Trigonometry Examples

$8 \sin^{2} (\frac{x}{2}) - 10 \sin (\frac{x}{2}) + 3 = 0$

Step 1

Factor the left side of the equation.

Tap for more steps...

Step 1.1

Let $u = \sin (\frac{x}{2})$ . Substitute $u$ for all occurrences of $\sin (\frac{x}{2})$ .

$8 u^{2} - 10 u + 3 = 0$

Step 1.2

Factor by grouping.

Tap for more steps...

Step 1.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 8 \cdot 3 = 24$ and whose sum is $b = - 10$ .

Tap for more steps...

Step 1.2.1.1

Factor $- 10$ out of $- 10 u$ .

$8 u^{2} - 10 u + 3 = 0$

Step 1.2.1.2

Rewrite $- 10$ as $- 4$ plus $- 6$

$8 u^{2} + (- 4 - 6) u + 3 = 0$

Step 1.2.1.3

Apply the distributive property.

$8 u^{2} - 4 u - 6 u + 3 = 0$

Step 1.2.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.2.2.1

Group the first two terms and the last two terms.

$(8 u^{2} - 4 u) - 6 u + 3 = 0$

Step 1.2.2.2

Factor out the greatest common factor (GCF) from each group.

$4 u (2 u - 1) - 3 (2 u - 1) = 0$

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor, $2 u - 1$ .

$(2 u - 1) (4 u - 3) = 0$

Step 1.3

Replace all occurrences of $u$ with $\sin (\frac{x}{2})$ .

$(2 \sin (\frac{x}{2}) - 1) (4 \sin (\frac{x}{2}) - 3) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 \sin (\frac{x}{2}) - 1 = 0$

$4 \sin (\frac{x}{2}) - 3 = 0$

Step 3

Set $2 \sin (\frac{x}{2}) - 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.1

Set $2 \sin (\frac{x}{2}) - 1$ equal to $0$ .

$2 \sin (\frac{x}{2}) - 1 = 0$

Step 3.2

Solve $2 \sin (\frac{x}{2}) - 1 = 0$ for $x$ .

Tap for more steps...

Step 3.2.1

Add $1$ to both sides of the equation.

$2 \sin (\frac{x}{2}) = 1$

Step 3.2.2

Divide each term in $2 \sin (\frac{x}{2}) = 1$ by $2$ and simplify.

Tap for more steps...

Step 3.2.2.1

Divide each term in $2 \sin (\frac{x}{2}) = 1$ by $2$ .

$\frac{2 \sin (\frac{x}{2})}{2} = \frac{1}{2}$

Step 3.2.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{2 \sin (\frac{x}{2})}{2} = \frac{1}{2}$

Step 3.2.2.2.1.2

Divide $\sin (\frac{x}{2})$ by $1$ .

$\sin (\frac{x}{2}) = \frac{1}{2}$

Step 3.2.3

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$\frac{x}{2} = \arcsin (\frac{1}{2})$

Step 3.2.4

Simplify the right side.

Tap for more steps...

Step 3.2.4.1

The exact value of $\arcsin (\frac{1}{2})$ is $\frac{π}{6}$ .

$\frac{x}{2} = \frac{π}{6}$

Step 3.2.5

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \frac{π}{6}$

Step 3.2.6

Simplify both sides of the equation.

Tap for more steps...

Step 3.2.6.1

Simplify the left side.

Tap for more steps...

Step 3.2.6.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.6.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \frac{π}{6}$

Step 3.2.6.1.1.2

Rewrite the expression.

$x = 2 \frac{π}{6}$

Step 3.2.6.2

Simplify the right side.

Tap for more steps...

Step 3.2.6.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.6.2.1.1

Factor $2$ out of $6$ .

$x = 2 \frac{π}{2 (3)}$

Step 3.2.6.2.1.2

Cancel the common factor.

$x = 2 \frac{π}{2 \cdot 3}$

Step 3.2.6.2.1.3

Rewrite the expression.

$x = \frac{π}{3}$

Step 3.2.7

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$\frac{x}{2} = π - \frac{π}{6}$

Step 3.2.8

Solve for $x$ .

Tap for more steps...

Step 3.2.8.1

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (π - \frac{π}{6})$

Step 3.2.8.2

Simplify both sides of the equation.

Tap for more steps...

Step 3.2.8.2.1

Simplify the left side.

Tap for more steps...

Step 3.2.8.2.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.8.2.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (π - \frac{π}{6})$

Step 3.2.8.2.1.1.2

Rewrite the expression.

$x = 2 (π - \frac{π}{6})$

Step 3.2.8.2.2

Simplify the right side.

Tap for more steps...

Step 3.2.8.2.2.1

Simplify $2 (π - \frac{π}{6})$ .

Tap for more steps...

Step 3.2.8.2.2.1.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x = 2 (π \cdot \frac{6}{6} - \frac{π}{6})$

Step 3.2.8.2.2.1.2

Simplify terms.

Tap for more steps...

Step 3.2.8.2.2.1.2.1

Combine $π$ and $\frac{6}{6}$ .

$x = 2 (\frac{π \cdot 6}{6} - \frac{π}{6})$

Step 3.2.8.2.2.1.2.2

Combine the numerators over the common denominator.

