Trigonometry Examples

$\sin (\frac{7 π}{12}) \cos (\frac{π}{4}) - \cos (\frac{7 π}{12}) \sin (\frac{π}{4})$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

The exact value of $\sin (\frac{7 π}{12})$ is $\frac{\sqrt{2 + \sqrt{3}}}{2}$ .

Tap for more steps...

Step 1.1.1

Rewrite $\frac{7 π}{12}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\sin (\frac{\frac{7 π}{6}}{2}) \cos (\frac{π}{4}) - \cos (\frac{7 π}{12}) \sin (\frac{π}{4})$

Step 1.1.2

Apply the sine half-angle identity.

$(\pm \sqrt{\frac{1 - \cos (\frac{7 π}{6})}{2}}) \cos (\frac{π}{4}) - \cos (\frac{7 π}{12}) \sin (\frac{π}{4})$

Step 1.1.3

Change the $\pm$ to $+$ because sine is positive in the second quadrant.

$\sqrt{\frac{1 - \cos (\frac{7 π}{6})}{2}} \cos (\frac{π}{4}) - \cos (\frac{7 π}{12}) \sin (\frac{π}{4})$

Step 1.1.4

Simplify $\sqrt{\frac{1 - \cos (\frac{7 π}{6})}{2}}$ .

Tap for more steps...

Step 1.1.4.1

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 third quadrant.

$\sqrt{\frac{1 - - \cos (\frac{π}{6})}{2}} \cos (\frac{π}{4}) - \cos (\frac{7 π}{12}) \sin (\frac{π}{4})$

Step 1.1.4.2

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$\sqrt{\frac{1 - - \frac{\sqrt{3}}{2}}{2}} \cos (\frac{π}{4}) - \cos (\frac{7 π}{12}) \sin (\frac{π}{4})$

Step 1.1.4.3

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

Tap for more steps...

Step 1.1.4.3.1

Multiply $- 1$ by $- 1$ .

$\sqrt{\frac{1 + 1 \frac{\sqrt{3}}{2}}{2}} \cos (\frac{π}{4}) - \cos (\frac{7 π}{12}) \sin (\frac{π}{4})$

Step 1.1.4.3.2

Multiply $\frac{\sqrt{3}}{2}$ by $1$ .

$\sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} \cos (\frac{π}{4}) - \cos (\frac{7 π}{12}) \sin (\frac{π}{4})$

Step 1.1.4.4

Write $1$ as a fraction with a common denominator.

$\sqrt{\frac{\frac{2}{2} + \frac{\sqrt{3}}{2}}{2}} \cos (\frac{π}{4}) - \cos (\frac{7 π}{12}) \sin (\frac{π}{4})$

Step 1.1.4.5

Combine the numerators over the common denominator.

$\sqrt{\frac{\frac{2 + \sqrt{3}}{2}}{2}} \cos (\frac{π}{4}) - \cos (\frac{7 π}{12}) \sin (\frac{π}{4})$

Step 1.1.4.6

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{\frac{2 + \sqrt{3}}{2} \cdot \frac{1}{2}} \cos (\frac{π}{4}) - \cos (\frac{7 π}{12}) \sin (\frac{π}{4})$

Step 1.1.4.7

Multiply $\frac{2 + \sqrt{3}}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 1.1.4.7.1

Multiply $\frac{2 + \sqrt{3}}{2}$ by $\frac{1}{2}$ .

$\sqrt{\frac{2 + \sqrt{3}}{2 \cdot 2}} \cos (\frac{π}{4}) - \cos (\frac{7 π}{12}) \sin (\frac{π}{4})$

Step 1.1.4.7.2

Multiply $2$ by $2$ .

$\sqrt{\frac{2 + \sqrt{3}}{4}} \cos (\frac{π}{4}) - \cos (\frac{7 π}{12}) \sin (\frac{π}{4})$

Step 1.1.4.8

Rewrite $\sqrt{\frac{2 + \sqrt{3}}{4}}$ as $\frac{\sqrt{2 + \sqrt{3}}}{\sqrt{4}}$ .

$\frac{\sqrt{2 + \sqrt{3}}}{\sqrt{4}} \cos (\frac{π}{4}) - \cos (\frac{7 π}{12}) \sin (\frac{π}{4})$

Step 1.1.4.9

Simplify the denominator.

