Question

sketch an angle θ in standard position such that θ has the least possible positive measure, and the point (3,√2) is on the terminal side of θ. then find the values of the six trigonometric functions for the angle. rationalize denominators if applicable. do not use a calculator. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. sin θ= (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) b. the function is undefined. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. cos θ= (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Find the radius $r$

For a point $(x,y)=(3,\sqrt{2})$ on the terminal - side of the angle $\theta$ in standard position, use the formula $r = \sqrt{x^{2}+y^{2}}$. So, $r=\sqrt{3^{2}+(\sqrt{2})^{2}}=\sqrt{9 + 2}=\sqrt{11}$.

Step2: Calculate $\sin\theta$

By the definition of the sine function $\sin\theta=\frac{y}{r}$. Substituting $y = \sqrt{2}$ and $r=\sqrt{11}$, we get $\sin\theta=\frac{\sqrt{2}}{\sqrt{11}}=\frac{\sqrt{22}}{11}$.

Step3: Calculate $\cos\theta$

By the definition of the cosine function $\cos\theta=\frac{x}{r}$. Substituting $x = 3$ and $r=\sqrt{11}$, we get $\cos\theta=\frac{3}{\sqrt{11}}=\frac{3\sqrt{11}}{11}$.

Step4: Calculate $\tan\theta$

By the definition of the tangent function $\tan\theta=\frac{y}{x}$. Substituting $x = 3$ and $y=\sqrt{2}$, we get $\tan\theta=\frac{\sqrt{2}}{3}$.

Step5: Calculate $\csc\theta$

Since $\csc\theta=\frac{r}{y}$, substituting $r=\sqrt{11}$ and $y=\sqrt{2}$, we get $\csc\theta=\frac{\sqrt{11}}{\sqrt{2}}=\frac{\sqrt{22}}{2}$.

Step6: Calculate $\sec\theta$

Since $\sec\theta=\frac{r}{x}$, substituting $r=\sqrt{11}$ and $x = 3$, we get $\sec\theta=\frac{\sqrt{11}}{3}$.

Step7: Calculate $\cot\theta$

Since $\cot\theta=\frac{x}{y}$, substituting $x = 3$ and $y=\sqrt{2}$, we get $\cot\theta=\frac{3}{\sqrt{2}}=\frac{3\sqrt{2}}{2}$.

For the sketch: The point $(3,\sqrt{2})$ is in the first - quadrant. The angle $\theta$ in standard position with the terminal - side passing through the point $(3,\sqrt{2})$ starts from the positive $x$ - axis and rotates counter - clockwise to the ray passing through $(3,\sqrt{2})$.

Answer:

A. $\sin\theta=\frac{\sqrt{22}}{11}$
A. $\cos\theta=\frac{3\sqrt{11}}{11}$
A. $\tan\theta=\frac{\sqrt{2}}{3}$
A. $\csc\theta=\frac{\sqrt{22}}{2}$
A. $\sec\theta=\frac{\sqrt{11}}{3}$
A. $\cot\theta=\frac{3\sqrt{2}}{2}$