Evaluate and round your answer to the nearest hundredth. csc(-120o)
csc 120° = 1/sin 120°
sin 120° = sin 60° = √3/2
so, csc 120° = 2/√3
To evaluate csc(-120°), we need to use the unit circle and the definition of csc.
The unit circle is a circle with a radius of 1, centered at the origin (0,0) on the Cartesian plane. When we measure an angle counterclockwise from the positive x-axis, we can find the sine (sin) and cosecant (csc) values for that angle.
To find the csc value, we take the reciprocal of the sin value.
For -120°, we look at the corresponding angle on the unit circle. The point on the unit circle for -120° lies in the third quadrant, at an angle of 60° from the negative x-axis.
In the third quadrant, the sine value (sin) is negative. So, we find the sin value for 60° and take its reciprocal to get the csc value.
Since sin(60°) = √3/2, the reciprocal is 2/√3.
To get the csc value to the nearest hundredth, we can divide 2 by √3 and round the result.
Using a calculator, we find:
csc(-120°) ≈ 2 / √3 ≈ 1.1547
So, when rounded to the nearest hundredth, csc(-120°) ≈ 1.15.
To evaluate csc(-120°) and round the answer to the nearest hundredth, we need to understand the concept of the cosecant (csc) function.
The cosecant (csc) function is the reciprocal of the sine function. It is defined as:
csc(x) = 1 / sin(x)
To evaluate csc(-120°), we first need to find the value of sin(-120°). To do this, we use the unit circle and remember that sine is the y-coordinate of the point on the unit circle corresponding to the given angle.
In the unit circle, the angle -120° is in the third quadrant. In the third quadrant, the y-coordinate is negative, so sin(-120°) is negative.
To find sin(-120°), we can use the symmetry property of the unit circle. Since sin is an odd function, we have sin(-120°) = -sin(120°).
In the first quadrant, sin(120°) is equal to the square root of 3 divided by 2. So, sin(-120°) = -sqrt(3) / 2.
Now, we can find csc(-120°) by taking the reciprocal of sin(-120°):
csc(-120°) = 1 / ( -sqrt(3) / 2)
To evaluate this expression, we can multiply the numerator and denominator by 2:
csc(-120°) = 2 / ( -sqrt(3) )
Next, we can rationalize the denominator by multiplying the numerator and denominator by 1 / sqrt(3):
csc(-120°) = (2 / ( -sqrt(3) )) * (1 / sqrt(3))
This simplifies to:
csc(-120°) = -2 / 3sqrt(3)
To round the answer to the nearest hundredth, we can use a calculator to approximate the value:
csc(-120°) ≈ -0.384
So, the rounded answer to the nearest hundredth is approximately -0.38.