Trigonometry Examples

${27.35}^{2} = {(21.71)}^{2} + {(17.96)}^{2} - 2 \cdot 21.71 \cdot 17.96 \cos (A)$

Step 1

Rewrite the equation as ${(21.71)}^{2} + {(17.96)}^{2} - 2 \cdot 21.71 \cdot (17.96 \cos (A)) = {27.35}^{2}$ .

${(21.71)}^{2} + {(17.96)}^{2} - 2 \cdot 21.71 \cdot (17.96 \cos (A)) = {27.35}^{2}$

Step 2

Simplify ${21.71}^{2} + {17.96}^{2} - 2 \cdot 21.71 \cdot (17.96 \cos (A))$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Raise $21.71$ to the power of $2$ .

$471.3241 + {17.96}^{2} - 2 \cdot 21.71 \cdot (17.96 \cos (A)) = {27.35}^{2}$

Step 2.1.2

Raise $17.96$ to the power of $2$ .

$471.3241 + 322.5616 - 2 \cdot 21.71 \cdot (17.96 \cos (A)) = {27.35}^{2}$

Step 2.1.3

Multiply $- 2$ by $21.71$ .

$471.3241 + 322.5616 - 43.42 \cdot (17.96 \cos (A)) = {27.35}^{2}$

Step 2.1.4

Multiply $17.96$ by $- 43.42$ .

$471.3241 + 322.5616 - 779.8232 \cos (A) = {27.35}^{2}$

Step 2.2

Add $471.3241$ and $322.5616$ .

$793.8857 - 779.8232 \cos (A) = {27.35}^{2}$

Step 3

Raise $27.35$ to the power of $2$ .

$793.8857 - 779.8232 \cos (A) = 748.0225$

Step 4

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

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Step 4.1

Subtract $793.8857$ from both sides of the equation.

$- 779.8232 \cos (A) = 748.0225 - 793.8857$

Step 4.2

Subtract $793.8857$ from $748.0225$ .

$- 779.8232 \cos (A) = - 45.8632$

Step 5

Divide each term in $- 779.8232 \cos (A) = - 45.8632$ by $- 779.8232$ and simplify.

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Step 5.1

Divide each term in $- 779.8232 \cos (A) = - 45.8632$ by $- 779.8232$ .

$\frac{- 779.8232 \cos (A)}{- 779.8232} = \frac{- 45.8632}{- 779.8232}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 779.8232$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 779.8232 \cos (A)}{- 779.8232} = \frac{- 45.8632}{- 779.8232}$

Step 5.2.1.2

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

$\cos (A) = \frac{- 45.8632}{- 779.8232}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $- 45.8632$ by $- 779.8232$ .

$\cos (A) = 0.0588123$

Step 6

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

$A = \arccos (0.0588123)$

Step 7

Simplify the right side.

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Step 7.1

Evaluate $\arccos (0.0588123)$ .

$A = 1.51195006$

Step 8

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.

$A = 2 (3.14159265) - 1.51195006$

Step 9

Simplify $2 (3.14159265) - 1.51195006$ .

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Step 9.1

Multiply $2$ by $3.14159265$ .

$A = 6.2831853 - 1.51195006$

Step 9.2

Subtract $1.51195006$ from $6.2831853$ .

$A = 4.77123524$

Step 10

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

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Step 10.1

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

$\frac{2 π}{| b |}$

Step 10.2

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

$\frac{2 π}{| 1 |}$

Step 10.3

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

$\frac{2 π}{1}$

Step 10.4

Divide $2 π$ by $1$ .

$2 π$

Step 11

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

$A = 1.51195006 + 2 π n, 4.77123524 + 2 π n$ , for any integer $n$