Find the exact value of each expression, if it exists:
the -1 are representing the inverse functions!
(a) sin -1 (-√2/2)
(b) cos−1 (−1)
(c) sin( sin−1 (π))
(d) cos−1(cos(−4π/ 3))
(e) tan−1 (tan(0.6))
(f) cos−1( cos13π/12))
(g) tan( sin−1 (8 /17))
(h) cot (cos−1 ( 5 /13))
these are what i have. I'm looking to make sure i have done them correctly. If i haven't if you could show me the work to the ones i got wrong that would be great!
a. -π/4
b. π
c. undefined because domain [-1,1]
d. -4π/3
e. .6
f. undefined because domain [0, π]
g. 8/15
h. 5/12
d: the arccos has principal vales from 0 to π.
cos(-4π/3) = -1/2
cos^-1(-1/2) = 2π/3
f:
cos(13π/12) = -cos(π/12) = -x
arccos(-x) = π-arccos(x) = 11π/12
The others look good. Be careful with the principal values stuff.
Let's go through each expression and find the exact value:
(a) sin^(-1)(-√2/2)
To find sin^(-1)(x), also known as arcsin(x), we need to find the angle whose sine is x. In this case, we have sin^(-1)(-√2/2). Looking at the unit circle, we see that this corresponds to an angle of -π/4. So the value is -π/4.
(b) cos^(-1)(-1)
The expression cos^(-1)(x), also known as arccos(x), represents the angle whose cosine is x. In this case, we have cos^(-1)(-1). Again, looking at the unit circle, we see that this corresponds to an angle of π. So the value is π.
(c) sin(sin^(-1)(π))
The expression sin(sin^(-1)(x)) simplifies to just x. In this case, we have sin(sin^(-1)(π)). Since the arcsin function only takes inputs between -1 and 1, and π is outside of this range, the expression is undefined.
(d) cos^(-1)(cos(-4π/3))
Similar to (c), the expression cos^(-1)(cos(x)) simplifies to just x. In this case, we have cos^(-1)(cos(-4π/3)). The cos function repeats every 2π, so we can add or subtract any multiple of 2π to the angle without changing its cosine. Therefore, -4π/3 is in the same position as 2π/3 on the unit circle, and cos(2π/3) = -1/2. So the value is -4π/3.
(e) tan^(-1)(tan(0.6))
The expression tan^(-1)(x), also known as arctan(x), represents the angle whose tangent is x. In this case, we have tan^(-1)(tan(0.6)). The tangent function also repeats every π, so we can add or subtract any multiple of π to the angle without changing its tangent. Therefore, 0.6 is in the same position as 0.6 - π, which is approximately 0.6 - 3.14 ≈ -2.54. So the value is approximately -2.54.
(f) cos^(-1)(cos(13π/12))
Similar to (d), the expression cos^(-1)(cos(x)) simplifies to just x. In this case, we have cos^(-1)(cos(13π/12)). The cos function repeats every 2π, so we can add or subtract any multiple of 2π to the angle without changing its cosine. Therefore, 13π/12 is in the same position as π/12 on the unit circle, and cos(π/12) = √3/2. So the value is √3/2.
(g) tan(sin^(-1)(8/17))
The expression sin^(-1)(x), also known as arcsin(x), represents the angle whose sine is x. In this case, we have sin^(-1)(8/17). The sine function only takes inputs between -1 and 1, and 8/17 is within this range. Therefore, we can find the angle whose sine is 8/17. Let's call this angle A. Then, tan(A) = (8/15)/(√(1 - (8/17)^2)). Calculating this, we find tan(A) ≈ 8/15. So the value is 8/15.
(h) cot(cos^(-1)(5/13))
The expression cos^(-1)(x), also known as arccos(x), represents the angle whose cosine is x. In this case, we have cos^(-1)(5/13). The cosine function only takes inputs between -1 and 1, and 5/13 is within this range. Therefore, we can find the angle whose cosine is 5/13. Let's call this angle B. Then, cot(B) = 1/(√(1 - (5/13)^2)). Calculating this, we find cot(B) = 5/12. So the value is 5/12.