Solve: sin^-1(cos 4pi/3)
usually you restrict yourself to the principal value, so that would be -π/6
To solve the equation sin^⁻1(cos(4π/3)), we need to find the angle whose cosine is equal to cos(4π/3).
First, let's find the reference angle for 4π/3:
Step 1: Reduce the angle to its smallest positive value in radians.
4π/3 = (4 * π/3) - (2π) = (4 - 6)π/3 = -2π/3
Step 2: Find the reference angle by taking the absolute value.
Reference angle = | -2π/3 | = 2π/3
Now that we have the reference angle, we can determine the angle whose cosine is equal to cos(4π/3). Since cosine is positive in the second quadrant, we can add the reference angle to π to find the solution:
Angle = π + 2π/3 = (3π/3) + (2π/3) = 5π/3
Therefore, sin^⁻1(cos(4π/3)) = sin^⁻1(cos(5π/3)) = 5π/3 (or in degrees, approximately 300°).
To solve the equation sin^(-1)(cos(4π/3)), we first need to understand the trigonometric relationship between sine and cosine functions.
The function sin^(-1), also denoted as arcsin or inverse sine, is the inverse function of the sine function. It means that if sin(x) = y, then sin^(-1)(y) = x.
Now, let's work on the given expression step by step:
1. Start with the angle 4π/3. Visualize this angle on the unit circle or refer to a trigonometric table to understand its position. The angle 4π/3 is located in the second quadrant of the unit circle.
2. We know that cosine (cos) represents the x-coordinate of a point on the unit circle, while sine (sin) represents the y-coordinate.
3. Determine the value of cos(4π/3). Since the angle is in the second quadrant, the x-coordinate (cos) will be negative. In the second quadrant, the value of cos(4π/3) is equal to -1/2.
4. We are now left with sin^(-1)(-1/2).
5. Rewriting sin^(-1)(-1/2) = x, we can convert it to sin(x) = -1/2.
6. Find the angle (x) whose sine is -1/2. This can be done by referring to a trigonometric table or by using a calculator's inverse sine function.
The solution for sin^(-1)(cos(4π/3)) is x = 7π/6 or -π/6.
Note: The inverse trigonometric functions have multiple solutions, and the given expression can have multiple valid answers.
cos (4 pi/3) = - cos pi/3 = -cos 60 degrees = - 1/2
could be 180+30 or 360-30
210 deg or 330 deg
that is pi + pi/6 = 7 pi/6
or
2 pi - pi/6 = 11 pi/6