Trigonometry Examples

$\frac{150}{400} = \cos (a)$

Step 1

Rewrite the equation as $\cos (a) = \frac{150}{400}$ .

$\cos (a) = \frac{150}{400}$

Step 2

Cancel the common factor of $150$ and $400$ .

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Step 2.1

Factor $50$ out of $150$ .

$\cos (a) = \frac{50 (3)}{400}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $50$ out of $400$ .

$\cos (a) = \frac{50 \cdot 3}{50 \cdot 8}$

Step 2.2.2

Cancel the common factor.

$\cos (a) = \frac{50 \cdot 3}{50 \cdot 8}$

Step 2.2.3

Rewrite the expression.

$\cos (a) = \frac{3}{8}$

Step 3

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

$a = \arccos (\frac{3}{8})$

Step 4

Simplify the right side.

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Step 4.1

Evaluate $\arccos (\frac{3}{8})$ .

$a = 1.18639955$

Step 5

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.

$a = 2 (3.14159265) - 1.18639955$

Step 6

Simplify $2 (3.14159265) - 1.18639955$ .

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Step 6.1

Multiply $2$ by $3.14159265$ .

$a = 6.2831853 - 1.18639955$

Step 6.2

Subtract $1.18639955$ from $6.2831853$ .

$a = 5.09678575$

Step 7

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

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Step 7.1

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

$\frac{2 π}{| b |}$

Step 7.2

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

$\frac{2 π}{| 1 |}$

Step 7.3

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

$\frac{2 π}{1}$

Step 7.4

Divide $2 π$ by $1$ .

$2 π$

Step 8

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

$a = 1.18639955 + 2 π n, 5.09678575 + 2 π n$ , for any integer $n$