Let (5,3) be a point on the terminal side of Theta Find the exact values of cosTheta ,cscTheta , and tanTheta cosTheta = cscTheta = tanTheta =

To find the exact values of the trigonometric functions for the angle $\Theta$ with a point $(5,3)$ on its terminal side, we can use the following steps:<br /><br />1. Find the radius (hypotenuse) of the right triangle formed by the point $(5,3)$ and the origin $(0,0)$.<br /> - The radius can be found using the Pythagorean theorem: $r = \sqrt{x^2 + y^2} = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34}$.<br /><br />2. Find the cosine of $\Theta$.<br /> - The cosine of $\Theta$ is the ratio of the adjacent side to the hypotenuse: $\cos\Theta = \frac{x}{r} = \frac{5}{\sqrt{34}}$.<br /><br />3. Find the cosecant of $\Theta$.<br /> - The cosecant of $\Theta$ is the reciprocal of the sine of $\Theta$: $\csc\Theta = \frac{1}{\sin\Theta}$.<br /> - The sine of $\Theta$ is the ratio of the opposite side to the hypotenuse: $\sin\Theta = \frac{y}{r} = \frac{3}{\sqrt{34}}$.<br /> - Therefore, $\csc\Theta = \frac{1}{\sin\Theta} = \frac{\sqrt{34}}{3}$.<br /><br />4. Find the tangent of $\Theta$.<br /> - The tangent of $\Theta$ is the ratio of the opposite side to the adjacent side: $\tan\Theta = \frac{y}{x} = \frac{3}{5}$.<br /><br />In summary, the exact values of the trigonometric functions for the angle $\Theta$ with a point $(5,3)$ on its terminal side are:<br />- $\cos\Theta = \frac{5}{\sqrt{34}}$<br />- $\csc\Theta = \frac{\sqrt{34}}{3}$<br />- $\tan\Theta = \frac{3