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

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

Solution

Step 1: Use the Pythagorean Identity

The Pythagorean identity states that for any angle $ x $:

\[ \sin^2 x + \cos^2 x = 1 \]

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

Step 2: Substitute the Given Value

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

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

Step 3: Solve for $\cos^2 x$

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

\[ \left(\frac{24}{25}\right)^2 = \frac{576}{625} \]

Substitute back into the equation:

\[ \frac{576}{625} + \cos^2 x = 1 \]

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

\[ \cos^2 x = 1 - \frac{576}{625} = \frac{625}{625} - \frac{576}{625} = \frac{49}{625} \]

Step 4: Solve for $\cos x$

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

\[ \cos x = \pm \sqrt{\frac{49}{625}} = \pm \frac{7}{25} \]

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

\[ \boxed{\cos x = \frac{7}{25} \text{ or } \cos x = -\frac{7}{25}} \]

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