Trigonometry Examples

$\sin^{2} (32 °) + \cos^{2} (m) = 1$

Step 1

Subtract $1$ from both sides of the equation.

$\sin^{2} (32 °) + \cos^{2} (m) - 1 = 0$

Step 2

Simplify $\sin^{2} (32 °) + \cos^{2} (m) - 1$ .

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Step 2.1

Move $- 1$ .

$\sin^{2} (32 °) - 1 + \cos^{2} (m) = 0$

Step 2.2

Reorder $\sin^{2} (32 °)$ and $- 1$ .

$- 1 + \sin^{2} (32 °) + \cos^{2} (m) = 0$

Step 2.3

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

$- 1 (1) + \sin^{2} (32 °) + \cos^{2} (m) = 0$

Step 2.4

Factor $- 1$ out of $\sin^{2} (32 °)$ .

$- 1 (1) - 1 (- \sin^{2} (32 °)) + \cos^{2} (m) = 0$

Step 2.5

Factor $- 1$ out of $- 1 (1) - 1 (- \sin^{2} (32 °))$ .

$- 1 (1 - \sin^{2} (32 °)) + \cos^{2} (m) = 0$

Step 2.6

Rewrite $- 1 (1 - \sin^{2} (32 °))$ as $- (1 - \sin^{2} (32 °))$ .

$- (1 - \sin^{2} (32 °)) + \cos^{2} (m) = 0$

Step 2.7

Apply pythagorean identity.

$- \cos^{2} (32 °) + \cos^{2} (m) = 0$

Step 2.8

Reorder $- \cos^{2} (32 °)$ and $\cos^{2} (m)$ .

$\cos^{2} (m) - \cos^{2} (32 °) = 0$

Step 2.9

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \cos (m)$ and $b = \cos (32 °)$ .

$(\cos (m) + \cos (32 °)) (\cos (m) - \cos (32 °)) = 0$

Step 2.10

Evaluate $\cos (32 °)$ .

$(\cos (m) + 0.84804809) (\cos (m) - \cos (32 °)) = 0$

Step 2.11

Simplify each term.

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Step 2.11.1

Evaluate $\cos (32 °)$ .

$(\cos (m) + 0.84804809) (\cos (m) - 1 \cdot 0.84804809) = 0$

Step 2.11.2

Multiply $- 1$ by $0.84804809$ .

$(\cos (m) + 0.84804809) (\cos (m) - 0.84804809) = 0$

Step 2.12

Expand $(\cos (m) + 0.84804809) (\cos (m) - 0.84804809)$ using the FOIL Method.

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Step 2.12.1

Apply the distributive property.

$\cos (m) (\cos (m) - 0.84804809) + 0.84804809 (\cos (m) - 0.84804809) = 0$

Step 2.12.2

Apply the distributive property.

$\cos (m) \cos (m) + \cos (m) \cdot - 0.84804809 + 0.84804809 (\cos (m) - 0.84804809) = 0$

Step 2.12.3

Apply the distributive property.

$\cos (m) \cos (m) + \cos (m) \cdot - 0.84804809 + 0.84804809 \cos (m) + 0.84804809 \cdot - 0.84804809 = 0$

Step 2.13

Combine the opposite terms in $\cos (m) \cos (m) + \cos (m) \cdot - 0.84804809 + 0.84804809 \cos (m) + 0.84804809 \cdot - 0.84804809$ .

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Step 2.13.1

Reorder the factors in the terms $\cos (m) \cdot - 0.84804809$ and $0.84804809 \cos (m)$ .

$\cos (m) \cos (m) - 0.84804809 \cos (m) + 0.84804809 \cos (m) + 0.84804809 \cdot - 0.84804809 = 0$

Step 2.13.2

Add $- 0.84804809 \cos (m)$ and $0.84804809 \cos (m)$ .

$\cos (m) \cos (m) + 0 \cos (m) + 0.84804809 \cdot - 0.84804809 = 0$

Step 2.14

Simplify each term.

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Step 2.14.1

Multiply $\cos (m) \cos (m)$ .

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Step 2.14.1.1

Raise $\cos (m)$ to the power of $1$ .