Tap for more steps...

Step 1.1.4.9.1

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

$\frac{\sqrt{2 + \sqrt{3}}}{\sqrt{2^{2}}} \cos (\frac{π}{4}) - \cos (\frac{7 π}{12}) \sin (\frac{π}{4})$

Step 1.1.4.9.2

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

$\frac{\sqrt{2 + \sqrt{3}}}{2} \cos (\frac{π}{4}) - \cos (\frac{7 π}{12}) \sin (\frac{π}{4})$

Step 1.2

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2 + \sqrt{3}}}{2} \cdot \frac{\sqrt{2}}{2} - \cos (\frac{7 π}{12}) \sin (\frac{π}{4})$

Step 1.3

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

Tap for more steps...

Step 1.3.1

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

$\frac{\sqrt{2 + \sqrt{3}} \sqrt{2}}{2 \cdot 2} - \cos (\frac{7 π}{12}) \sin (\frac{π}{4})$

Step 1.3.2

Combine using the product rule for radicals.

$\frac{\sqrt{(2 + \sqrt{3}) \cdot 2}}{2 \cdot 2} - \cos (\frac{7 π}{12}) \sin (\frac{π}{4})$

Step 1.3.3

Multiply $2$ by $2$ .

$\frac{\sqrt{(2 + \sqrt{3}) \cdot 2}}{4} - \cos (\frac{7 π}{12}) \sin (\frac{π}{4})$

Step 1.4

The exact value of $\cos (\frac{7 π}{12})$ is $- \frac{\sqrt{2 - \sqrt{3}}}{2}$ .

Tap for more steps...

Step 1.4.1

Rewrite $\frac{7 π}{12}$ as an angle where the values of the six trigonometric functions are known divided by $2$ .

$\frac{\sqrt{(2 + \sqrt{3}) \cdot 2}}{4} - \cos (\frac{\frac{7 π}{6}}{2}) \sin (\frac{π}{4})$

Step 1.4.2

Apply the cosine half-angle identity $\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

$\frac{\sqrt{(2 + \sqrt{3}) \cdot 2}}{4} - (\pm \sqrt{\frac{1 + \cos (\frac{7 π}{6})}{2}}) \sin (\frac{π}{4})$

Step 1.4.3

Change the $\pm$ to $-$ because cosine is negative in the second quadrant.

$\frac{\sqrt{(2 + \sqrt{3}) \cdot 2}}{4} - - \sqrt{\frac{1 + \cos (\frac{7 π}{6})}{2}} \sin (\frac{π}{4})$

Step 1.4.4

Simplify $- \sqrt{\frac{1 + \cos (\frac{7 π}{6})}{2}}$ .

Tap for more steps...

Step 1.4.4.1

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 third quadrant.

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

Step 1.4.4.2

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

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

Step 1.4.4.3

Write $1$ as a fraction with a common denominator.

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

Step 1.4.4.4

Combine the numerators over the common denominator.

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

Step 1.4.4.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.4.4.6

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

Tap for more steps...

Step 1.4.4.6.1

Multiply $\frac{2 - \sqrt{3}}{2}$ by $\frac{1}{2}$ .

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

Step 1.4.4.6.2

Multiply $2$ by $2$ .

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

Step 1.4.4.7

Rewrite $\sqrt{\frac{2 - \sqrt{3}}{4}}$ as $\frac{\sqrt{2 - \sqrt{3}}}{\sqrt{4}}$ .

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

Step 1.4.4.8

Simplify the denominator.

Tap for more steps...

Step 1.4.4.8.1

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

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

Step 1.4.4.8.2

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

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

Step 1.5

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

Tap for more steps...

Step 1.5.1

Multiply $- 1$ by $- 1$ .

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

Step 1.5.2

Multiply $\frac{\sqrt{2 - \sqrt{3}}}{2}$ by $1$ .

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

Step 1.6

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

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

Step 1.7

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

Tap for more steps...

Step 1.7.1

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

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

Step 1.7.2

Combine using the product rule for radicals.

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

Step 1.7.3

Multiply $2$ by $2$ .

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.86602540 \dots$