2021-08-20

P(x, y) is the endpoint of the unit circle determined by t. Then .

joshyoung05M

According to the definition, $\mathrm{sin}t$ is equivalent to y. $\mathrm{cos}t$ is equivalent to x. $\mathrm{tan}t$ is equivalent to $\frac{y}{x}$
The unidentified terms are $x,y,\frac{y}{x}$.

Eliza Beth13

$\mathrm{sin}t=y$
$\mathrm{cos}t=x$
$\mathrm{tan}t=\frac{\mathrm{sin}t}{\mathrm{cos}t}$
Explanation:
Let's solve the problem step by step:
1. Sine of angle t ($\mathrm{sin}t$):
We know that the sine of an angle is defined as the y-coordinate of the corresponding point on the unit circle. Therefore, $\mathrm{sin}t=y$.
2. Cosine of angle t ($\mathrm{cos}t$):
Similarly, the cosine of an angle is defined as the x-coordinate of the corresponding point on the unit circle. Hence, $\mathrm{cos}t=x$.
3. Tangent of angle t ($\mathrm{tan}t$):
Tangent is the ratio of sine to cosine, so $\mathrm{tan}t=\frac{\mathrm{sin}t}{\mathrm{cos}t}$.

Let's assume that the angle t is measured counterclockwise from the positive x-axis to the line segment connecting the origin to the point P(x, y).
To find sin t, we can use the y-coordinate of the point P. Since P lies on the unit circle, the y-coordinate is equal to sin t.
Therefore, $\mathrm{sin}t=y$.
To find cos t, we can use the x-coordinate of the point P. Similarly, since P lies on the unit circle, the x-coordinate is equal to cos t.
Therefore, $\mathrm{cos}t=x$.
Finally, to find tan t, we can divide the y-coordinate by the x-coordinate, as tan t is defined as the ratio of sin t to cos t.
Therefore, $\mathrm{tan}t=\frac{y}{x}$.
These are the solutions for sin t, cos t, and tan t based on the given endpoint P(x, y) on the unit circle.

Mr Solver

Step 1:
To solve the problem, we need to find the values of $\mathrm{sin}t$, $\mathrm{cos}t$, and $\mathrm{tan}t$ given that $P\left(x,y\right)$ is the endpoint of the unit circle determined by $t$.
In a unit circle, the coordinates of a point on the circle can be expressed as $\left(\mathrm{cos}t,\mathrm{sin}t\right)$, where $t$ is the angle measured from the positive x-axis to the terminal side of the angle.
Step 2:
Therefore, in this case, we have $P\left(x,y\right)=\left(\mathrm{cos}t,\mathrm{sin}t\right)$.
Hence, we can determine the values of $\mathrm{sin}t$, $\mathrm{cos}t$, and $\mathrm{tan}t$ as follows:
$\mathrm{sin}t=y$
$\mathrm{cos}t=x$
$\mathrm{tan}t=\frac{\mathrm{sin}t}{\mathrm{cos}t}=\frac{y}{x}$
Thus, the values of $\mathrm{sin}t$, $\mathrm{cos}t$, and $\mathrm{tan}t$ for the endpoint $P\left(x,y\right)$ of the unit circle determined by $t$ are $\mathrm{sin}t=y$, $\mathrm{cos}t=x$, and $\mathrm{tan}t=\frac{y}{x}$.

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