$\cos^{1} (m) \cos (m) + 0 \cos (m) + 0.84804809 \cdot - 0.84804809 = 0$

Step 2.14.1.2

Raise $\cos (m)$ to the power of $1$ .

$\cos^{1} (m) \cos^{1} (m) + 0 \cos (m) + 0.84804809 \cdot - 0.84804809 = 0$

Step 2.14.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${\cos (m)}^{1 + 1} + 0 \cos (m) + 0.84804809 \cdot - 0.84804809 = 0$

Step 2.14.1.4

Add $1$ and $1$ .

$\cos^{2} (m) + 0 \cos (m) + 0.84804809 \cdot - 0.84804809 = 0$

Step 2.14.2

Multiply $0$ by $\cos (m)$ .

$\cos^{2} (m) + 0 + 0.84804809 \cdot - 0.84804809 = 0$

Step 2.14.3

Multiply $0.84804809$ by $- 0.84804809$ .

$\cos^{2} (m) + 0 - 0.71918557 = 0$

Step 2.15

Add $\cos^{2} (m)$ and $0$ .

$\cos^{2} (m) - 0.71918557 = 0$

Step 3

Solve for $m$ .

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Step 3.1

Add $0.71918557$ to both sides of the equation.

$\cos^{2} (m) = 0.71918557$

Step 3.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\cos (m) = \pm \sqrt{0.71918557}$

Step 3.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.3.1

First, use the positive value of the $\pm$ to find the first solution.

$\cos (m) = \sqrt{0.71918557}$

Step 3.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$\cos (m) = - \sqrt{0.71918557}$

Step 3.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$\cos (m) = \sqrt{0.71918557}, - \sqrt{0.71918557}$

Step 3.4

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

$\cos (m) = \sqrt{0.71918557}$

$\cos (m) = - \sqrt{0.71918557}$

Step 3.5

Solve for $m$ in $\cos (m) = \sqrt{0.71918557}$ .

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Step 3.5.1

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

$m = \arccos (\sqrt{0.71918557})$

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Evaluate $\arccos (\sqrt{0.71918557})$ .

$m = 32$

Step 3.5.3

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $360$ to find the solution in the fourth quadrant.

$m = 360 - 32$

Step 3.5.4

Subtract $32$ from $360$ .

$m = 328$

Step 3.5.5

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

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Step 3.5.5.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 3.5.5.2

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

$\frac{360}{| 1 |}$

Step 3.5.5.3

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

$\frac{360}{1}$

Step 3.5.5.4

Divide $360$ by $1$ .

$360$

Step 3.5.6

The period of the $\cos (m)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$m = 32 + 360 n, 328 + 360 n$ , for any integer $n$

Step 3.6

Solve for $m$ in $\cos (m) = - \sqrt{0.71918557}$ .

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Step 3.6.1

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

$m = \arccos (- \sqrt{0.71918557})$

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

Evaluate $\arccos (- \sqrt{0.71918557})$ .

$m = 148$

Step 3.6.3

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $360$ to find the solution in the third quadrant.

$m = 360 - 148$

Step 3.6.4

Subtract $148$ from $360$ .

$m = 212$

Step 3.6.5

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

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Step 3.6.5.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 3.6.5.2

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

$\frac{360}{| 1 |}$

Step 3.6.5.3

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

$\frac{360}{1}$

Step 3.6.5.4

Divide $360$ by $1$ .

$360$

Step 3.6.6

The period of the $\cos (m)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$m = 148 + 360 n, 212 + 360 n$ , for any integer $n$

Step 3.7

List all of the solutions.

$m = 32 + 360 n, 328 + 360 n, 148 + 360 n, 212 + 360 n$ , for any integer $n$

Step 3.8

Consolidate the solutions.

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Step 3.8.1

Consolidate $32 + 360 n$ and $212 + 360 n$ to $32 + 180 n$ .

$m = 32 + 180 n, 328 + 360 n, 148 + 360 n$ , for any integer $n$

Step 3.8.2

Consolidate $328 + 360 n$ and $148 + 360 n$ to $148 + 180 n$ .

$m = 32 + 180 n, 148 + 180 n$ , for any integer $n$