Trigonometry Examples

$15 \cos^{2} (x) = 13 + \sin (x)$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $13$ from both sides of the equation.

$15 \cos^{2} (x) - 13 = \sin (x)$

Step 1.2

Subtract $\sin (x)$ from both sides of the equation.

$15 \cos^{2} (x) - 13 - \sin (x) = 0$

Step 2

Replace the $15 \cos^{2} (x)$ with $15 (1 - \sin^{2} (x))$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

$15 (1 - \sin^{2} (x)) - 13 - \sin (x) = 0$

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

$15 \cdot 1 + 15 (- \sin^{2} (x)) - 13 - \sin (x) = 0$

Step 3.2

Multiply $15$ by $1$ .

$15 + 15 (- \sin^{2} (x)) - 13 - \sin (x) = 0$

Step 3.3

Multiply $- 1$ by $15$ .

$15 - 15 \sin^{2} (x) - 13 - \sin (x) = 0$

Step 4

Subtract $13$ from $15$ .

$- 15 \sin^{2} (x) + 2 - \sin (x) = 0$

Step 5

Reorder the polynomial.

$- 15 \sin^{2} (x) - \sin (x) + 2 = 0$

Step 6

Substitute $u$ for $\sin (x)$ .

$- 15 {(u)}^{2} - (u) + 2 = 0$

Step 7

Factor the left side of the equation.

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Step 7.1

Factor $- 1$ out of $- 15 u^{2} - u + 2$ .

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Step 7.1.1

Factor $- 1$ out of $- 15 u^{2}$ .

$- (15 u^{2}) - u + 2 = 0$

Step 7.1.2

Factor $- 1$ out of $- u$ .

$- (15 u^{2}) - u + 2 = 0$

Step 7.1.3

Rewrite $2$ as $- 1 (- 2)$ .

$- (15 u^{2}) - u - 1 \cdot - 2 = 0$

Step 7.1.4

Factor $- 1$ out of $- (15 u^{2}) - u$ .

$- (15 u^{2} + u) - 1 \cdot - 2 = 0$

Step 7.1.5

Factor $- 1$ out of $- (15 u^{2} + u) - 1 (- 2)$ .

$- (15 u^{2} + u - 2) = 0$

Step 7.2

Factor.

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Step 7.2.1

Factor by grouping.

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Step 7.2.1.1

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 $a \cdot c = 15 \cdot - 2 = - 30$ and whose sum is $b = 1$ .

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Step 7.2.1.1.1

Multiply by $1$ .

$- (15 u^{2} + 1 u - 2) = 0$

Step 7.2.1.1.2

Rewrite $1$ as $- 5$ plus $6$

$- (15 u^{2} + (- 5 + 6) u - 2) = 0$

Step 7.2.1.1.3

Apply the distributive property.

$- (15 u^{2} - 5 u + 6 u - 2) = 0$

Step 7.2.1.2

Factor out the greatest common factor from each group.

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Step 7.2.1.2.1

Group the first two terms and the last two terms.

$- (15 u^{2} - 5 u + 6 u - 2) = 0$

Step 7.2.1.2.2

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

$- (5 u (3 u - 1) + 2 (3 u - 1)) = 0$

Step 7.2.1.3

Factor the polynomial by factoring out the greatest common factor, $3 u - 1$ .

$- ((3 u - 1) (5 u + 2)) = 0$

Step 7.2.2

Remove unnecessary parentheses.

$- (3 u - 1) (5 u + 2) = 0$

Step 8

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

$3 u - 1 = 0$

$5 u + 2 = 0$

Step 9

Set $3 u - 1$ equal to $0$ and solve for $u$ .

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Step 9.1

Set $3 u - 1$ equal to $0$ .

$3 u - 1 = 0$

Step 9.2

Solve $3 u - 1 = 0$ for $u$ .

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Step 9.2.1

Add $1$ to both sides of the equation.

$3 u = 1$

Step 9.2.2

Divide each term in $3 u = 1$ by $3$ and simplify.

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Step 9.2.2.1

Divide each term in $3 u = 1$ by $3$ .

$\frac{3 u}{3} = \frac{1}{3}$

Step 9.2.2.2

Simplify the left side.

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Step 9.2.2.2.1

Cancel the common factor of $3$ .

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Step 9.2.2.2.1.1

Cancel the common factor.

