cos x = -0.6 , 0 <= theta <= 2pi
Use inverse trigonometric functions to find a solution to the equation in the given interval. Then find all other solutions to the equation on this interval.
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To solve the equation cos(x) = -0.6 in the given interval 0 <= theta <= 2pi, you can use the inverse trigonometric function, specifically the arccosine function denoted as acos().
1. Find the initial solution using arccosine:
To find the initial solution, plug in -0.6 into the arccosine function:
x = acos(-0.6)
Use a calculator or a trigonometric table to find the arccosine of -0.6. The result, let's say x_initial, will be in radians.
2. Determine if there are other solutions within the given interval:
Since the cosine function has a periodicity of 2pi, there may be more solutions within the given interval. To find them, consider the relationship between cosine and the unit circle:
- The cosine function is positive for an angle in Quadrant I (0 to pi/2) and Quadrant IV (3pi/2 to 2pi).
- The cosine function is negative for an angle in Quadrant II (pi/2 to pi) and Quadrant III (pi to 3pi/2).
Therefore, we need to determine if there are additional solutions in Quadrants II and III that fall within the given interval (0 to 2pi).
3. Find additional solutions in Quadrant II:
In Quadrant II, the angle is between pi/2 and pi, and the cosine is negative. So, find the second solution within this range using arccosine:
x_2 = pi - x_initial
4. Find additional solutions in Quadrant III:
In Quadrant III, the angle is between pi and 3pi/2, and the cosine is negative. So, find the third solution within this range using arccosine:
x_3 = pi + x_initial
5. Check for solutions in the interval (0 to 2pi):
Now, check if the initial solution (x_initial), second solution (x_2), and third solution (x_3) fall within the interval (0 to 2pi). Adjust them if necessary.
- If x_initial is within the interval (0 to 2pi), then x = x_initial is a valid solution.
- If x_2 is within the interval (0 to 2pi), then x = x_2 is a valid solution.
- If x_3 is within the interval (0 to 2pi), then x = x_3 is a valid solution.
After checking, you will have the solutions to the equation cos(x) = -0.6 within the given interval.