Trigonometry Examples

$\frac{1 + \sin (θ)}{\tan (θ)} = \cos (θ) + \cot (θ)$

Step 1

Simplify $\frac{1 + \sin (θ)}{\tan (θ)}$ .

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Step 1.1

Rewrite $\tan (θ)$ in terms of sines and cosines.

$\frac{1 + \sin (θ)}{\frac{\sin (θ)}{\cos (θ)}} = \cos (θ) + \cot (θ)$

Step 1.2

Multiply the numerator by the reciprocal of the denominator.

$(1 + \sin (θ)) \frac{\cos (θ)}{\sin (θ)} = \cos (θ) + \cot (θ)$

Step 1.3

Apply the distributive property.

$1 \frac{\cos (θ)}{\sin (θ)} + \sin (θ) \frac{\cos (θ)}{\sin (θ)} = \cos (θ) + \cot (θ)$

Step 1.4

Multiply $\frac{\cos (θ)}{\sin (θ)}$ by $1$ .

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

Step 1.5

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

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Step 1.5.1

Cancel the common factor.

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

Step 1.5.2

Rewrite the expression.

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

Step 1.6

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

$\cot (θ) + \cos (θ) = \cos (θ) + \cot (θ)$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$θ \approx - 0.1, 0.1, - \frac{1}{5}, \frac{1}{5}, 0.3, \frac{2}{5}, - \frac{1}{2}$

Step 3