Trigonometry Examples

$x \cos (60) + y \sin (60) - 15 = 0$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

The exact value of $\cos (60)$ is $\frac{1}{2}$ .

$x \frac{1}{2} + y \sin (60) - 15 = 0$

Step 1.2

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

$\frac{x}{2} + y \sin (60) - 15 = 0$

Step 1.3

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

$\frac{x}{2} + y \frac{\sqrt{3}}{2} - 15 = 0$

Step 1.4

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

$\frac{x}{2} + \frac{y \sqrt{3}}{2} - 15 = 0$

Step 2

Move all terms not containing $y$ to the right side of the equation.

Tap for more steps...

Step 2.1

Subtract $\frac{x}{2}$ from both sides of the equation.

$\frac{y \sqrt{3}}{2} - 15 = - \frac{x}{2}$

Step 2.2

Add $15$ to both sides of the equation.

$\frac{y \sqrt{3}}{2} = - \frac{x}{2} + 15$

Step 3

Multiply both sides of the equation by $\frac{2}{\sqrt{3}}$ .

$\frac{2}{\sqrt{3}} \cdot \frac{y \sqrt{3}}{2} = \frac{2}{\sqrt{3}} (- \frac{x}{2} + 15)$

Step 4

Simplify both sides of the equation.

Tap for more steps...

Step 4.1

Simplify the left side.

Tap for more steps...

Step 4.1.1

Simplify $\frac{2}{\sqrt{3}} \cdot \frac{y \sqrt{3}}{2}$ .

Tap for more steps...

Step 4.1.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.1.1.1.1

Cancel the common factor.

$\frac{2}{\sqrt{3}} \cdot \frac{y \sqrt{3}}{2} = \frac{2}{\sqrt{3}} (- \frac{x}{2} + 15)$

Step 4.1.1.1.2

Rewrite the expression.

$\frac{1}{\sqrt{3}} (y \sqrt{3}) = \frac{2}{\sqrt{3}} (- \frac{x}{2} + 15)$

Step 4.1.1.2

Cancel the common factor of $\sqrt{3}$ .

Tap for more steps...

Step 4.1.1.2.1

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

$\frac{1}{\sqrt{3}} (\sqrt{3} y) = \frac{2}{\sqrt{3}} (- \frac{x}{2} + 15)$

Step 4.1.1.2.2

Cancel the common factor.

$\frac{1}{\sqrt{3}} (\sqrt{3} y) = \frac{2}{\sqrt{3}} (- \frac{x}{2} + 15)$

Step 4.1.1.2.3

Rewrite the expression.

$y = \frac{2}{\sqrt{3}} (- \frac{x}{2} + 15)$

Step 4.2

Simplify the right side.

Tap for more steps...

Step 4.2.1

Simplify $\frac{2}{\sqrt{3}} (- \frac{x}{2} + 15)$ .

Tap for more steps...

Step 4.2.1.1

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

$y = \frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} (- \frac{x}{2} + 15)$

Step 4.2.1.2

Combine and simplify the denominator.

Tap for more steps...

Step 4.2.1.2.1

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

$y = \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}} (- \frac{x}{2} + 15)$

Step 4.2.1.2.2

Raise $\sqrt{3}$ to the power of $1$ .

$y = \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} (- \frac{x}{2} + 15)$

Step 4.2.1.2.3

Raise $\sqrt{3}$ to the power of $1$ .

$y = \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} (- \frac{x}{2} + 15)$

Step 4.2.1.2.4

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

$y = \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} (- \frac{x}{2} + 15)$

Step 4.2.1.2.5

Add $1$ and $1$ .

$y = \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}} (- \frac{x}{2} + 15)$

Step 4.2.1.2.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 4.2.1.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$y = \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} (- \frac{x}{2} + 15)$

Step 4.2.1.2.6.2

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

$y = \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} (- \frac{x}{2} + 15)$

Step 4.2.1.2.6.3

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

$y = \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} (- \frac{x}{2} + 15)$

Step 4.2.1.2.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.1.2.6.4.1

Cancel the common factor.

$y = \frac{2 \sqrt{3}}{3^{\frac{2}{2}}} (- \frac{x}{2} + 15)$

Step 4.2.1.2.6.4.2

Rewrite the expression.

$y = \frac{2 \sqrt{3}}{3^{1}} (- \frac{x}{2} + 15)$

Step 4.2.1.2.6.5

Evaluate the exponent.

$y = \frac{2 \sqrt{3}}{3} (- \frac{x}{2} + 15)$

Step 4.2.1.3

Apply the distributive property.

$y = \frac{2 \sqrt{3}}{3} (- \frac{x}{2}) + \frac{2 \sqrt{3}}{3} \cdot 15$

Step 4.2.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.1.4.1

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

$y = \frac{2 \sqrt{3}}{3} \cdot \frac{- x}{2} + \frac{2 \sqrt{3}}{3} \cdot 15$

Step 4.2.1.4.2

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

$y = \frac{2 (\sqrt{3})}{3} \cdot \frac{- x}{2} + \frac{2 \sqrt{3}}{3} \cdot 15$

Step 4.2.1.4.3

Cancel the common factor.

$y = \frac{2 \sqrt{3}}{3} \cdot \frac{- x}{2} + \frac{2 \sqrt{3}}{3} \cdot 15$

Step 4.2.1.4.4

Rewrite the expression.

$y = \frac{\sqrt{3}}{3} (- x) + \frac{2 \sqrt{3}}{3} \cdot 15$

Step 4.2.1.5

Combine $\frac{\sqrt{3}}{3}$ and $x$ .

$y = - \frac{\sqrt{3} x}{3} + \frac{2 \sqrt{3}}{3} \cdot 15$

Step 4.2.1.6

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.2.1.6.1

Factor $3$ out of $15$ .

$y = - \frac{\sqrt{3} x}{3} + \frac{2 \sqrt{3}}{3} \cdot (3 (5))$

Step 4.2.1.6.2

Cancel the common factor.

$y = - \frac{\sqrt{3} x}{3} + \frac{2 \sqrt{3}}{3} \cdot (3 \cdot 5)$

Step 4.2.1.6.3

Rewrite the expression.

$y = - \frac{\sqrt{3} x}{3} + 2 \sqrt{3} \cdot 5$

Step 4.2.1.7

Multiply $5$ by $2$ .

$y = - \frac{\sqrt{3} x}{3} + 10 \sqrt{3}$