Question

1. (12 points) the arc shown below is a portion of the unit circle, ( x^{2}+y^{2}=1 ). express the following quantities in terms of ( \theta ). (a) ( oa ) (b) ( ab )

Explanation:

Step1: Recall the unit - circle property

For the unit circle \(x^{2}+y^{2}=1\), the radius \(r = 1\). In right - triangle \(OAB\), \(\cos\theta=\frac{OA}{OB}\).

Step2: Find \(OA\)

Since \(OB = 1\) (radius of the unit circle), from \(\cos\theta=\frac{OA}{OB}\), we can substitute \(OB = 1\) into the formula. So \(OA=\cos\theta\).

Step3: Find \(AB\)

In right - triangle \(OAB\), \(\sin\theta=\frac{AB}{OB}\). Substituting \(OB = 1\) (radius of the unit circle) into the formula \(\sin\theta=\frac{AB}{OB}\), we get \(AB=\sin\theta\).

Answer:

(a) \(OA=\cos\theta\)
(b) \(AB = \sin\theta\)