Find the Exact Value sin(15 degrees )cos(45 degrees )+cos(15 degrees )sin(45 degrees )

$\sin (15 °) \cos (45 °) + \cos (15 °) \sin (45 °)$

Step 1

Simplify each term.

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.

$\sin (45 - 30) \cos (45 °) + \cos (15 °) \sin (45 °)$

Separate negation.

$\sin (45 - (30)) \cos (45 °) + \cos (15 °) \sin (45 °)$

Apply the difference of angles identity.

$(\sin (45) \cos (30) - \cos (45) \sin (30)) \cos (45 °) + \cos (15 °) \sin (45 °)$

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

$(\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30)) \cos (45 °) + \cos (15 °) \sin (45 °)$

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

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

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

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

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

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

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

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

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

$(\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (45 °) + \cos (15 °) \sin (45 °)$

Multiply $2$ by $2$ .

$(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \cos (45 °) + \cos (15 °) \sin (45 °)$

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

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

$(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}) \cos (45 °) + \cos (15 °) \sin (45 °)$

Multiply $2$ by $2$ .

$(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}) \cos (45 °) + \cos (15 °) \sin (45 °)$

Combine the numerators over the common denominator.

$\frac{\sqrt{6} - \sqrt{2}}{4} \cos (45 °) + \cos (15 °) \sin (45 °)$

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

$\frac{\sqrt{6} - \sqrt{2}}{4} \cdot \frac{\sqrt{2}}{2} + \cos (15 °) \sin (45 °)$

Multiply $\frac{\sqrt{6} - \sqrt{2}}{4} \cdot \frac{\sqrt{2}}{2}$ .

Multiply $\frac{\sqrt{6} - \sqrt{2}}{4}$ by $\frac{\sqrt{2}}{2}$ .

$\frac{(\sqrt{6} - \sqrt{2}) \sqrt{2}}{4 \cdot 2} + \cos (15 °) \sin (45 °)$

Multiply $4$ by $2$ .

$\frac{(\sqrt{6} - \sqrt{2}) \sqrt{2}}{8} + \cos (15 °) \sin (45 °)$

Apply the distributive property.

$\frac{\sqrt{6} \sqrt{2} - \sqrt{2} \sqrt{2}}{8} + \cos (15 °) \sin (45 °)$

Combine using the product rule for radicals.

$\frac{\sqrt{6 \cdot 2} - \sqrt{2} \sqrt{2}}{8} + \cos (15 °) \sin (45 °)$

Multiply $- \sqrt{2} \sqrt{2}$ .

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{6 \cdot 2} - ({\sqrt{2}}^{1} \sqrt{2})}{8} + \cos (15 °) \sin (45 °)$

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{6 \cdot 2} - ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{8} + \cos (15 °) \sin (45 °)$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt{6 \cdot 2} - {\sqrt{2}}^{1 + 1}}{8} + \cos (15 °) \sin (45 °)$

Add $1$ and $1$ .

$\frac{\sqrt{6 \cdot 2} - {\sqrt{2}}^{2}}{8} + \cos (15 °) \sin (45 °)$

Simplify each term.

Multiply $6$ by $2$ .

$\frac{\sqrt{12} - {\sqrt{2}}^{2}}{8} + \cos (15 °) \sin (45 °)$

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$\frac{\sqrt{4 (3)} - {\sqrt{2}}^{2}}{8} + \cos (15 °) \sin (45 °)$

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2} \cdot 3} - {\sqrt{2}}^{2}}{8} + \cos (15 °) \sin (45 °)$

Pull terms out from under the radical.

$\frac{2 \sqrt{3} - {\sqrt{2}}^{2}}{8} + \cos (15 °) \sin (45 °)$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{2 \sqrt{3} - {(2^{\frac{1}{2}})}^{2}}{8} + \cos (15 °) \sin (45 °)$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 \sqrt{3} - 2^{\frac{1}{2} \cdot 2}}{8} + \cos (15 °) \sin (45 °)$

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

$\frac{2 \sqrt{3} - 2^{\frac{2}{2}}}{8} + \cos (15 °) \sin (45 °)$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 \sqrt{3} - 2^{\frac{2}{2}}}{8} + \cos (15 °) \sin (45 °)$

Rewrite the expression.

$\frac{2 \sqrt{3} - 2^{1}}{8} + \cos (15 °) \sin (45 °)$

Evaluate the exponent.

$\frac{2 \sqrt{3} - 1 \cdot 2}{8} + \cos (15 °) \sin (45 °)$

Multiply $- 1$ by $2$ .

$\frac{2 \sqrt{3} - 2}{8} + \cos (15 °) \sin (45 °)$

Cancel the common factor of $2 \sqrt{3} - 2$ and $8$ .

Factor $2$ out of $2 \sqrt{3}$ .

