Trigonometry Examples

$144 = w^{2} + 20^{2} - 2 w \cdot 20 \cdot 0.857$

Step 1

Rewrite the equation as $w^{2} + 20^{2} - 2 w \cdot 20 \cdot 0.857 = 144$ .

$w^{2} + 20^{2} - 2 w \cdot 20 \cdot 0.857 = 144$

Step 2

Simplify each term.

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Step 2.1

Raise $20$ to the power of $2$ .

$w^{2} + 400 - 2 w \cdot 20 \cdot 0.857 = 144$

Step 2.2

Multiply $20$ by $- 2$ .

$w^{2} + 400 - 40 w \cdot 0.857 = 144$

Step 2.3

Multiply $0.857$ by $- 40$ .

$w^{2} + 400 - 34.28 w = 144$

Step 3

Move all terms to the left side of the equation and simplify.

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Step 3.1

Subtract $144$ from both sides of the equation.

$w^{2} + 400 - 34.28 w - 144 = 0$

Step 3.2

Subtract $144$ from $400$ .

$w^{2} - 34.28 w + 256 = 0$

Step 4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5

Substitute the values $a = 1$ , $b = - 34.28$ , and $c = 256$ into the quadratic formula and solve for $w$ .

$\frac{34.28 \pm \sqrt{{(- 34.28)}^{2} - 4 \cdot (1 \cdot 256)}}{2 \cdot 1}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 34.28$ to the power of $2$ .

$w = \frac{34.28 \pm \sqrt{1175.1184 - 4 \cdot 1 \cdot 256}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot 256$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$w = \frac{34.28 \pm \sqrt{1175.1184 - 4 \cdot 256}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $256$ .

$w = \frac{34.28 \pm \sqrt{1175.1184 - 1024}}{2 \cdot 1}$

Step 6.1.3

Subtract $1024$ from $1175.1184$ .

$w = \frac{34.28 \pm \sqrt{151.1184}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$w = \frac{34.28 \pm \sqrt{151.1184}}{2}$

Step 7

The final answer is the combination of both solutions.

$w = 23.2865112, 10.99348879$