Find the Exact Value 4/3(sin(255)+sin(15))

$\frac{4}{3} (\sin (255) + \sin (15))$

Simplify each term.

The exact value of $\sin (255)$ is $- \frac{\sqrt{2} + \sqrt{6}}{4}$ .

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 third quadrant.

$\frac{4}{3} \cdot (- \sin (75) + \sin (15))$

Split $75$ into two angles where the values of the six trigonometric functions are known.

$\frac{4}{3} \cdot (- \sin (30 + 45) + \sin (15))$

Apply the sum of angles identity.

$\frac{4}{3} \cdot (- (\sin (30) \cos (45) + \cos (30) \sin (45)) + \sin (15))$

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$\frac{4}{3} \cdot (- (\frac{1}{2} \cos (45) + \cos (30) \sin (45)) + \sin (15))$

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

$\frac{4}{3} \cdot (- (\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (30) \sin (45)) + \sin (15))$

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$\frac{4}{3} \cdot (- (\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \sin (45)) + \sin (15))$

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

$\frac{4}{3} \cdot (- (\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) + \sin (15))$

Simplify $- (\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2})$ .

Simplify each term.

Multiply $\frac{1}{2} \cdot \frac{\sqrt{2}}{2}$ .

Multiply $\frac{1}{2}$ by $\frac{\sqrt{2}}{2}$ .

$\frac{4}{3} \cdot (- (\frac{\sqrt{2}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) + \sin (15))$

Multiply $2$ by $2$ .

$\frac{4}{3} \cdot (- (\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) + \sin (15))$

Multiply $\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}$ .

Multiply $\frac{\sqrt{3}}{2}$ by $\frac{\sqrt{2}}{2}$ .

$\frac{4}{3} \cdot (- (\frac{\sqrt{2}}{4} + \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}) + \sin (15))$

Combine using the product rule for radicals.

$\frac{4}{3} \cdot (- (\frac{\sqrt{2}}{4} + \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}) + \sin (15))$

Multiply $3$ by $2$ .

$\frac{4}{3} \cdot (- (\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{2 \cdot 2}) + \sin (15))$

Multiply $2$ by $2$ .

$\frac{4}{3} \cdot (- (\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}) + \sin (15))$

Combine the numerators over the common denominator.

$\frac{4}{3} \cdot (- \frac{\sqrt{2} + \sqrt{6}}{4} + \sin (15))$

The exact value of $\sin (15)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

Split $15$ into two angles where the values of the six trigonometric functions are known.

$\frac{4}{3} \cdot (- \frac{\sqrt{2} + \sqrt{6}}{4} + \sin (45 - 30))$

Separate negation.

$\frac{4}{3} \cdot (- \frac{\sqrt{2} + \sqrt{6}}{4} + \sin (45 - (30)))$

Apply the difference of angles identity.

$\frac{4}{3} \cdot (- \frac{\sqrt{2} + \sqrt{6}}{4} + \sin (45) \cos (30) - \cos (45) \sin (30))$

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

$\frac{4}{3} \cdot (- \frac{\sqrt{2} + \sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30))$

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$\frac{4}{3} \cdot (- \frac{\sqrt{2} + \sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30))$

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

$\frac{4}{3} \cdot (- \frac{\sqrt{2} + \sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30))$

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$\frac{4}{3} \cdot (- \frac{\sqrt{2} + \sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

Simplify each term.

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$ .

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{3}}{2}$ .

$\frac{4}{3} \cdot (- \frac{\sqrt{2} + \sqrt{6}}{4} + \frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Combine using the product rule for radicals.

$\frac{4}{3} \cdot (- \frac{\sqrt{2} + \sqrt{6}}{4} + \frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Multiply $2$ by $3$ .

$\frac{4}{3} \cdot (- \frac{\sqrt{2} + \sqrt{6}}{4} + \frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Multiply $2$ by $2$ .

$\frac{4}{3} \cdot (- \frac{\sqrt{2} + \sqrt{6}}{4} + \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Multiply $- \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

Multiply $\frac{1}{2}$ by $\frac{\sqrt{2}}{2}$ .

$\frac{4}{3} \cdot (- \frac{\sqrt{2} + \sqrt{6}}{4} + \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2})$

Multiply $2$ by $2$ .

$\frac{4}{3} \cdot (- \frac{\sqrt{2} + \sqrt{6}}{4} + \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4})$

Combine the numerators over the common denominator.

$\frac{4}{3} \cdot (- \frac{\sqrt{2} + \sqrt{6}}{4} + \frac{\sqrt{6} - \sqrt{2}}{4})$

Simplify terms.

Combine the numerators over the common denominator.

$\frac{4}{3} \cdot \frac{- (\sqrt{2} + \sqrt{6}) + \sqrt{6} - \sqrt{2}}{4}$

Apply the distributive property.

$\frac{4}{3} \cdot \frac{- \sqrt{2} - \sqrt{6} + \sqrt{6} - \sqrt{2}}{4}$

Subtract $\sqrt{2}$ from $- \sqrt{2}$ .

$\frac{4}{3} \cdot \frac{- 2 \sqrt{2} - \sqrt{6} + \sqrt{6}}{4}$

Add $- \sqrt{6}$ and $\sqrt{6}$ .

$\frac{4}{3} \cdot \frac{- 2 \sqrt{2} + 0}{4}$

Add $- 2 \sqrt{2}$ and $0$ .

$\frac{4}{3} \cdot \frac{- 2 \sqrt{2}}{4}$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4}{3} \cdot \frac{- 2 \sqrt{2}}{4}$

Rewrite the expression.

$\frac{1}{3} \cdot (- 2 \sqrt{2})$

Combine $- 2$ and $\frac{1}{3}$ .

$\frac{- 2}{3} \cdot \sqrt{2}$

Combine $\frac{- 2}{3}$ and $\sqrt{2}$ .

$\frac{- 2 \sqrt{2}}{3}$

Move the negative in front of the fraction.

$- \frac{2 \sqrt{2}}{3}$

The result can be shown in multiple forms.

Exact Form:

$- \frac{2 \sqrt{2}}{3}$

Decimal Form:

$- 0.94280904 \dots$

Find the Exact Value 4/3(sin(255)+sin(15))

Related Questions

Support LCMGCF.com by disabling your adblocker.