determine the exact value for sin[cos^-1(4/5)] if XE [0,2pi]
cos^-1(4/5)
tells me the angle is in either I or IV, since the cosine is positive in the first and fourth quadrants.
make a sketch in the first quadrant, right-angled triangle with hypotenuse 5 and base of 4 (r=5, x=4)
by Pythagoras the other side is 3, or y=3
so in the first quadrant sin[cos^-1(4/5)] = 3/5
and in the fourth quadrant sin[cos^-1(4/5)] = -3/5
OK great! thank you so much
To determine the exact value for sin[cos^-1(4/5)], it is helpful to understand the inverse trigonometric functions and the relationships between them.
In this case, we have cos^-1(4/5), which represents the angle whose cosine is equal to 4/5. Let's call this angle θ.
To find the value of sin[cos^-1(4/5)], we need to evaluate sin(θ).
To do this, we can use the Pythagorean identity: sin^2(θ) + cos^2(θ) = 1.
Since we already know the value of cos(θ) (which is 4/5), we can substitute it in the equation and solve for sin(θ).
sin^2(θ) + (4/5)^2 = 1
sin^2(θ) + 16/25 = 1
sin^2(θ) = 1 - 16/25
sin^2(θ) = 9/25
Taking the square root of both sides, we get:
sin(θ) = ±√(9/25)
sin(θ) = ±3/5
Since θ lies in the range [0, 2π], we need to determine the sign of sin(θ).
To do that, we go back to the definition of the inverse cosine function. Since the cosine value is positive (4/5), the angle θ must lie in the first or fourth quadrant.
In the first quadrant (0 to π/2), sin(θ) is positive (3/5). However, in the fourth quadrant (3π/2 to 2π), sin(θ) is negative (-3/5).
Since we need to find the value of sin(θ) in the given range, we take the positive value:
sin[cos^-1(4/5)] = 3/5.