Question

solve for x. do not convert your answer to decimal form and rationalize the denominator if necessary. draw show your work here hint: to add the square root symbol (√□), type
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Explanation:

Step1: Identify the trigonometric ratio

Since we know an angle (\(45^{\circ}\)) and the adjacent side (\(20\)) and we want to find the hypotenuse (\(x\)), we use the cosine ratio. The cosine of an angle in a right - triangle is defined as \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). So, \(\cos45^{\circ}=\frac{20}{x}\).

Step2: Substitute the value of \(\cos45^{\circ}\)

We know that \(\cos45^{\circ}=\frac{\sqrt{2}}{2}\). Substituting this into the equation \(\frac{\sqrt{2}}{2}=\frac{20}{x}\).

Step3: Solve for \(x\)

Cross - multiply to get \(x\sqrt{2}=20\times2 = 40\). Then \(x=\frac{40}{\sqrt{2}}\).

Step4: Rationalize the denominator

Multiply the numerator and denominator by \(\sqrt{2}\). So \(x=\frac{40\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{40\sqrt{2}}{2}=20\sqrt{2}\).

Answer:

\(20\sqrt{2}\)