If $\theta=\frac{15 \pi}{4}$, then find exact values for the following:
$\sec (\theta)$ equals
$\csc (\theta)$ equals
$\tan (\theta)$ equals
$\cot (\theta)$ equals

Step 1 ：First, we need to find the equivalent angle of \(\frac{15 \pi}{4}\) in the interval \([0, 2\pi)\). This is because the trigonometric functions repeat every \(2\pi\). We can do this by taking the remainder when \(\frac{15 \pi}{4}\) is divided by \(2\pi\).

Step 2 ：Once we have the equivalent angle, we can find the exact values of the trigonometric functions. The secant function is the reciprocal of the cosine function, the cosecant function is the reciprocal of the sine function, the tangent function is the sine function divided by the cosine function, and the cotangent function is the cosine function divided by the sine function.

Step 3 ：The values of the trigonometric functions are not exact because they are approximations. However, we can see that the values are very close to the exact values. The secant and cosecant functions are approximately \(\sqrt{2}\) and \(-\sqrt{2}\), respectively. The tangent and cotangent functions are approximately \(-1\) and \(-1\), respectively.

Step 4 ：Final Answer: \(\sec (\theta) = \boxed{\sqrt{2}}\), \(\csc (\theta) = \boxed{-\sqrt{2}}\), \(\tan (\theta) = \boxed{-1}\), \(\cot (\theta) = \boxed{-1}\)