Trigonometry Examples

$\frac{45}{- 144} = \cos (y)$

Rewrite the equation as $\cos (y) = \frac{45}{- 144}$ .

$\cos (y) = \frac{45}{- 144}$

Simplify $\frac{45}{- 144}$ .

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Cancel the common factor of $45$ and $- 144$ .

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Factor $9$ out of $45$ .

$\cos (y) = \frac{9 (5)}{- 144}$

Cancel the common factors.

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Factor $9$ out of $- 144$ .

$\cos (y) = \frac{9 \cdot 5}{9 \cdot - 16}$

Cancel the common factor.

$\cos (y) = \frac{9 \cdot 5}{9 \cdot - 16}$

Rewrite the expression.

$\cos (y) = \frac{5}{- 16}$

Move the negative in front of the fraction.

$\cos (y) = - \frac{5}{16}$

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

$y = \arccos (- \frac{5}{16})$

Simplify the right side.

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Evaluate $\arccos (- \frac{5}{16})$ .

$y = 1.88862003$

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$y = 2 (3.14159265) - 1.88862003$

Simplify $2 (3.14159265) - 1.88862003$ .

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Multiply $2$ by $3.14159265$ .

$y = 6.2831853 - 1.88862003$

Subtract $1.88862003$ from $6.2831853$ .

$y = 4.39456527$

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

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 1 |}$

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

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

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

$y = 1.88862003 + 2 π n, 4.39456527 + 2 π n$ , for any integer $n$