Questions: The terminal point P(x, y) determined by a real number t is given. Find sin t, cos t, and tan t. 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 $\tan t$. \[ \begin{array}{l} \sin t=\square \\ \cos t=\square \\ \tan t=\square \end{array} \]

Solution

Solution Steps

To find $\sin t$, $\cos t$, and $\tan t$ given the terminal point $P(x, y)$, we can use the following trigonometric definitions:

$\sin t = \frac{y}{r}$
$\cos t = \frac{x}{r}$
$\tan t = \frac{y}{x}$

where $r = \sqrt{x^2 + y^2}$ is the distance from the origin to the point $P(x, y)$.

Step 1: Calculate the distance $ r $

Given the terminal point $ P(x, y) $ with $ x = 3 $ and $ y = 4 $, we first calculate the distance $ r $ from the origin to the point $ P $ using the Pythagorean theorem: \[ r = \sqrt{x^2 + y^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5.0 \]

Step 2: Calculate $\sin t$

Using the definition $\sin t = \frac{y}{r}$, we find: \[ \sin t = \frac{4}{5.0} = 0.8000 \]

Step 3: Calculate $\cos t$

Using the definition $\cos t = \frac{x}{r}$, we find: \[ \cos t = \frac{3}{5.0} = 0.6000 \]

Step 4: Calculate $\tan t$

Using the definition $\tan t = \frac{y}{x}$, we find: \[ \tan t = \frac{4}{3} = 1.3333 \]

Final Answer

\[ \begin{array}{l} \sin t = \frac{4}{5} \\ \cos t = \frac{3}{5} \\ \tan t = \frac{4}{3} \end{array} \]

\boxed{ \begin{array}{l} \sin t = \frac{4}{5} \\ \cos t = \frac{3}{5} \\ \tan t = \frac{4}{3} \end{array} }

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