Calculus Examples

$lim_{t \to \frac{3 π}{2}} \sin (\frac{π}{2} \cdot \sin (t))$

Step 1

Evaluate the limit.

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Step 1.1

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

$\sin (lim_{t \to \frac{3 π}{2}} \frac{π}{2} \cdot \sin (t))$

Step 1.2

Move the term $\frac{π}{2}$ outside of the limit because it is constant with respect to $t$ .

$\sin (\frac{π}{2} lim_{t \to \frac{3 π}{2}} \sin (t))$

Step 1.3

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

$\sin (\frac{π}{2} \sin (lim_{t \to \frac{3 π}{2}} t))$

Step 2

Evaluate the limit of $t$ by plugging in $\frac{3 π}{2}$ for $t$ .

$\sin (\frac{π}{2} \sin (\frac{3 π}{2}))$

Step 3

Simplify the answer.

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Step 3.1

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

$\sin (\frac{π}{2} (- \sin (\frac{π}{2})))$

Step 3.2

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$\sin (\frac{π}{2} (- 1 \cdot 1))$

Step 3.3

Multiply $- 1$ by $1$ .

$\sin (\frac{π}{2} \cdot - 1)$

Step 3.4

Combine $\frac{π}{2}$ and $- 1$ .

$\sin (\frac{π \cdot - 1}{2})$

Step 3.5

Move $- 1$ to the left of $π$ .

$\sin (\frac{- 1 \cdot π}{2})$

Step 3.6

Move the negative in front of the fraction.

$\sin (- \frac{π}{2})$

Step 3.7

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\sin (\frac{3 π}{2})$

Step 3.8

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

$- \sin (\frac{π}{2})$

Step 3.9

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$- 1 \cdot 1$

Step 3.10

Multiply $- 1$ by $1$ .

$- 1$