$\frac{3 u}{3} = \frac{1}{3}$

Step 9.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{1}{3}$

Step 10

Set $5 u + 2$ equal to $0$ and solve for $u$ .

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Step 10.1

Set $5 u + 2$ equal to $0$ .

$5 u + 2 = 0$

Step 10.2

Solve $5 u + 2 = 0$ for $u$ .

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Step 10.2.1

Subtract $2$ from both sides of the equation.

$5 u = - 2$

Step 10.2.2

Divide each term in $5 u = - 2$ by $5$ and simplify.

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Step 10.2.2.1

Divide each term in $5 u = - 2$ by $5$ .

$\frac{5 u}{5} = \frac{- 2}{5}$

Step 10.2.2.2

Simplify the left side.

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Step 10.2.2.2.1

Cancel the common factor of $5$ .

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Step 10.2.2.2.1.1

Cancel the common factor.

$\frac{5 u}{5} = \frac{- 2}{5}$

Step 10.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{- 2}{5}$

Step 10.2.2.3

Simplify the right side.

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Step 10.2.2.3.1

Move the negative in front of the fraction.

$u = - \frac{2}{5}$

Step 11

The final solution is all the values that make $- (3 u - 1) (5 u + 2) = 0$ true.

$u = \frac{1}{3}, - \frac{2}{5}$

Step 12

Substitute $\sin (x)$ for $u$ .

$\sin (x) = \frac{1}{3}, - \frac{2}{5}$

Step 13

Set up each of the solutions to solve for $x$ .

$\sin (x) = \frac{1}{3}$

$\sin (x) = - \frac{2}{5}$

Step 14

Solve for $x$ in $\sin (x) = \frac{1}{3}$ .

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Step 14.1

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

$x = \arcsin (\frac{1}{3})$

Step 14.2

Simplify the right side.

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Step 14.2.1

Evaluate $\arcsin (\frac{1}{3})$ .

$x = 0.3398369$

Step 14.3

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.

$x = (3.14159265) - 0.3398369$

Step 14.4

Solve for $x$ .

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Step 14.4.1

Remove parentheses.

$x = 3.14159265 - 0.3398369$

Step 14.4.2

Remove parentheses.

$x = (3.14159265) - 0.3398369$

Step 14.4.3

Subtract $0.3398369$ from $3.14159265$ .

$x = 2.80175574$

Step 14.5

Find the period of $\sin (x)$ .

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Step 14.5.1

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

$\frac{2 π}{| b |}$

Step 14.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 14.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 14.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 14.6

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

$x = 0.3398369 + 2 π n, 2.80175574 + 2 π n$ , for any integer $n$

Step 15

Solve for $x$ in $\sin (x) = - \frac{2}{5}$ .

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Step 15.1

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

$x = \arcsin (- \frac{2}{5})$

Step 15.2

Simplify the right side.

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Step 15.2.1

Evaluate $\arcsin (- \frac{2}{5})$ .

$x = - 0.41151684$

Step 15.3

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$x = 2 (3.14159265) + 0.41151684 + 3.14159265$

Step 15.4

Simplify the expression to find the second solution.

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Step 15.4.1

Subtract $2 π$ from $2 (3.14159265) + 0.41151684 + 3.14159265$ .

$x = 2 (3.14159265) + 0.41151684 + 3.14159265 - 2 π$

Step 15.4.2

The resulting angle of $3.55310949$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) + 0.41151684 + 3.14159265$ .

$x = 3.55310949$

Step 15.5

Find the period of $\sin (x)$ .

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Step 15.5.1

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

$\frac{2 π}{| b |}$

Step 15.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 15.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 15.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 15.6

Add $2 π$ to every negative angle to get positive angles.

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Step 15.6.1

Add $2 π$ to $- 0.41151684$ to find the positive angle.

$- 0.41151684 + 2 π$

Step 15.6.2

Subtract $0.41151684$ from $2 π$ .

$5.87166846$

Step 15.6.3

List the new angles.

$x = 5.87166846$

Step 15.7

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

$x = 3.55310949 + 2 π n, 5.87166846 + 2 π n$ , for any integer $n$

Step 16

List all of the solutions.

$x = 0.3398369 + 2 π n, 2.80175574 + 2 π n, 3.55310949 + 2 π n, 5.87166846 + 2 π n$ , for any integer $n$