Trigonometry Examples

$(2 \cos (x) - 3 \sin^{2} (x)) = 13$

Step 1

Subtract $13$ from both sides of the equation.

$2 \cos (x) - 3 \sin^{2} (x) - 13 = 0$

Step 2

Replace the $- 3 \sin^{2} (x)$ with $- 3 (1 - \cos^{2} (x))$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

$2 \cos (x) - 3 (1 - \cos^{2} (x)) - 13 = 0$

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

$2 \cos (x) - 3 \cdot 1 - 3 (- \cos^{2} (x)) - 13 = 0$

Step 3.2

Multiply $- 3$ by $1$ .

$2 \cos (x) - 3 - 3 (- \cos^{2} (x)) - 13 = 0$

Step 3.3

Multiply $- 1$ by $- 3$ .

$2 \cos (x) - 3 + 3 \cos^{2} (x) - 13 = 0$

Step 4

Subtract $13$ from $- 3$ .

$2 \cos (x) + 3 \cos^{2} (x) - 16 = 0$

Step 5

Reorder the polynomial.

$3 \cos^{2} (x) + 2 \cos (x) - 16 = 0$

Step 6

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

$3 {(u)}^{2} + 2 (u) - 16 = 0$

Step 7

Factor by grouping.

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Step 7.1

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 $a \cdot c = 3 \cdot - 16 = - 48$ and whose sum is $b = 2$ .

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Step 7.1.1

Factor $2$ out of $2 u$ .

$3 u^{2} + 2 u - 16 = 0$

Step 7.1.2

Rewrite $2$ as $- 6$ plus $8$

$3 u^{2} + (- 6 + 8) u - 16 = 0$

Step 7.1.3

Apply the distributive property.

$3 u^{2} - 6 u + 8 u - 16 = 0$

Step 7.2

Factor out the greatest common factor from each group.

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Step 7.2.1

Group the first two terms and the last two terms.

$3 u^{2} - 6 u + 8 u - 16 = 0$

Step 7.2.2

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

$3 u (u - 2) + 8 (u - 2) = 0$

Step 7.3

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

$(u - 2) (3 u + 8) = 0$

Step 8

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

$u - 2 = 0$

$3 u + 8 = 0$

Step 9

Set $u - 2$ equal to $0$ and solve for $u$ .

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Step 9.1

Set $u - 2$ equal to $0$ .

$u - 2 = 0$

Step 9.2

Add $2$ to both sides of the equation.

$u = 2$

Step 10

Set $3 u + 8$ equal to $0$ and solve for $u$ .

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Step 10.1

Set $3 u + 8$ equal to $0$ .

$3 u + 8 = 0$

Step 10.2

Solve $3 u + 8 = 0$ for $u$ .

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Step 10.2.1

Subtract $8$ from both sides of the equation.

$3 u = - 8$

Step 10.2.2

Divide each term in $3 u = - 8$ by $3$ and simplify.

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Step 10.2.2.1

Divide each term in $3 u = - 8$ by $3$ .

$\frac{3 u}{3} = \frac{- 8}{3}$

Step 10.2.2.2

Simplify the left side.

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Step 10.2.2.2.1

Cancel the common factor of $3$ .

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Step 10.2.2.2.1.1

Cancel the common factor.

$\frac{3 u}{3} = \frac{- 8}{3}$

Step 10.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{- 8}{3}$

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.

$u = - \frac{8}{3}$

Step 11

The final solution is all the values that make $(u - 2) (3 u + 8) = 0$ true.

$u = 2, - \frac{8}{3}$

Step 12

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

$\cos (x) = 2, - \frac{8}{3}$

Step 13

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

$\cos (x) = 2$

$\cos (x) = - \frac{8}{3}$

Step 14

Solve for $x$ in $\cos (x) = 2$ .

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Step 14.1

The range of cosine is $- 1 \leq y \leq 1$ . Since $2$ does not fall in this range, there is no solution.

No solution

Step 15

Solve for $x$ in $\cos (x) = - \frac{8}{3}$ .

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Step 15.1

The range of cosine is $- 1 \leq y \leq 1$ . Since $- \frac{8}{3}$ does not fall in this range, there is no solution.

No solution