Trigonometry Examples

$\cos (x) = \frac{20}{29}$ , $\frac{3 π}{2} < x < 2 π$

Step 1

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

$x = \arccos (\frac{20}{29})$

Step 2

Simplify the right side.

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Step 2.1

Evaluate $\arccos (\frac{20}{29})$ .

$x = 0.80978357$

Step 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.

$x = 2 (3.14159265) - 0.80978357$

Step 4

Solve for $x$ .

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Step 4.1

Remove parentheses.

$x = 2 (3.14159265) - 0.80978357$

Step 4.2

Simplify $2 (3.14159265) - 0.80978357$ .

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Step 4.2.1

Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 0.80978357$

Step 4.2.2

Subtract $0.80978357$ from $6.2831853$ .

$x = 5.47340173$

Step 5

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

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Step 5.1

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

$\frac{2 π}{| b |}$

Step 5.2

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

$\frac{2 π}{| 1 |}$

Step 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 5.4

Divide $2 π$ by $1$ .

$2 π$

Step 6

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

$x = 0.80978357 + 2 π n, 5.47340173 + 2 π n$ , for any integer $n$

Step 7

Plug in $0$ for $n$ and simplify to see if the solution is contained in $(\frac{3 π}{2}, 2 π)$ .

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Step 7.1

Plug in $0$ for $n$ .

$5.47340173 + 2 π (0)$

Step 7.2

Simplify.

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Step 7.2.1

Multiply $2 π (0)$ .

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Step 7.2.1.1

Multiply $0$ by $2$ .

$5.47340173 + 0 π$

Step 7.2.1.2

Multiply $0$ by $π$ .

$5.47340173 + 0$

Step 7.2.2

Add $5.47340173$ and $0$ .

$5.47340173$

Step 7.3

The interval $(\frac{3 π}{2}, 2 π)$ contains $5.47340173$ .

$x = 5.47340173$