Trigonometry Examples

$87^{2} = 8^{2} + 5^{2} - 2 (8 \cdot 5) \cos (B)$

Step 1

Rewrite the equation as $8^{2} + 5^{2} - 2 \cdot 8 \cdot (5 \cos (B)) = 87^{2}$ .

$8^{2} + 5^{2} - 2 \cdot 8 \cdot (5 \cos (B)) = 87^{2}$

Step 2

Simplify $8^{2} + 5^{2} - 2 \cdot 8 \cdot (5 \cos (B))$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Raise $8$ to the power of $2$ .

$64 + 5^{2} - 2 \cdot 8 \cdot (5 \cos (B)) = 87^{2}$

Step 2.1.2

Raise $5$ to the power of $2$ .

$64 + 25 - 2 \cdot 8 \cdot (5 \cos (B)) = 87^{2}$

Step 2.1.3

Multiply $- 2$ by $8$ .

$64 + 25 - 16 \cdot (5 \cos (B)) = 87^{2}$

Step 2.1.4

Multiply $5$ by $- 16$ .

$64 + 25 - 80 \cos (B) = 87^{2}$

Step 2.2

Add $64$ and $25$ .

$89 - 80 \cos (B) = 87^{2}$

Step 3

Raise $87$ to the power of $2$ .

$89 - 80 \cos (B) = 7569$

Step 4

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

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Step 4.1

Subtract $89$ from both sides of the equation.

$- 80 \cos (B) = 7569 - 89$

Step 4.2

Subtract $89$ from $7569$ .

$- 80 \cos (B) = 7480$

Step 5

Divide each term in $- 80 \cos (B) = 7480$ by $- 80$ and simplify.

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Step 5.1

Divide each term in $- 80 \cos (B) = 7480$ by $- 80$ .

$\frac{- 80 \cos (B)}{- 80} = \frac{7480}{- 80}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 80$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 80 \cos (B)}{- 80} = \frac{7480}{- 80}$

Step 5.2.1.2

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

$\cos (B) = \frac{7480}{- 80}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Cancel the common factor of $7480$ and $- 80$ .

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Step 5.3.1.1

Factor $40$ out of $7480$ .

$\cos (B) = \frac{40 (187)}{- 80}$

Step 5.3.1.2

Cancel the common factors.

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Step 5.3.1.2.1

Factor $40$ out of $- 80$ .

$\cos (B) = \frac{40 \cdot 187}{40 \cdot - 2}$

Step 5.3.1.2.2

Cancel the common factor.

$\cos (B) = \frac{40 \cdot 187}{40 \cdot - 2}$

Step 5.3.1.2.3

Rewrite the expression.

$\cos (B) = \frac{187}{- 2}$

Step 5.3.2

Move the negative in front of the fraction.

$\cos (B) = - \frac{187}{2}$

Step 6

The range of cosine is $- 1 \leq y \leq 1$ . Since $- \frac{187}{2}$ does not fall in this range, there is no solution.

No solution