$\frac{2 (\sqrt{3}) - 2}{8} + \cos (15 °) \sin (45 °)$

Factor $2$ out of $- 2$ .

$\frac{2 (\sqrt{3}) + 2 (- 1)}{8} + \cos (15 °) \sin (45 °)$

Factor $2$ out of $2 (\sqrt{3}) + 2 (- 1)$ .

$\frac{2 (\sqrt{3} - 1)}{8} + \cos (15 °) \sin (45 °)$

Cancel the common factors.

Factor $2$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

The exact value of $\cos (15 °)$ is $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

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

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

Separate negation.

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

Apply the difference of angles identity $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

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

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

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

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

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

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

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

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

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

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

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

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

$\frac{\sqrt{3} - 1}{4} + (\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \sin (45 °)$

Multiply $2$ by $2$ .

$\frac{\sqrt{3} - 1}{4} + (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) \sin (45 °)$

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

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

$\frac{\sqrt{3} - 1}{4} + (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2}) \sin (45 °)$

Multiply $2$ by $2$ .

$\frac{\sqrt{3} - 1}{4} + (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}) \sin (45 °)$

Combine the numerators over the common denominator.

$\frac{\sqrt{3} - 1}{4} + \frac{\sqrt{6} + \sqrt{2}}{4} \sin (45 °)$

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

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

Multiply $\frac{\sqrt{6} + \sqrt{2}}{4} \cdot \frac{\sqrt{2}}{2}$ .

Multiply $\frac{\sqrt{6} + \sqrt{2}}{4}$ by $\frac{\sqrt{2}}{2}$ .

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

Multiply $4$ by $2$ .

$\frac{\sqrt{3} - 1}{4} + \frac{(\sqrt{6} + \sqrt{2}) \sqrt{2}}{8}$

Apply the distributive property.

$\frac{\sqrt{3} - 1}{4} + \frac{\sqrt{6} \sqrt{2} + \sqrt{2} \sqrt{2}}{8}$

Combine using the product rule for radicals.

$\frac{\sqrt{3} - 1}{4} + \frac{\sqrt{6 \cdot 2} + \sqrt{2} \sqrt{2}}{8}$

Combine using the product rule for radicals.

$\frac{\sqrt{3} - 1}{4} + \frac{\sqrt{6 \cdot 2} + \sqrt{2 \cdot 2}}{8}$

Simplify each term.

Multiply $6$ by $2$ .

$\frac{\sqrt{3} - 1}{4} + \frac{\sqrt{12} + \sqrt{2 \cdot 2}}{8}$

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$\frac{\sqrt{3} - 1}{4} + \frac{\sqrt{4 (3)} + \sqrt{2 \cdot 2}}{8}$

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{3} - 1}{4} + \frac{\sqrt{2^{2} \cdot 3} + \sqrt{2 \cdot 2}}{8}$

Pull terms out from under the radical.

$\frac{\sqrt{3} - 1}{4} + \frac{2 \sqrt{3} + \sqrt{2 \cdot 2}}{8}$

Multiply $2$ by $2$ .

$\frac{\sqrt{3} - 1}{4} + \frac{2 \sqrt{3} + \sqrt{4}}{8}$

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{3} - 1}{4} + \frac{2 \sqrt{3} + \sqrt{2^{2}}}{8}$

Pull terms out from under the radical, assuming positive real numbers.

$\frac{\sqrt{3} - 1}{4} + \frac{2 \sqrt{3} + 2}{8}$

Cancel the common factor of $2 \sqrt{3} + 2$ and $8$ .

Factor $2$ out of $2 \sqrt{3}$ .

$\frac{\sqrt{3} - 1}{4} + \frac{2 (\sqrt{3}) + 2}{8}$

Factor $2$ out of $2$ .

$\frac{\sqrt{3} - 1}{4} + \frac{2 (\sqrt{3}) + 2 (1)}{8}$

Factor $2$ out of $2 (\sqrt{3}) + 2 (1)$ .

$\frac{\sqrt{3} - 1}{4} + \frac{2 (\sqrt{3} + 1)}{8}$

Cancel the common factors.

Factor $2$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{\sqrt{3} - 1}{4} + \frac{\sqrt{3} + 1}{4}$

Step 2

Simplify terms.

Combine the numerators over the common denominator.

$\frac{\sqrt{3} - 1 + \sqrt{3} + 1}{4}$

Add $\sqrt{3}$ and $\sqrt{3}$ .

$\frac{- 1 + 2 \sqrt{3} + 1}{4}$

Simplify by adding numbers.

Add $- 1$ and $1$ .

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

Add $0$ and $2 \sqrt{3}$ .

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

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2 \sqrt{3}$ .

$\frac{2 (\sqrt{3})}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.86602540 \dots$

Find the Exact Value sin(15 degrees )cos(45 degrees )+cos(15 degrees )sin(45 degrees )

Related Questions

Support LCMGCF.com by disabling your adblocker.