Trigonometry Examples

$24 \cos (135) i + 24 \sin (135) j$

Step 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.

$24 (- \cos (45)) i + 24 \sin (135) j$

Step 2

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$24 (- \frac{\sqrt{2}}{2}) i + 24 \sin (135) j$

Step 3

Cancel the common factor of $2$ .

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Step 3.1

Move the leading negative in $- \frac{\sqrt{2}}{2}$ into the numerator.

$24 \frac{- \sqrt{2}}{2} i + 24 \sin (135) j$

Step 3.2

Factor $2$ out of $24$ .

$2 (12) \frac{- \sqrt{2}}{2} i + 24 \sin (135) j$

Step 3.3

Cancel the common factor.

$2 \cdot 12 \frac{- \sqrt{2}}{2} i + 24 \sin (135) j$

Step 3.4

Rewrite the expression.

$12 (- \sqrt{2}) i + 24 \sin (135) j$

Step 4

Multiply $- 1$ by $12$ .

$- 12 \sqrt{2} i + 24 \sin (135) j$

Step 5

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$- 12 \sqrt{2} i + 24 \sin (45) j$

Step 6

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$- 12 \sqrt{2} i + 24 \frac{\sqrt{2}}{2} j$

Step 7

Cancel the common factor of $2$ .

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Step 7.1

Factor $2$ out of $24$ .

$- 12 \sqrt{2} i + 2 (12) \frac{\sqrt{2}}{2} j$

Step 7.2

Cancel the common factor.

$- 12 \sqrt{2} i + 2 \cdot 12 \frac{\sqrt{2}}{2} j$

Step 7.3

Rewrite the expression.

$- 12 \sqrt{2} i + 12 \sqrt{2} j$