$x = 2 \frac{π \cdot 6 - π}{6}$

Step 3.2.8.2.2.1.2.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.8.2.2.1.2.3.1

Factor $2$ out of $6$ .

$x = 2 \frac{π \cdot 6 - π}{2 (3)}$

Step 3.2.8.2.2.1.2.3.2

Cancel the common factor.

$x = 2 \frac{π \cdot 6 - π}{2 \cdot 3}$

Step 3.2.8.2.2.1.2.3.3

Rewrite the expression.

$x = \frac{π \cdot 6 - π}{3}$

Step 3.2.8.2.2.1.3

Simplify the numerator.

Tap for more steps...

Step 3.2.8.2.2.1.3.1

Move $6$ to the left of $π$ .

$x = \frac{6 \cdot π - π}{3}$

Step 3.2.8.2.2.1.3.2

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{3}$

Step 3.2.9

Find the period of $\sin (\frac{x}{2})$ .

Tap for more steps...

Step 3.2.9.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.2.9.2

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

Step 3.2.9.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Step 3.2.9.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 3.2.9.5

Multiply $2$ by $2$ .

$4 π$

Step 3.2.10

The period of the $\sin (\frac{x}{2})$ function is $4 π$ so values will repeat every $4 π$ radians in both directions.

$x = \frac{π}{3} + 4 π n, \frac{5 π}{3} + 4 π n$ , for any integer $n$

Step 4

Set $4 \sin (\frac{x}{2}) - 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.1

Set $4 \sin (\frac{x}{2}) - 3$ equal to $0$ .

$4 \sin (\frac{x}{2}) - 3 = 0$

Step 4.2

Solve $4 \sin (\frac{x}{2}) - 3 = 0$ for $x$ .

Tap for more steps...

Step 4.2.1

Add $3$ to both sides of the equation.

$4 \sin (\frac{x}{2}) = 3$

Step 4.2.2

Divide each term in $4 \sin (\frac{x}{2}) = 3$ by $4$ and simplify.

Tap for more steps...

Step 4.2.2.1

Divide each term in $4 \sin (\frac{x}{2}) = 3$ by $4$ .

$\frac{4 \sin (\frac{x}{2})}{4} = \frac{3}{4}$

Step 4.2.2.2

Simplify the left side.

Tap for more steps...

Step 4.2.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{4 \sin (\frac{x}{2})}{4} = \frac{3}{4}$

Step 4.2.2.2.1.2

Divide $\sin (\frac{x}{2})$ by $1$ .

$\sin (\frac{x}{2}) = \frac{3}{4}$

Step 4.2.3

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$\frac{x}{2} = \arcsin (\frac{3}{4})$

Step 4.2.4

Simplify the right side.

Tap for more steps...

Step 4.2.4.1

Evaluate $\arcsin (\frac{3}{4})$ .

$\frac{x}{2} = 0.84806207$

Step 4.2.5

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \cdot 0.84806207$

Step 4.2.6

Simplify both sides of the equation.

Tap for more steps...

Step 4.2.6.1

Simplify the left side.

Tap for more steps...

Step 4.2.6.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.6.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \cdot 0.84806207$

Step 4.2.6.1.1.2

Rewrite the expression.

$x = 2 \cdot 0.84806207$

Step 4.2.6.2

Simplify the right side.

Tap for more steps...

Step 4.2.6.2.1

Multiply $2$ by $0.84806207$ .

$x = 1.69612415$

Step 4.2.7

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$\frac{x}{2} = (3.14159265) - 0.84806207$

Step 4.2.8

Solve for $x$ .

Tap for more steps...

Step 4.2.8.1

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 ((3.14159265) - 0.84806207)$

Step 4.2.8.2

Simplify both sides of the equation.

Tap for more steps...

Step 4.2.8.2.1

Simplify the left side.

Tap for more steps...

Step 4.2.8.2.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.8.2.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 ((3.14159265) - 0.84806207)$

Step 4.2.8.2.1.1.2

Rewrite the expression.

$x = 2 ((3.14159265) - 0.84806207)$

Step 4.2.8.2.2

Simplify the right side.

Tap for more steps...

Step 4.2.8.2.2.1

Simplify $2 ((3.14159265) - 0.84806207)$ .

Tap for more steps...

Step 4.2.8.2.2.1.1

Subtract $0.84806207$ from $3.14159265$ .

$x = 2 \cdot 2.29353057$

Step 4.2.8.2.2.1.2

Multiply $2$ by $2.29353057$ .

$x = 4.58706114$

Step 4.2.9

Find the period of $\sin (\frac{x}{2})$ .

Tap for more steps...

Step 4.2.9.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 4.2.9.2

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

Step 4.2.9.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Step 4.2.9.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 4.2.9.5

Multiply $2$ by $2$ .

$4 π$

Step 4.2.10

The period of the $\sin (\frac{x}{2})$ function is $4 π$ so values will repeat every $4 π$ radians in both directions.

$x = 1.69612415 + 4 π n, 4.58706114 + 4 π n$ , for any integer $n$

Step 5

The final solution is all the values that make $(2 \sin (\frac{x}{2}) - 1) (4 \sin (\frac{x}{2}) - 3) = 0$ true.

$x = \frac{π}{3} + 4 π n, \frac{5 π}{3} + 4 π n, 1.69612415 + 4 π n, 4.58706114 + 4 π n$ , for any integer $n$