find all real numbers that satisfy the equation: sinx= -\sqrt3/ 2
review your "standard angles."
sin π/3 = √3/2
sine is negative in QIII and QIV. That means that
sin(π+π/3) = sin(2π-π/3) = -√3/2
Now you can add any multiple of 2π to those values, and you get all real numbers as required.
If you draw a circle of radius r around the origin, and mark the point (x,y) on the circle, at an angle θ, measured from the positive x-axis,
sin θ = y/r
So, since r is always positive, you need y < 0 to have a negative sine.
To find all real numbers that satisfy the equation sin(x) = -√3/2, we need to find the values of x that make the sine of x equal to -√3/2.
Since the sine function has a periodicity of 2π, we can find the solution within the interval [0, 2π].
The reference angle for -√3/2 is π/3 radians, which is the angle in the first quadrant where sine is equal to -√3/2.
To find the solutions, we need to consider two cases:
Case 1: x is in the first and second quadrants.
In the first quadrant, the angle is π/3 radians. In the second quadrant, the angle is π - π/3 = 2π/3 radians.
So the solutions for this case are x = π/3 and x = 2π/3.
Case 2: x is in the third and fourth quadrants.
In the third quadrant, the angle is π + π/3 = 4π/3 radians. In the fourth quadrant, the angle is 2π - π/3 = 5π/3 radians.
So the solutions for this case are x = 4π/3 and x = 5π/3.
Therefore, the real numbers that satisfy the equation sin(x) = -√3/2 are x = π/3, x = 2π/3, x = 4π/3, and x = 5π/3, within the interval [0, 2π].
To find all real numbers that satisfy the equation sin(x) = -√3 / 2, we can use the unit circle and trigonometric ratios.
Step 1: Determine the angle
We know that sin(x) equals -√3 / 2. In the unit circle, this value represents two angles: 7π/6 and 11π/6.
Step 2: Find the general solutions
To find the general solutions, we need to add 2π to each angle we found in step 1.
For the angle 7π/6, the general solution is (7π/6) + 2πn, where n is an integer.
For the angle 11π/6, the general solution is (11π/6) + 2πn, where n is an integer.
Step 3: Simplify the general solutions
To simplify the general solutions, we need to write them in a more commonly used format.
For the angle 7π/6, the general solution can be simplified as follows:
7π/6 + 2πn = (12π + 7π) / 6 = 19π / 6
For the angle 11π/6, the general solution can be simplified as follows:
11π/6 + 2πn = (12π + 11π) / 6 = 23π / 6
Step 4: Finalize the solutions
Combining the simplified general solutions:
x = 19π/6, 23π/6, where n is an integer.
So, the real numbers that satisfy the equation sin(x) = -√3 / 2 are given by the solutions: x = 19π/6 + 2πn and x = 23π/6 + 2πn, where n is an integer.