Calculus Examples

$lim_{θ \to π} \frac{\sin (θ)}{\tan (θ)}$

Step 1

Apply trigonometric identities.

Tap for more steps...

Step 1.1

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

$lim_{θ \to π} \frac{\sin (θ)}{\frac{\sin (θ)}{\cos (θ)}}$

Step 1.2

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (θ)}{\cos (θ)}$ .

$lim_{θ \to π} \sin (θ) \frac{\cos (θ)}{\sin (θ)}$

Step 1.3

Write $\sin (θ)$ as a fraction with denominator $1$ .

$lim_{θ \to π} \frac{\sin (θ)}{1} \cdot \frac{\cos (θ)}{\sin (θ)}$

Step 1.4

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

Tap for more steps...

Step 1.4.1

Cancel the common factor.

$lim_{θ \to π} \frac{\sin (θ)}{1} \cdot \frac{\cos (θ)}{\sin (θ)}$

Step 1.4.2

Rewrite the expression.

$lim_{θ \to π} \cos (θ)$

Step 2

Move the limit inside the trig function because cosine is continuous.

$\cos (lim_{θ \to π} θ)$

Step 3

Evaluate the limit of $θ$ by plugging in $π$ for $θ$ .

$\cos (π)$

Step 4

Simplify the answer.

Tap for more steps...

Step 4.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$- \cos (0)$

Step 4.2

The exact value of $\cos (0)$ is $1$ .

$- 1 \cdot 1$

Step 4.3

Multiply $- 1$ by $1$ .

$- 1$