Question

2-82. examine the diagram at right. if the y - coordinate of p is 0.8, answer the following questions. homework help a. what does sin(θ) equal? b. what does cos(θ) equal?

Explanation:

Step1: Recall the definition of sine in a unit - circle

In a unit - circle with radius $r = 1$, for a point $P(x,y)$ on the circle and the angle $\theta$ measured counter - clockwise from the positive $x$ - axis, $\sin(\theta)=y$.
Since $y = 0.8$, $\sin(\theta)=0.8$.

Step2: Use the Pythagorean identity

The Pythagorean identity for trigonometric functions is $\sin^{2}\theta+\cos^{2}\theta = 1$.
We know that $\sin(\theta)=0.8$, so $\cos^{2}\theta=1 - \sin^{2}\theta$.
Substitute $\sin(\theta)=0.8$ into the identity: $\cos^{2}\theta=1-(0.8)^{2}=1 - 0.64 = 0.36$.
Then $\cos\theta=\pm\sqrt{0.36}=\pm0.6$.
Since the point $P$ has a positive $y$ - coordinate, it is in either the first or second quadrant. In the first quadrant $\cos\theta>0$, and in the second quadrant $\cos\theta<0$. Without further information about the angle $\theta$ (to determine the quadrant precisely), we have two possible values for $\cos\theta$.

Answer:

a. $\sin(\theta)=0.8$
b. $\cos(\theta)=\pm0.6$