Trigonometry Examples

$17 \sin^{2} (x) - 3 \sin (x) = \cos^{2} (x)$

Step 1

Subtract $\cos^{2} (x)$ from both sides of the equation.

$17 \sin^{2} (x) - 3 \sin (x) - \cos^{2} (x) = 0$

Step 2

Replace $\cos^{2} (x)$ with $1 - \sin^{2} (x)$ .

$17 \sin^{2} (x) - 3 \sin (x) - (1 - \sin^{2} (x)) = 0$

Step 3

Solve for $x$ .

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Step 3.1

Factor the left side of the equation.

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Step 3.1.1

Let $u = \sin (x)$ . Substitute $u$ for all occurrences of $\sin (x)$ .

$18 u^{2} - 3 u - 1 = 0$

Step 3.1.2

Factor by grouping.

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Step 3.1.2.1

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 $a \cdot c = 18 \cdot - 1 = - 18$ and whose sum is $b = - 3$ .

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Step 3.1.2.1.1

Factor $- 3$ out of $- 3 u$ .

$18 u^{2} - 3 u - 1 = 0$

Step 3.1.2.1.2

Rewrite $- 3$ as $3$ plus $- 6$

$18 u^{2} + (3 - 6) u - 1 = 0$

Step 3.1.2.1.3

Apply the distributive property.

$18 u^{2} + 3 u - 6 u - 1 = 0$

Step 3.1.2.2

Factor out the greatest common factor from each group.

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Step 3.1.2.2.1

Group the first two terms and the last two terms.

$(18 u^{2} + 3 u) - 6 u - 1 = 0$

Step 3.1.2.2.2

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

$3 u (6 u + 1) - (6 u + 1) = 0$

Step 3.1.2.3

Factor the polynomial by factoring out the greatest common factor, $6 u + 1$ .

$(6 u + 1) (3 u - 1) = 0$

Step 3.1.3

Replace all occurrences of $u$ with $\sin (x)$ .

$(6 \sin (x) + 1) (3 \sin (x) - 1) = 0$

Step 3.2

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

$6 \sin (x) + 1 = 0$

$3 \sin (x) - 1 = 0$

Step 3.3

Set $6 \sin (x) + 1$ equal to $0$ and solve for $x$ .

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Step 3.3.1

Set $6 \sin (x) + 1$ equal to $0$ .

$6 \sin (x) + 1 = 0$

Step 3.3.2

Solve $6 \sin (x) + 1 = 0$ for $x$ .

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Step 3.3.2.1

Subtract $1$ from both sides of the equation.

$6 \sin (x) = - 1$

Step 3.3.2.2

Divide each term in $6 \sin (x) = - 1$ by $6$ and simplify.

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Step 3.3.2.2.1

Divide each term in $6 \sin (x) = - 1$ by $6$ .

$\frac{6 \sin (x)}{6} = \frac{- 1}{6}$

Step 3.3.2.2.2

Simplify the left side.

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Step 3.3.2.2.2.1

Cancel the common factor of $6$ .

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Step 3.3.2.2.2.1.1

Cancel the common factor.

$\frac{6 \sin (x)}{6} = \frac{- 1}{6}$

Step 3.3.2.2.2.1.2

Divide $\sin (x)$ by $1$ .

$\sin (x) = \frac{- 1}{6}$

Step 3.3.2.2.3

Simplify the right side.

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Step 3.3.2.2.3.1

Move the negative in front of the fraction.

$\sin (x) = - \frac{1}{6}$

Step 3.3.2.3

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

$x = \arcsin (- \frac{1}{6})$

Step 3.3.2.4

Simplify the right side.

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Step 3.3.2.4.1

Evaluate $\arcsin (- \frac{1}{6})$ .

$x = - 0.16744807$

Step 3.3.2.5

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.

$x = 2 (3.14159265) + 0.16744807 + 3.14159265$

Step 3.3.2.6

Simplify the expression to find the second solution.

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Step 3.3.2.6.1

Subtract $2 π$ from $2 (3.14159265) + 0.16744807 + 3.14159265$ .

$x = 2 (3.14159265) + 0.16744807 + 3.14159265 - 2 π$

Step 3.3.2.6.2

The resulting angle of $3.30904073$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) + 0.16744807 + 3.14159265$ .

$x = 3.30904073$

Step 3.3.2.7

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

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Step 3.3.2.7.1

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

$\frac{2 π}{| b |}$

Step 3.3.2.7.2

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

$\frac{2 π}{| 1 |}$

Step 3.3.2.7.3

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

$\frac{2 π}{1}$

Step 3.3.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.3.2.8

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

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Step 3.3.2.8.1

Add $2 π$ to $- 0.16744807$ to find the positive angle.

$- 0.16744807 + 2 π$

Step 3.3.2.8.2

Subtract $0.16744807$ from $2 π$ .

$6.11573722$

Step 3.3.2.8.3

List the new angles.

$x = 6.11573722$

Step 3.3.2.9

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

$x = 3.30904073 + 2 π n, 6.11573722 + 2 π n$ , for any integer $n$

Step 3.4

Set $3 \sin (x) - 1$ equal to $0$ and solve for $x$ .

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Step 3.4.1

Set $3 \sin (x) - 1$ equal to $0$ .

$3 \sin (x) - 1 = 0$

Step 3.4.2

Solve $3 \sin (x) - 1 = 0$ for $x$ .

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Step 3.4.2.1

Add $1$ to both sides of the equation.

$3 \sin (x) = 1$

Step 3.4.2.2

Divide each term in $3 \sin (x) = 1$ by $3$ and simplify.

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Step 3.4.2.2.1

Divide each term in $3 \sin (x) = 1$ by $3$ .

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

Step 3.4.2.2.2

Simplify the left side.

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Step 3.4.2.2.2.1

Cancel the common factor of $3$ .

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Step 3.4.2.2.2.1.1

Cancel the common factor.

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

Step 3.4.2.2.2.1.2

Divide $\sin (x)$ by $1$ .

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

Step 3.4.2.3

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

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

Step 3.4.2.4

Simplify the right side.

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Step 3.4.2.4.1

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

$x = 0.3398369$

Step 3.4.2.5

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 3.4.2.6

Solve for $x$ .

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Step 3.4.2.6.1

Remove parentheses.

$x = 3.14159265 - 0.3398369$

Step 3.4.2.6.2

Remove parentheses.

$x = (3.14159265) - 0.3398369$

Step 3.4.2.6.3

Subtract $0.3398369$ from $3.14159265$ .

$x = 2.80175574$

Step 3.4.2.7

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

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Step 3.4.2.7.1

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

$\frac{2 π}{| b |}$

Step 3.4.2.7.2

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

$\frac{2 π}{| 1 |}$

Step 3.4.2.7.3

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

$\frac{2 π}{1}$

Step 3.4.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.4.2.8

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 3.5

The final solution is all the values that make $(6 \sin (x) + 1) (3 \sin (x) - 1) = 0$ true.

$x = 3.30904073 + 2 π n, 6.11573722 + 2 π n, 0.3398369 + 2 π n, 2.80175574 + 2 π n$ , for any integer $n$