Question

pretest unit 4
given the diagram below, which of the following ordered pairs represents the vertex, a?
diagram: triangle with c at origin (intersection of x-axis and y-axis), a with a vertical dashed line (right angle) to x-axis, b on x-axis. diagram not drawn to scale.
options:
a. $(a\cos c, b\sin c)$
b. $(b\cos c, b\sin c)$
c. $(a\cos b, a\sin b)$
d. $(b\cos c, c\sin c)$

Explanation:

Step1: Recall Trigonometric Ratios

In a right - triangle (the dashed line from A to the x - axis forms a right triangle with side AC and the x - axis), for an angle \(C\) (at point C), if we consider the length of the adjacent side to angle \(C\) (along the x - axis) and the opposite side (vertical side from the x - axis to A) and the hypotenuse (length \(b\) maybe, assuming the length of AC is \(b\)).

We know that in a right - triangle, \(\cos C=\frac{\text{adjacent}}{\text{hypotenuse}}\) and \(\sin C = \frac{\text{opposite}}{\text{hypotenuse}}\). Let the hypotenuse (length of AC) be \(b\).

Step2: Find x - coordinate of A

The x - coordinate of point A is the length of the adjacent side to angle \(C\) in the right - triangle. Using the cosine ratio \(\cos C=\frac{x}{b}\) (where \(x\) is the x - coordinate of A and \(b\) is the length of AC), we get \(x = b\cos C\).

Step3: Find y - coordinate of A

The y - coordinate of point A is the length of the opposite side to angle \(C\) in the right - triangle. Using the sine ratio \(\sin C=\frac{y}{b}\) (where \(y\) is the y - coordinate of A and \(b\) is the length of AC), we get \(y = b\sin C\).

So the ordered pair representing point A is \((b\cos C,b\sin C)\).

Answer:

B. \((b\cos C, b\sin C)\)