Trigonometry Examples

$\cos (x) = \frac{10}{15}$

Step 1

Cancel the common factor of $10$ and $15$ .

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Step 1.1

Factor $5$ out of $10$ .

$\cos (x) = \frac{5 (2)}{15}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $5$ out of $15$ .

$\cos (x) = \frac{5 \cdot 2}{5 \cdot 3}$

Step 1.2.2

Cancel the common factor.

$\cos (x) = \frac{5 \cdot 2}{5 \cdot 3}$

Step 1.2.3

Rewrite the expression.

$\cos (x) = \frac{2}{3}$

Step 2

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

$x = \arccos (\frac{2}{3})$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arccos (\frac{2}{3})$ .

$x = 0.84106867$

Step 4

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.

$x = 2 (3.14159265) - 0.84106867$

Step 5

Simplify $2 (3.14159265) - 0.84106867$ .

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Step 5.1

Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 0.84106867$

Step 5.2

Subtract $0.84106867$ from $6.2831853$ .

$x = 5.44211663$

Step 6

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

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Step 6.1

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

$\frac{2 π}{| b |}$

Step 6.2

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

$\frac{2 π}{| 1 |}$

Step 6.3

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

$\frac{2 π}{1}$

Step 6.4

Divide $2 π$ by $1$ .

$2 π$

Step 7

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

$x = 0.84106867 + 2 π n, 5.44211663 + 2 π n$ , for any integer $n$