how do i evaluate the function csc 11 pie/4 without using a calculator

Subtract 2 pi from (11/4) pi and you get (3/4)pi. That is in the second quadrant, where the sin and csc functions are positive.

The reference angle is
pi - (3/4)pi = pi/4
Thus we have
csc(11/4)pi = csc(3/4)pi = csc(pi/4)
= 1/sin(pi/4) = sqrt2
The sine is 1/sqrt2 because the reference angle is 45 degrees. You can get sine 45 from the Pythagorean theorem.

To evaluate the function csc(11π/4) without using a calculator, we need to understand the properties of the cosecant function and the angle 11π/4.

1. First, let's determine the reference angle: 11π/4 is equivalent to 8π/4 + 3π/4, which means it's in the second quadrant.
The reference angle for the second quadrant is π - the given angle's equivalent angle in the first quadrant.
Therefore, the reference angle is π - 3π/4= π/4.

2. The cosecant function is defined as the reciprocal of the sine function. So, we need to find the sine of the reference angle π/4.
The sine of π/4 is 1/√2 or √2/2 (if you rationalize the denominator).

3. Finally, we take the reciprocal of the sine to get the cosecant of the angle:
csc(11π/4) = 1/(√2/2) or 2/√2 (rationalized) or √2.

So, csc(11π/4) is equal to √2.

To evaluate the function csc(11π/4) without using a calculator, we need to recall the properties of the cosecant function.

The cosecant function (csc) is the reciprocal of the sine function (sin). So, csc(x) = 1/sin(x).

Let's start by finding the value of sin(11π/4). Here's how we can do it:

1. The angle 11π/4 is equivalent to 45 degrees or π/4 in radians. Dividing 11π/4 by 2π, we get 11/8, which tells us that 11π/4 completes fewer than three full rotations, or 2π.

2. We need to determine which quadrant 11π/4 lies in. Starting from the positive x-axis (right side), we rotate counterclockwise until we reach the angle 11π/4. Since 11π/4 is greater than 2π, it lies in the third quadrant.

3. In the third quadrant, the value of the sine function is negative. Specifically, sin(x) is negative in the third and fourth quadrants.

4. Since sin is negative in the third quadrant, we know that sin(11π/4) is negative.

Now, to find csc(11π/4), we can apply the reciprocal property:

csc(11π/4) = 1/sin(11π/4)

Since sin(11π/4) is negative, we have:

csc(11π/4) = 1/(-sin(11π/4))

So, to evaluate csc(11π/4) without a calculator, we simply find the reciprocal of sin(11π/4) and change the sign:

csc(11π/4) = -1/sin(11π/4)

Please note that it is always recommended to use a calculator or reference table for greater accuracy. However, understanding the concepts and properties of the trigonometric functions allows us to approximate values without using a calculator.