The point P=(20,-15) on the circle x^2+y^2=r^2 Is also on the terminal side of an angle B in standard position. Find sinTheta ,cosTheta ,tanTheta ,cscTheta sinTheta = (Type an integer or a simplified fraction.) cosTheta =square (Type an integer or a simplifed fraction.) tanTheta =square (Type an integer or a simplifee fraction.) cscTheta =square (lype an integer or a simplifed fraction) secTheta =square (Iype an integer or a simplifeed fraction.) cotTheta =square (Type aninteger or a simplifee fraction.)

To find the trigonometric functions of angle $\Theta$ where point $P=(20,-15)$ lies on the terminal side of $\Theta$, we first need to find the radius $r$ of the circle using the Pythagorean theorem, since $P$ is on the circle $x^2 + y^2 = r^2$. The radius $r$ is the distance from the origin to point $P$, which is the hypotenuse of a right triangle with legs of lengths 20 and 15.<br /><br />Using the Pythagorean theorem:<br />\[ r = \sqrt{20^2 + (-15)^2} = \sqrt{400 + 225} = \sqrt{625} = 25 \]<br /><br />Now, we can find the trigonometric functions:<br /><br />1. **Sine ($\sin\Theta$)**:<br />\[ \sin\Theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{-15}{25} = -\frac{3}{5} \]<br /><br />2. **Cosine ($\cos\Theta$)**:<br />\[ \cos\Theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{20}{25} = \frac{4}{5} \]<br /><br />3. **Tangent ($\tan\Theta$)**:<br />\[ \tan\Theta = \frac{\sin\Theta}{\cos\Theta} = \frac{-\frac{3}{5}}{\frac{4}{5}} = -\frac{3}{4} \]<br /><br />4. **Cosecant ($\csc\Theta$)**:<br />\[ \csc\Theta = \frac{1}{\sin\Theta} = \frac{1}{-\frac{3}{5}} = -\frac{5}{3} \]<br /><br />5. **Secant ($\sec\Theta$)**:<br />\[ \sec\Theta = \frac{1}{\cos\Theta} = \frac{1}{\frac{4}{5}} = \frac{5}{4} \]<br /><br />6. **Cotangent ($\cot\Theta$)**:<br />\[ \cot\Theta = \frac{1}{\tan\Theta} = \frac{1}{-\frac{3}{4}} = -\frac{4}{3} \]<br /><br />So, the values are:<br />\[ \sin\Theta = -\frac{3}{5} \]<br />\[ \cos\Theta = \frac{4}{5} \]<br />\[ \tan\Theta = -\frac{3}{4} \]<br />\[ \csc\Theta = -\frac{5}{3} \]<br />\[ \sec\Theta = \frac{5}{4} \]<br />\[ \cot\Theta = -\frac{4}{3} \]