Write the expression and evaluate. There is only one answer whIch should match the range of the inverse trig function.
Sec^-1(-2)
Sec^-1(-sqrt2)
Tan^-1(-sqrt3)
Tan^-1(sqrt3)
I will do the 2nd one:
sec^-1 (-√2) = Ø , such that sec Ø = -√2
or
cos Ø = -1/√2
I know that cos 45° = 1/√2
but the cosine is negative in II and III
so Ø = 180-45° = 135°
or Ø = 180+45 = 225°
They probably expect the answer of 135° or 3π/4 radians
How did you come up with 135 being the answer? I understand what you did but you said it could also be 225, so what makes you pick 135?
Also doesnt the cos45 = sqrt2/2 not 1/sqrt 2?
yes, because sqrt2/2 = 1/sqrt2
To evaluate expressions involving inverse trigonometric functions, we'll use the following steps:
1. Identify the inverse trigonometric function: In this case, we have sec^-1 and tan^-1.
2. Recall the ranges of inverse trigonometric functions:
- sec^-1 (arcsec) has a range [0, π] and [π, 2π]
- tan^-1 (arctan) has a range (-π/2, π/2)
3. Determine the angle that corresponds to the given expression:
a) sec^-1(-2):
The range of sec^-1 is [0, π] and [π, 2π], so we need to find the angle whose secant is -2 within that range. Using the reciprocal identity for secant and cosine, we have:
sec^-1(-2) = cos^-1(1 / -2) = cos^-1(-1/2)
The angle with a cosine value of -1/2 lies in the second quadrant, so it's π/3 or 120 degrees.
b) sec^-1(-√2):
Following the same steps, we have:
sec^-1(-√2) = cos^-1(1 / -√2) = cos^-1(-1/√2)
The angle with a cosine value of -1/√2 lies in the third quadrant, so it's (2π - π/4) or 7π/4 radians.
c) tan^-1(-√3):
The range of tan^-1 is (-π/2, π/2), so we need to find the angle whose tangent is -√3 within that range. Using the tangent identity, we have:
tan^-1(-√3) = tan^-1(√3) + π
The angle with a tangent value of √3 lies in the second quadrant, so it's (π/3 + π) or 4π/3 radians.
d) tan^-1(√3):
Following the same steps, we have:
tan^-1(√3) = tan^-1(√3)
The angle with a tangent value of √3 lies in the first quadrant, so it's (π/3) or π/3 radians.
Hence, the evaluated expressions are:
a) sec^-1(-2) = π/3 or 120 degrees
b) sec^-1(-√2) = 7π/4 radians
c) tan^-1(-√3) = 4π/3 radians
d) tan^-1(√3) = π/3 radians