Trigonometry Examples

$14 (1 - \cos (θ)) = \sin^{2} (θ)$

Step 1

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

$14 (1 - \cos (θ)) - \sin^{2} (θ) = 0$

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

$14 \cdot 1 + 14 (- \cos (θ)) - \sin^{2} (θ) = 0$

Step 2.2

Multiply $14$ by $1$ .

$14 + 14 (- \cos (θ)) - \sin^{2} (θ) = 0$

Step 2.3

Multiply $- 1$ by $14$ .

$14 - 14 \cos (θ) - \sin^{2} (θ) = 0$

Step 3

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

$14 - 14 \cos (θ) - (1 - \cos^{2} (θ)) = 0$

Step 4

Solve for $θ$ .

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Step 4.1

Substitute $u$ for $\cos (θ)$ .

$14 - 14 u - (1 - {(u)}^{2}) = 0$

Step 4.2

Simplify $14 - 14 u - (1 - u^{2})$ .

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Apply the distributive property.

$14 - 14 u - 1 \cdot 1 - - u^{2} = 0$

Step 4.2.1.2

Multiply $- 1$ by $1$ .

$14 - 14 u - 1 - - u^{2} = 0$

Step 4.2.1.3

Multiply $- - u^{2}$ .

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Step 4.2.1.3.1

Multiply $- 1$ by $- 1$ .

$14 - 14 u - 1 + 1 u^{2} = 0$

Step 4.2.1.3.2

Multiply $u^{2}$ by $1$ .

$14 - 14 u - 1 + u^{2} = 0$

Step 4.2.2

Subtract $1$ from $14$ .

$- 14 u + 13 + u^{2} = 0$

Step 4.3

Factor $- 14 u + 13 + u^{2}$ using the AC method.

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Step 4.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $13$ and whose sum is $- 14$ .

$- 13, - 1$

Step 4.3.2

Write the factored form using these integers.

$(u - 13) (u - 1) = 0$

Step 4.4

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

$u - 13 = 0$

$u - 1 = 0$

Step 4.5

Set $u - 13$ equal to $0$ and solve for $u$ .

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Step 4.5.1

Set $u - 13$ equal to $0$ .

$u - 13 = 0$

Step 4.5.2

Add $13$ to both sides of the equation.

$u = 13$

Step 4.6

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

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Step 4.6.1

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

$u - 1 = 0$

Step 4.6.2

Add $1$ to both sides of the equation.

$u = 1$

Step 4.7

The final solution is all the values that make $(u - 13) (u - 1) = 0$ true.

$u = 13, 1$

Step 4.8

Substitute $\cos (θ)$ for $u$ .

$\cos (θ) = 13, 1$

Step 4.9

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

$\cos (θ) = 13$

$\cos (θ) = 1$

Step 4.10

Solve for $θ$ in $\cos (θ) = 13$ .

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Step 4.10.1

The range of cosine is $- 1 \leq y \leq 1$ . Since $13$ does not fall in this range, there is no solution.

No solution

Step 4.11

Solve for $θ$ in $\cos (θ) = 1$ .

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Step 4.11.1

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

$θ = \arccos (1)$

Step 4.11.2

Simplify the right side.

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Step 4.11.2.1

The exact value of $\arccos (1)$ is $0$ .

$θ = 0$

Step 4.11.3

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

$θ = 2 π - 0$

Step 4.11.4

Subtract $0$ from $2 π$ .

$θ = 2 π$

Step 4.11.5

Find the period of $\cos (θ)$ .

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Step 4.11.5.1

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

$\frac{2 π}{| b |}$

Step 4.11.5.2

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

$\frac{2 π}{| 1 |}$

Step 4.11.5.3

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

$\frac{2 π}{1}$

Step 4.11.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 4.11.6

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

$θ = 2 π n, 2 π + 2 π n$ , for any integer $n$

Step 4.12

List all of the solutions.

$θ = 2 π n, 2 π + 2 π n$ , for any integer $n$

Step 4.13

Consolidate the answers.

$θ = 2 π n$ , for any integer $n$