calculated using $\frac{180}{| b |}$ .

$\frac{180}{| b |}$

Step 9.2

Replace $b$ with $1$ in the formula for period.

$\frac{180}{| 1 |}$

Step 9.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{180}{1}$

Step 9.4

Divide $180$ by $1$ .

$180$

Step 10

The period of the $\tan (θ)$ function is $180$ so values will repeat every $180$ degrees in both directions.

$θ = 45 + 180 n, 225 + 180 n$ , for any integer $n$

Step 11

Consolidate the answers.

$θ = 45 + 180 n$ , for any integer $n$