solve for exact solutions over the interval of 0, 2pi by first solving the trigonometric function 6 sin x +12=9
6sinx + 12 = 9
6sinx = -3
sinx = -1/2
your reference angle is π/6
so place that angle in QII, QIV where sinx < 0
To solve the equation 6 sin(x) + 12 = 9 over the interval 0 to 2π, we need to isolate the sine function on one side of the equation. Here are the steps:
1. Subtract 12 from both sides of the equation:
6 sin(x) = 9 - 12
6 sin(x) = -3
2. Divide both sides of the equation by 6:
sin(x) = -3/6
sin(x) = -1/2
Now, we need to find the values of x where sin(x) equals -1/2 over the interval 0 to 2π.
To find the solutions, we can use the unit circle or refer to common values of sin(x) for specific angles. The angles where sin(x) equals -1/2 are:
- π/6 (30 degrees) and 7π/6 (210 degrees).
Note that the unit circle repeats after 2π, so we need to check if there are any more solutions within the given interval. We can do this by adding or subtracting 2π to the angles we found.
Adding 2π to each angle gives us:
- π/6 + 2π = 11π/6 and 7π/6 + 2π = 19π/6
Subtracting 2π from each angle gives us:
- π/6 - 2π = -11π/6 and 7π/6 - 2π = -17π/6
Now we have all the solutions within the interval 0 to 2π, which are:
- π/6, 7π/6, 11π/6, 19π/6, -11π/6, and -17π/6.
To solve the trigonometric equation 6 sin x + 12 = 9 over the interval 0 to 2π, we need to isolate the sin x term and then solve for x.
Step 1: Begin by subtracting 12 from both sides of the equation:
6 sin x + 12 - 12 = 9 - 12
6 sin x = -3
Step 2: Divide both sides of the equation by 6:
(6 sin x)/6 = (-3)/6
sin x = -1/2
Step 3: Now, we need to find the values of x for which sin x is equal to -1/2 within the given interval of 0 to 2π.
We can start by looking at the unit circle, which represents the values of sin x and cos x for different angles.
We can see that sin x = -1/2 occurs at two angles in the unit circle: π/6 (30 degrees) and 5π/6 (150 degrees).
However, since the interval is specified as 0 to 2π, we need to consider the angles within this range.
The angle π/6 (30 degrees) falls within the specified range.
To find the other angle within the given interval, we can use the symmetry of the unit circle. Since sin x = -1/2, we can look for the second angle with the same sin value but in the third quadrant.
To find this angle, we subtract the angle from π (180 degrees).
π - π/6 = 6π/6 - π/6 = 5π/6 (150 degrees)
So, the exact solutions to the equation 6 sin x + 12 = 9 over the interval 0 to 2π are x = π/6 and x = 5π/6.