Questions: The terminal point P(x, y) determined by a real number t is given. Find sin(t), cos(t), and tan(t). (-2/5, sqrt(21)/5) sin(t)= cos(t)= tan(t)=

Transcript text: The terminal point $P(x, y)$ determined by a real number $t$ is given. Find $\sin (t), \cos (t)$, and $\tan (t)$. \[ \begin{array}{l} \left(-\frac{2}{5}, \frac{\sqrt{21}}{5}\right) \\ \sin (t)=\square \\ \cos (t)=\square \\ \tan (t)=\square \end{array} \]

Solution

Solution Steps

To find $\sin(t)$, $\cos(t)$, and $\tan(t)$ given the terminal point $P(x, y)$, we use the definitions of these trigonometric functions in terms of the coordinates of the point on the unit circle. Specifically, $\sin(t) = y$, $\cos(t) = x$, and $\tan(t) = \frac{y}{x}$.

Step 1: Determine $\sin(t)$

Given the terminal point $P\left(-\frac{2}{5}, \frac{\sqrt{21}}{5}\right)$, we find $\sin(t)$ using the $y$-coordinate: \[ \sin(t) = y = \frac{\sqrt{21}}{5} \approx 0.9165 \]

Step 2: Determine $\cos(t)$

Next, we find $\cos(t)$ using the $x$-coordinate: \[ \cos(t) = x = -\frac{2}{5} = -0.4 \]

Step 3: Determine $\tan(t)$

Finally, we calculate $\tan(t)$ using the ratio of $y$ to $x$: \[ \tan(t) = \frac{y}{x} = \frac{\frac{\sqrt{21}}{5}}{-\frac{2}{5}} = -\frac{\sqrt{21}}{2} \approx -2.2913 \]

Final Answer

\[ \sin(t) \approx 0.9165, \quad \cos(t) = -0.4, \quad \tan(t) \approx -2.2913 \] Thus, the final answers are: \[ \boxed{\sin(t) \approx 0.9165, \quad \cos(t) = -0.4, \quad \tan(t) \approx -2.2913} \]

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