Trigonometry Examples

$64 = 144 + 49 - 168 \cos (x)$

Step 1

Rewrite the equation as $144 + 49 - 168 \cos (x) = 64$ .

$144 + 49 - 168 \cos (x) = 64$

Step 2

Add $144$ and $49$ .

$193 - 168 \cos (x) = 64$

Step 3

Move all terms not containing $\cos (x)$ to the right side of the equation.

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Step 3.1

Subtract $193$ from both sides of the equation.

$- 168 \cos (x) = 64 - 193$

Step 3.2

Subtract $193$ from $64$ .

$- 168 \cos (x) = - 129$

Step 4

Divide each term in $- 168 \cos (x) = - 129$ by $- 168$ and simplify.

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Step 4.1

Divide each term in $- 168 \cos (x) = - 129$ by $- 168$ .

$\frac{- 168 \cos (x)}{- 168} = \frac{- 129}{- 168}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 168$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 168 \cos (x)}{- 168} = \frac{- 129}{- 168}$

Step 4.2.1.2

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

$\cos (x) = \frac{- 129}{- 168}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Cancel the common factor of $- 129$ and $- 168$ .

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Step 4.3.1.1

Factor $- 3$ out of $- 129$ .

$\cos (x) = \frac{- 3 (43)}{- 168}$

Step 4.3.1.2

Cancel the common factors.

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Step 4.3.1.2.1

Factor $- 3$ out of $- 168$ .

$\cos (x) = \frac{- 3 \cdot 43}{- 3 \cdot 56}$

Step 4.3.1.2.2

Cancel the common factor.

$\cos (x) = \frac{- 3 \cdot 43}{- 3 \cdot 56}$

Step 4.3.1.2.3

Rewrite the expression.

$\cos (x) = \frac{43}{56}$

Step 5

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

$x = \arccos (\frac{43}{56})$

Step 6

Simplify the right side.

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Step 6.1

Evaluate $\arccos (\frac{43}{56})$ .

$x = 0.69530688$

Step 7

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.

$x = 2 (3.14159265) - 0.69530688$

Step 8

Simplify $2 (3.14159265) - 0.69530688$ .

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Step 8.1

Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 0.69530688$

Step 8.2

Subtract $0.69530688$ from $6.2831853$ .

$x = 5.58787842$

Step 9

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

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Step 9.1

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

$\frac{2 π}{| b |}$

Step 9.2

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

$\frac{2 π}{| 1 |}$

Step 9.3

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

$\frac{2 π}{1}$

Step 9.4

Divide $2 π$ by $1$ .

$2 π$

Step 10

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

$x = 0.69530688 + 2 π n, 5.58787842 + 2 π n$ , for any integer $n$