hello, i don't quite understand how to do this type of problem:

1. find the exact value of sin (pi/3).

even more confusing is

2. find the exact value of csc^-1 (1).

i would appreciate any kind of help! thank you!

I normally like to replace pi with 180.

So...
sin(pi/3)
= sin(180/3)
= sin(60)
What is sin(60)? sqrt(3)/2

As for csc^-1(1), I like to think of this as its inverse, 1/sin^-1(1). Where in the unit circle is sin=1? That is, where is y=1? 90 degrees

Since they seem to want angles to be expressed in radians, it might be better to give pi/2 as the answer to csc^-1(1) . 90 degrees is also correct, of course.

Hello! I'd be happy to help you understand how to solve these types of problems.

1. To find the exact value of sin(pi/3), we can use the unit circle. The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It can help us find the trigonometric values (such as sine, cosine, etc.) for angles.

First, let's look at the angle pi/3. This angle corresponds to a point on the unit circle that is located in the first quadrant. If we draw a line from the origin to that point, it makes an angle of pi/3 with the positive x-axis.

The y-coordinate of this point on the unit circle gives us the value of sin(pi/3). So, we need to find the y-coordinate of this point.

Using the formula for the y-coordinate on the unit circle, we have:
y-coordinate = sin(angle) = sin(pi/3).

Next, we need to determine the coordinates of the point on the unit circle. One way to do this is by using the values of the special angles (30°, 45°, 60°, etc.) and their corresponding sin and cosine values.

The special angle closest to pi/3 is 60°, which corresponds to the point (1/2, √3/2) on the unit circle. Since pi/3 is smaller than 60°, the corresponding point will be somewhere between the positive x-axis and the point (1/2, √3/2).

So, the exact value of sin(pi/3) is equal to the y-coordinate of this point, which is √3/2.

2. The notation csc^-1 (1) is asking for the value of the inverse cosecant function of 1. The inverse cosecant function (csc^-1) is the function that gives us the angle whose cosecant is equal to a given value.

In this case, we are looking for the angle whose cosecant is equal to 1.

To solve this, we need to find the angle whose sine is equal to 1/1, which is 1. The sine function is the reciprocal of the cosecant function.

So, we can rewrite the problem as finding the angle whose sine is equal to 1/1. Based on our knowledge of the unit circle, we know that sine takes its maximum value of 1 at the angle π/2.

Therefore, the exact value of csc^-1 (1) is π/2.

I hope this explanation helps you understand how to solve these types of problems! Let me know if you have any further questions.