Find the values of x in the interval 0<=x<360 that satisfy the equation x=arcsin squareroot of 2/2. Express your answer in degrees.
360 degrees - 45=315 degrees.
Is this right?
x = arcsin (√2/2)
since √2/2 is positive and the sine of an angle is positive in the I and II quadrant
x = 45 degrees or 135 degrees.
sin 315 = - √2/2
Not quite. The angles that satisfy that equation are 45 and 135 degrees. the sine of 315 degrees is -sqrt2/2
The Sine function is positive in the first and second quadrants
Thanks
To solve the equation x = arcsin(sqrt(2)/2) and express the answer in degrees, you need to follow these steps:
1. Start by understanding the inverse sine function, also known as arcsin. The inverse sine function gives you the angle whose sine value is equal to a given value. In this case, we are looking for an angle whose sine value is sqrt(2)/2.
2. Remember that the sine function has a positive value of sqrt(2)/2 at both 45 degrees (π/4 radians) and 135 degrees (3π/4 radians) on the unit circle. Since we need to express the answer in the interval 0 <= x < 360 degrees, we will focus on the first quadrant where the angle lies between 0 and 90 degrees.
3. Set up the equation: x = arcsin(sqrt(2)/2). Since the inverse sine function only returns angles between -90 degrees (-π/2 radians) and 90 degrees (π/2 radians), we know that the angle we are looking for is in the first or second quadrant.
4. To find the value of x, we can take the inverse sine of sqrt(2)/2. In this case, arcsin(sqrt(2)/2) is equal to 45 degrees or π/4 radians. Therefore, one solution to the equation is x = 45 degrees.
5. Since we are looking for solutions in the interval 0 <= x < 360 degrees, we need to consider all possible solutions. In this case, we can add multiples of 360 degrees to the initial solution.
6. Adding multiples of 360 degrees to the initial solution, we get: 45 degrees + 360n, where n is any integer.
7. Therefore, the values of x in the interval 0 <= x < 360 that satisfy the equation x = arcsin(sqrt(2)/2) are: 45 degrees, 45 degrees + 360 degrees, 45 degrees + 2*360 degrees, and so on.
So, your initial answer of 315 degrees is incorrect. The correct values of x in the given interval are 45 degrees, 405 degrees, 765 degrees, and so on.