1. cos^-1((-sqrt3)/2)
5pi/6
2. arccos(cos7pi/2)
-pi/2
3. sin^-1(sin7pi/4)
-pi/4
4. csc(arctan11/60)
61/11
1. ok
#2
why are you giving the negative result?
cos 7π/2 = 0
so we need cos^-1 (0) = π/2
see: http://www.wolframalpha.com/input/?i=arccos%28cos%287pi%2F2%29%29+
#4. arctan (11/60)
construct a right-angled triangle with opposite as 60 and adjacent as 11
r^2 = 11^2 + 60^2 = 3721
r = √3720
so sin(arctan (11/60)) = 11/√3720
then csc(arctan (11/60)) = √3720/11
For #4, why did you take the square root of 3720 when r^2 = 3721?
To find the answers to these trigonometric expressions, you will be using the inverse trigonometric functions. Here's how you can calculate each result:
1. cos^-1((-sqrt3)/2):
The inverse cosine function, denoted as cos^(-1) or acos, gives you the angle whose cosine is the given value. In this case, the cosine of the angle is (-sqrt3)/2. To find the angle, evaluate the inverse cosine function of (-sqrt3)/2:
cos^(-1)(-sqrt3/2)
The answer is 5π/6.
2. arccos(cos(7π/2)):
Here, you have the cosine of an angle as input and need to find that angle. But before that, simplify the expression inside the arccos function using the properties of cosine:
cos(7π/2) = cos(3π/2 + 2π)
Since the cosine function is periodic with a period of 2π, you are looking for the angle within the same period that has the same cosine value. For (3π/2 + 2π), this corresponds to an angle of (-π/2).
Therefore, the answer is -π/2.
3. sin^(-1)(sin(7π/4)):
Similar to the previous question, simplify the expression inside the arcsin function first:
sin(7π/4) = sin(5π/4 + 2π)
Since the sine function is periodic with a period of 2π, you are looking for the angle within the same period that has the same sine value. For (5π/4 + 2π), this corresponds to an angle of (-π/4).
Therefore, the answer is -π/4.
4. csc(arctan(11/60)):
In this case, you have to evaluate the expression inside the arcsine function first:
arctan(11/60) = θ (let this be the angle)
Now, compute the cosecant of θ:
csc(θ) = 1/sin(θ)
Since the sine of an angle in the first and fourth quadrants is positive, you need to find the equivalent angle within the first quadrant where the sine value is positive:
sin(θ) = sin(θ + 2π)
For this angle, the sine value is 11/61. Therefore:
csc(arctan(11/60)) = 1/(11/61) = 61/11.
The results of the calculations are as follows:
1. cos^-1((-sqrt3)/2) = 5π/6
2. arccos(cos(7π/2)) = -π/2
3. sin^-1(sin(7π/4)) = -π/4
4. csc(arctan(11/60)) = 61/11