Evaluate please:

a.) arcsin(sin 13pi/16)
b.) arccos(cos[-pi/18])
c.) arcsin(sin pi/18
d.) arcsec(-√2)
e.) arctan(-√3/3)

a) isn't arcsin(sin(?)) the inverse of itself?

so 13π/16

b) and c) same thing, except by definition arcsin( ..) will give you the smallest positive angle

so arccos(cos(-π/18) = π/18

d) arcsec(-√2)
the angle Ø so that secØ = -√2
or cosØ = (-1/√2)
Ø = 135° or 3π/4

e) -π/6

To evaluate the given expressions, we'll use the properties of inverse trigonometric functions. Recall that inverse trigonometric functions "undo" the effect of their corresponding trigonometric functions.

a) arcsin(sin 13π/16):
To evaluate arcsin(sin 13π/16), we need to find an angle whose sine is equal to sin 13π/16. Since sin is positive in the second and third quadrants, we need to find an angle in those quadrants with the same sine value.

Step 1: Reduce the angle inside the sine function:
13π/16 = (8π/16) + (5π/16) = π/2 + 5π/16

Step 2: Identify an equivalent angle within the range of arcsin (between -π/2 and π/2):
To find an equivalent angle within the range, we can subtract multiples of 2π until we find an angle in the desired range.

π/2 + 5π/16 = π/2 + (40π/16) + (5π/16) = π/2 + 2π + (5π/16) = 9π/2 + (5π/16)

Step 3: Determine the equivalent angle within the range:
To obtain an equivalent angle within the range, subtract 2π until we get an angle between -π/2 and π/2.

9π/2 + (5π/16) - 2π = (9π/2 - 4π/2) + (5π/16) = (5π/2) + (5π/16) = (40π/16) + (5π/16) = 45π/16

So, arcsin(sin 13π/16) = 45π/16.

b) arccos(cos[-π/18]):
To evaluate arccos(cos[-π/18]), we need to find an angle whose cosine is equal to cos[-π/18]. Since cos is positive in the first and fourth quadrants, we need to find an angle in those quadrants with the same cosine value.

Step 1: Reduce the angle inside the cosine function:
[-π/18] = -[π/18] = -π/18

Step 2: Identify an equivalent angle within the range of arccos (between 0 and π):
To find an equivalent angle within the range, we can add multiples of 2π until we find an angle in the desired range.

-π/18 + 2π = (36π/18) + (36π/18) = 72π/18 = 4π

Step 3: Determine the equivalent angle within the range:
To get an equivalent angle within the range, subtract 2π until we get an angle between 0 and π.

4π - 2π = 2π

So, arccos(cos[-π/18]) = 2π.

c) arcsin(sin π/18):
To evaluate arcsin(sin π/18), we need to find an angle whose sine is equal to sin π/18. Since sin is positive in the first and second quadrants, we need to find an angle in those quadrants with the same sine value.

Step 1: Reduce the angle inside the sine function:
π/18

Step 2: Identify an equivalent angle within the range of arcsin (between -π/2 and π/2):
Since π/18 is already within the range, we don't need to make any adjustments.

So, arcsin(sin π/18) = π/18.

d) arcsec(-√2):
To evaluate arcsec(-√2), we need to find an angle whose secant is equal to -√2. Since sec is negative in the second and third quadrants, we need to find an angle in those quadrants with the same secant value.

Step 1: Calculate the cosine of the angle using the identity sec^2θ = 1/cos^2θ:
cos^2θ = 1/(-√2)^2 = 1/2
cosθ = ±√(1/2) = ±√2/2

Step 2: Determine the angles with the given cosine:
Since cos is negative in the second and third quadrants, the angle can be either in the second or the third quadrant. We need to find the equivalent angle with cosine equal to ±√2/2.

In the second quadrant, cosθ = -√2/2, which corresponds to an angle of 3π/4.
In the third quadrant, cosθ = √2/2, which corresponds to an angle of 5π/4.

So, arcsec(-√2) can be 3π/4 or 5π/4.

e) arctan(-√3/3):
To evaluate arctan(-√3/3), we need to find an angle whose tangent is equal to -√3/3.

Step 1: Calculate the angle using the inverse tangent function:
arctan(-√3/3) = -π/6

So, arctan(-√3/3) = -π/6.