Find the exact value of the trigonometric function at the given real number
Csc(-3pie/4)
Since csc is the reciprocal of sin, we can first find the value of sin(-3*pi/4) and then take its reciprocal.
To find the value of sin(-3*pi/4), we can use the fact that sin has symmetry about the y-axis. This means that sin(-theta) = -sin(theta) for any angle theta. Therefore, sin(-3*pi/4) = -sin(3*pi/4).
Since 3*pi/4 is in the second quadrant, we can use the unit circle to find its sine. The point on the unit circle that corresponds to 3*pi/4 is (-sqrt(2)/2, sqrt(2)/2), so sin(3*pi/4) = sqrt(2)/2.
Therefore, sin(-3*pi/4) = -sqrt(2)/2.
Taking the reciprocal, we have:
csc(-3*pi/4) = 1/sin(-3*pi/4) = 1/(-sqrt(2)/2) = -sqrt(2)/2.
To find the exact value of the trigonometric function csc(-3π/4), we need to use the definition of the csc function.
csc(x) = 1/sin(x)
In this case, x = -3π/4.
First, let's find the value of sin(-3π/4). To do this, we need to know the unit circle and the trigonometric values associated with -3π/4.
The angle -3π/4 is in the third quadrant, where the sine function is negative.
Looking at the unit circle, we see that the corresponding angle in the first quadrant for -3π/4 is π/4.
The sine function value at π/4 is 1/√2:
sin(π/4) = 1/√2
Since sin(-x) = -sin(x), we have:
sin(-3π/4) = -sin(3π/4)
Now we can find the value of csc(-3π/4):
csc(-3π/4) = 1/sin(-3π/4)
= 1/(-sin(3π/4))
= 1/(-1/√2)
= -√2
Therefore, the exact value of csc(-3π/4) is -√2.
To find the exact value of the trigonometric function csc(-3π/4), we need to recall the definitions and properties of the trigonometric functions.
The cosecant function (csc) is defined as the reciprocal of the sine function. In other words, csc(x) = 1/sin(x).
Now, let's determine the value of sin(-3π/4).
The sine function has the following properties:
- sin(-θ) = -sin(θ)
- sin(θ + 2π) = sin(θ)
Using these properties, we can simplify sin(-3π/4) as follows:
sin(-3π/4) = -sin(3π/4)
Since sin(π/4) = sin(45°) = 1/√2, we can substitute this value into the expression:
sin(-3π/4) = -sin(3π/4) = -1/√2
Now, let's find the value of csc(-3π/4), which is the reciprocal of sin(-3π/4).
csc(-3π/4) = 1/sin(-3π/4) = 1/(-1/√2) = -√2
Therefore, the exact value of csc(-3π/4) is -√2.