Trigonometry Examples

$\sin (75)$

Step 1

Convert the radian measure to degrees.

Tap for more steps...

Step 1.1

To convert radians to degrees, multiply by $\frac{180}{π}$ , since a full circle is $360 °$ or $2 π$ radians.

$(\sin (75)) \cdot \frac{180 °}{π}$

Step 1.2

The exact value of $\sin (75)$ is $\frac{\sqrt{2} + \sqrt{6}}{4}$ .

Tap for more steps...

Step 1.2.1

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

$\sin (30 + 45) \cdot \frac{180}{π}$

Step 1.2.2

Apply the sum of angles identity.

$(\sin (30) \cos (45) + \cos (30) \sin (45)) \cdot \frac{180}{π}$

Step 1.2.3

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

$(\frac{1}{2} \cos (45) + \cos (30) \sin (45)) \cdot \frac{180}{π}$

Step 1.2.4

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

$(\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (30) \sin (45)) \cdot \frac{180}{π}$

Step 1.2.5

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

$(\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \sin (45)) \cdot \frac{180}{π}$

Step 1.2.6

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

$(\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) \cdot \frac{180}{π}$

Step 1.2.7

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

Tap for more steps...

Step 1.2.7.1

Simplify each term.

Tap for more steps...

Step 1.2.7.1.1

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

Tap for more steps...

Step 1.2.7.1.1.1

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

$(\frac{\sqrt{2}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) \cdot \frac{180}{π}$

Step 1.2.7.1.1.2

Multiply $2$ by $2$ .

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}) \cdot \frac{180}{π}$

Step 1.2.7.1.2

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

Tap for more steps...

Step 1.2.7.1.2.1

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

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}) \cdot \frac{180}{π}$

Step 1.2.7.1.2.2

Combine using the product rule for radicals.

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}) \cdot \frac{180}{π}$

Step 1.2.7.1.2.3

Multiply $3$ by $2$ .

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{2 \cdot 2}) \cdot \frac{180}{π}$

Step 1.2.7.1.2.4

Multiply $2$ by $2$ .

$(\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}) \cdot \frac{180}{π}$

Step 1.2.7.2

Combine the numerators over the common denominator.

$\frac{\sqrt{2} + \sqrt{6}}{4} \cdot \frac{180}{π}$

Step 1.3

Simplify terms.

Tap for more steps...

Step 1.3.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 1.3.1.1

Factor $4$ out of $180$ .

$\frac{\sqrt{2} + \sqrt{6}}{4} \cdot \frac{4 (45)}{π}$

Step 1.3.1.2

Cancel the common factor.

$\frac{\sqrt{2} + \sqrt{6}}{4} \cdot \frac{4 \cdot 45}{π}$

Step 1.3.1.3

Rewrite the expression.

$(\sqrt{2} + \sqrt{6}) \cdot \frac{45}{π}$

Step 1.3.2

Apply the distributive property.

$\sqrt{2} \frac{45}{π} + \sqrt{6} \frac{45}{π}$

Step 1.3.3

Combine $\sqrt{2}$ and $\frac{45}{π}$ .

$\frac{\sqrt{2} \cdot 45}{π} + \sqrt{6} \frac{45}{π}$

Step 1.3.4

Combine $\sqrt{6}$ and $\frac{45}{π}$ .

$\frac{\sqrt{2} \cdot 45}{π} + \frac{\sqrt{6} \cdot 45}{π}$

Step 1.4

Simplify each term.

Tap for more steps...

Step 1.4.1

Move $45$ to the left of $\sqrt{2}$ .

$\frac{45 \sqrt{2}}{π} + \frac{\sqrt{6} \cdot 45}{π}$

Step 1.4.2

Move $45$ to the left of $\sqrt{6}$ .

$\frac{45 \sqrt{2}}{π} + \frac{45 \sqrt{6}}{π}$

Step 1.5

$π$ is approximately equal to $3.14159265$ .

$\frac{45 \sqrt{2}}{3.14159265} + \frac{45 \sqrt{6}}{3.14159265}$

Step 1.6

Convert to a decimal.

$55.34347316 °$

Step 2

The angle is in the first quadrant.

Quadrant $1$

Step 3