Answer to Question #91661 in Trigonometry for Haley Lucas

Question

Use the unit circle to find the value of sin 3(pi)/2 and cos 3(pi)/2

Answer 1

Solution.

(a)x²+y²=1

This is the equation of unit circle,where radius equals 1.

Let us consider the right triangle ABC.Hypotenuse AB is radius itself.As long as R=1,AB=1.

(b)

$$\sin\alpha=\frac{opposite}{hypotenuse}$$

$$\sin\alpha=\frac{BC}{AB}$$

BC is AB projection on y-axis,so that BC=y.

Then:

$$\sin\alpha=\frac{y}{1}$$ $$\sin\alpha=y$$

(c)

$$\cos\alpha=\frac{adjacent}{hypotenuse}$$

$$\cos\alpha=\frac{AC}{AB}$$

AC is AB projection on x-axis,so that AC=x.Then:

$$\cos\alpha=x$$

At the point where

$$\alpha=\frac{3\pi}{2}$$ $$x=0$$

According to this,

$$\cos\frac{3\pi}{2}=0$$

$$y=-1$$

And that is why

$$\sin\frac{3\pi}{2}=-1$$

Answer:

$$\sin\frac{3\pi}{2}=-1$$

$$\cos\frac{3\pi}{2}=0$$