Let's begin by using the given information and applying the Pythagorean identity for trigonometric functions, $\cos^2 \theta + \sin^2 \theta = 1$, to find the values of $x$ and $y$.

Step 1: Use the identity to find $x$

You are given the point $(x, \frac{24}{25})$ on the unit circle, where the $y$-coordinate corresponds to $\sin \theta$ and the $x$-coordinate corresponds to $\cos \theta$. Since this point lies on the unit circle, it satisfies the equation:

$\cos^2 \theta + \sin^2 \theta = 1$

Substitute $\sin \theta = \frac{24}{25}$ into this identity:

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

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

$x^2 + \frac{576}{625} = 1$

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

$x^2 = 1 - \frac{576}{625} = \frac{625}{625} - \frac{576}{625} = \frac{49}{625}$

Take the square root of both sides:

$x = \pm \frac{7}{25}$

Thus, $x$ can be either $\frac{7}{25}$ or $-\frac{7}{25}$.

Step 2: Determine the circular function values for $s = (x, \frac{24}{25})$

Assume that $s = (x, \frac{24}{25})$ represents a point on the unit circle. For this point, we can now calculate the values of the six circular functions (sine, cosine, tangent, cotangent, secant, and cosecant).

1. Sine ($\sin \theta$): This is simply the $y$-coordinate of the point on the unit circle. $\sin \theta = \frac{24}{25}$

2. Cosine ($\cos \theta$): This is the $x$-coordinate of the point on the unit circle. As we found earlier: $\cos \theta = \pm \frac{7}{25}$

3. Tangent ($\tan \theta$): This is the ratio of sine to cosine. $\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{24}{25}}{\pm \frac{7}{25}} = \pm \frac{24}{7}$

4. Cotangent ($\cot \theta$): This is the reciprocal of tangent. $\cot \theta = \frac{1}{\tan \theta} = \pm \frac{7}{24}$

5. Secant ($\sec \theta$): This is the reciprocal of cosine. $\sec \theta = \frac{1}{\cos \theta} = \pm \frac{25}{7}$

6. Cosecant ($\csc \theta$): This is the reciprocal of sine. $\csc \theta = \frac{1}{\sin \theta} = \frac{25}{24}$

Final Values for the Circular Functions

• Sine: $\sin \theta = \frac{24}{25}$
• Cosine: $\cos \theta = \pm \frac{7}{25}$
• Tangent: $\tan \theta = \pm \frac{24}{7}$
• Cotangent: $\cot \theta = \pm \frac{7}{24}$
• Secant: $\sec \theta = \pm \frac{25}{7}$
• Cosecant: $\csc \theta = \frac{25}{24}$

Notes:

• The signs of the functions depend on the quadrant of the point $(x, \frac{24}{25})$ on the unit circle.
• If $x = \frac{7}{25}$, the point is in the first or fourth quadrant, where sine is positive and cosine is positive or negative.
• If $x = -\frac{7}{25}$, the point is in the second or third quadrant, where sine is positive and cosine is negative.

Would you like any further clarification or help with a specific part of this solution?

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Related Questions:

1. How do the signs of trigonometric functions vary across different quadrants on the unit circle?
2. Can you explain the unit circle and how trigonometric functions are derived from it?
3. What is the significance of the Pythagorean identity in trigonometry?
4. How do you compute the six trigonometric functions from a given point on the unit circle?
5. What happens if the point lies on the axes of the unit circle (like (1,0) or (0,1))?

Tip: Always remember to check the quadrant when working with trigonometric identities on the unit circle, as it directly affects the signs of the functions.