Precalc question

Question

Answer 1

First, draw a picture.

The inequality tells us that the angle is in the first quadrant. The cos is 2/5, meaning that the hypotenuse is 2.5 times the shortest side (which is on the x axis.) Get the proportions on your drawing approximately correct, so you can visualize what the sine will be.

Then just plug the numbers into your calculator:

angle = arccos (2/5)

then take the sine of the angle

If your drawing is fairly accurate, you'll see that the sine should be approximately 0.9

Answer 2

We first set up a right triangle. Since the cosine of an angle is equal to the adjacent side divided by the hypotenuse (in relation to the angle theta), we set up a triangle this way. Using Pythagorean Theorem, we find the third side to be sqrt(21). This is because we have 5^2=2^2 + x^2 and solve for x. This is the length of the side of the triangle across from angle theta. Since sine of an angle equals the opposite side divided by the hypotenuse, we will have sqrt(21)/5.

Also, since we have bounds on the angle, we know that it is in Quadrant 1. So the sine should be positive (this means sqrt(21) must be positive).