Questions: Use the Pythagorean identity to find cos x. sin x = 24/25 cos x = [?]

Use the Pythagorean identity to find cos x.
sin x = 24/25
cos x = [?]
Transcript text: Use the Pythagorean identity to find $\cos x$. \[ \begin{array}{l} \sin x=\frac{24}{25} \\ \cos x=[?] \end{array} \]
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Solution

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Solution Steps

Step 1: Use the Pythagorean Identity

The Pythagorean identity states that for any angle x x :

sin2x+cos2x=1 \sin^2 x + \cos^2 x = 1

Given that sinx=2425 \sin x = \frac{24}{25} , we can substitute this value into the identity.

Step 2: Substitute the Given Value

Substitute sinx=2425 \sin x = \frac{24}{25} into the Pythagorean identity:

(2425)2+cos2x=1 \left(\frac{24}{25}\right)^2 + \cos^2 x = 1

Step 3: Solve for cos2x\cos^2 x

Calculate (2425)2\left(\frac{24}{25}\right)^2:

(2425)2=576625 \left(\frac{24}{25}\right)^2 = \frac{576}{625}

Substitute back into the equation:

576625+cos2x=1 \frac{576}{625} + \cos^2 x = 1

Subtract 576625\frac{576}{625} from both sides:

cos2x=1576625=625625576625=49625 \cos^2 x = 1 - \frac{576}{625} = \frac{625}{625} - \frac{576}{625} = \frac{49}{625}

Step 4: Solve for cosx\cos x

Take the square root of both sides to find cosx\cos x:

cosx=±49625=±725 \cos x = \pm \sqrt{\frac{49}{625}} = \pm \frac{7}{25}

Final Answer

Since the problem does not specify the quadrant, both positive and negative values are possible. Therefore, the solutions are:

cosx=725 or cosx=725 \boxed{\cos x = \frac{7}{25} \text{ or } \cos x = -\frac{7}{25}}

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