Trigonometry Examples

$- 110 \cos (x) = 40 \cos^{2} (x) + 60$

Step 1

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

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Step 1.1

Subtract $40 \cos^{2} (x)$ from both sides of the equation.

$- 110 \cos (x) - 40 \cos^{2} (x) = 60$

Step 1.2

Subtract $60$ from both sides of the equation.

$- 110 \cos (x) - 40 \cos^{2} (x) - 60 = 0$

Step 2

Factor $- 110 \cos (x) - 40 \cos^{2} (x) - 60$ .

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Step 2.1

Factor $10$ out of $- 110 \cos (x) - 40 \cos^{2} (x) - 60$ .

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Step 2.1.1

Factor $10$ out of $- 110 \cos (x)$ .

$10 (- 11 \cos (x)) - 40 \cos^{2} (x) - 60 = 0$

Step 2.1.2

Factor $10$ out of $- 40 \cos^{2} (x)$ .

$10 (- 11 \cos (x)) + 10 (- 4 \cos^{2} (x)) - 60 = 0$

Step 2.1.3

Factor $10$ out of $- 60$ .

$10 (- 11 \cos (x)) + 10 (- 4 \cos^{2} (x)) + 10 (- 6) = 0$

Step 2.1.4

Factor $10$ out of $10 (- 11 \cos (x)) + 10 (- 4 \cos^{2} (x))$ .

$10 (- 11 \cos (x) - 4 \cos^{2} (x)) + 10 (- 6) = 0$

Step 2.1.5

Factor $10$ out of $10 (- 11 \cos (x) - 4 \cos^{2} (x)) + 10 (- 6)$ .

$10 (- 11 \cos (x) - 4 \cos^{2} (x) - 6) = 0$

Step 2.2

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

$10 (- 11 u - 4 u^{2} - 6) = 0$

Step 2.3

Factor by grouping.

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Step 2.3.1

Reorder terms.

$10 (- 4 u^{2} - 11 u - 6) = 0$

Step 2.3.2

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 $a \cdot c = - 4 \cdot - 6 = 24$ and whose sum is $b = - 11$ .

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Step 2.3.2.1

Factor $- 11$ out of $- 11 u$ .

$10 (- 4 u^{2} - 11 u - 6) = 0$

Step 2.3.2.2

Rewrite $- 11$ as $- 3$ plus $- 8$

$10 (- 4 u^{2} + (- 3 - 8) u - 6) = 0$

Step 2.3.2.3

Apply the distributive property.

$10 (- 4 u^{2} - 3 u - 8 u - 6) = 0$

Step 2.3.3

Factor out the greatest common factor from each group.

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Step 2.3.3.1

Group the first two terms and the last two terms.

$10 ((- 4 u^{2} - 3 u) - 8 u - 6) = 0$

Step 2.3.3.2

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

$10 (u (- 4 u - 3) + 2 (- 4 u - 3)) = 0$

Step 2.3.4

Factor the polynomial by factoring out the greatest common factor, $- 4 u - 3$ .

$10 ((- 4 u - 3) (u + 2)) = 0$

Step 2.4

Factor.

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Step 2.4.1

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

$10 ((- 4 \cos (x) - 3) (\cos (x) + 2)) = 0$

Step 2.4.2

Remove unnecessary parentheses.

$10 (- 4 \cos (x) - 3) (\cos (x) + 2) = 0$

Step 3

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

$- 4 \cos (x) - 3 = 0$

$\cos (x) + 2 = 0$

Step 4

Set $- 4 \cos (x) - 3$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $- 4 \cos (x) - 3$ equal to $0$ .

$- 4 \cos (x) - 3 = 0$

Step 4.2

Solve $- 4 \cos (x) - 3 = 0$ for $x$ .

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Step 4.2.1

Add $3$ to both sides of the equation.

$- 4 \cos (x) = 3$

Step 4.2.2

Divide each term in $- 4 \cos (x) = 3$ by $- 4$ and simplify.

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Step 4.2.2.1

Divide each term in $- 4 \cos (x) = 3$ by $- 4$ .

$\frac{- 4 \cos (x)}{- 4} = \frac{3}{- 4}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $- 4$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{- 4 \cos (x)}{- 4} = \frac{3}{- 4}$

Step 4.2.2.2.1.2

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

$\cos (x) = \frac{3}{- 4}$

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Move the negative in front of the fraction.

$\cos (x) = - \frac{3}{4}$

Step 4.2.3

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

$x = \arccos (- \frac{3}{4})$

Step 4.2.4

Simplify the right side.

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Step 4.2.4.1

Evaluate $\arccos (- \frac{3}{4})$ .

$x = 2.4188584$

Step 4.2.5

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$x = 2 (3.14159265) - 2.4188584$

Step 4.2.6

Solve for $x$ .

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Step 4.2.6.1

Remove parentheses.

$x = 2 (3.14159265) - 2.4188584$

Step 4.2.6.2

Simplify $2 (3.14159265) - 2.4188584$ .

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Step 4.2.6.2.1

Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 2.4188584$

Step 4.2.6.2.2

Subtract $2.4188584$ from $6.2831853$ .

$x = 3.8643269$

Step 4.2.7

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

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Step 4.2.7.1

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

$\frac{2 π}{| b |}$

Step 4.2.7.2

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

$\frac{2 π}{| 1 |}$

Step 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 4.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 4.2.8

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

$x = 2.4188584 + 2 π n, 3.8643269 + 2 π n$ , for any integer $n$

Step 5

Set $\cos (x) + 2$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $\cos (x) + 2$ equal to $0$ .

$\cos (x) + 2 = 0$

Step 5.2

Solve $\cos (x) + 2 = 0$ for $x$ .

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Step 5.2.1

Subtract $2$ from both sides of the equation.

$\cos (x) = - 2$

Step 5.2.2

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 6

The final solution is all the values that make $10 (- 4 \cos (x) - 3) (\cos (x) + 2) = 0$ true.

$x = 2.4188584 + 2 π n, 3.8643269 + 2 π n$ , for any integer $n$