Question

question
the terminal ray of some angle \\(\theta\\) in standard position intersects a circle of radius 5 at point d with coordinates \\((p, -q)\\). write an expression for the sine of \\(\theta\\) and fill in the sentence to justify your answer below.
answer attempt 1 out of 2
\\(\sin(\theta) = \square\\) because sine is the ratio of point ds \\(\square\\) to its \\(
\

$$\begin{aligned} &\\text{horizontal displacement from the y-axis} \\\\ &\\text{vertical displacement from the x-axis} \\\\ &\\text{distance from the origin} \\end{aligned}$$

\\)
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Explanation:

Step1: Recall sine definition on unit circle

For a point $(x,y)$ on a circle of radius $r$, $\sin(\theta)=\frac{y}{r}$

Step2: Identify values from the problem

Here, $y=-q$, $r=5$

Step3: Substitute values into formula

$\sin(\theta)=\frac{-q}{5}$

Step4: Justify the ratio

Sine uses vertical displacement over radius.

Answer:

$\sin(\theta) = \frac{-q}{5}$ because sine is the ratio of Point D's vertical displacement from the x-axis to its distance from the origin