Solve the following equation for all solutions: sin(6x)cos(11x)-cos(6x)sin(11x)=-.1
Recall that sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B)
Thus,
sin(6x)cos(11x) - cos(6x)sin(11x) = -0.1
sin(6x - 11x) = -0.1
sin(-5x) = -0.1
-sin(5x) = -0.1
sin(5x) = 0.1
5x = sin^-1 (0.1)
x = [ sin^-1 (0.1) ] / 5
Well, well, well, looks like we have some trigonometry here! Let's see if we can unravel this equation together, shall we?
First, we notice that we have a difference of sines on one side and a difference of cosines on the other. This immediately rings a bell, reminding us of the sine of a difference formula and the cosine of a difference formula. So, let's apply those:
sin(a - b) = sin(a)cos(b) - cos(a)sin(b)
cos(a - b) = cos(a)cos(b) + sin(a)sin(b)
Now, we see that our equation can be rewritten as:
sin(6x - 11x) = -0.1
Simplifying further:
sin(-5x) = -0.1
Now, we just need to find the solutions for -5x. To do that, we want to find the arcsine of -0.1. However, my dear friend, I regret to inform you that the arcsine of -0.1 is not a "sine-ful" number. In fact, it falls outside the range of -1 to 1. Therefore, there are no real solutions to this equation.
In conclusion, this equa-sin won't lead us to any solutions. But hey, at least we had a good laugh along the way, right? Keep those math questions coming, and I'll be here to sprinkle some more humor in your life!
To solve the equation sin(6x)cos(11x)-cos(6x)sin(11x)=-.1, we can use the trigonometric identity for the sine of a difference of angles:
sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
By comparing this identity with the equation, we can see that A = 11x and B = 6x. Therefore, we can rewrite the equation as:
sin(11x - 6x) = -0.1
simplifying the left side gives:
sin(5x) = -0.1
To solve for x, we need to find the values of x that satisfy this equation.
Step 1: Finding the principal value:
The principal value of sin(5x) = -0.1 is between -1 and 1. However, -0.1 is not within the range of the sine function. So, this equation does not have any solutions in the principal value range.
Step 2: Using the periodicity of the sine function:
The sine function has a periodicity of 2π. To find all the solutions, we can add any multiple of 2π to the principal value.
So, the general solution is:
5x = arcsin(-0.1) + 2πn, where n is an integer.
Step 3: Solving for x:
To isolate x, divide both sides of the equation by 5:
x = (arcsin(-0.1) + 2πn) / 5, where n is an integer.
Note: Since sinusoidal functions have an infinite number of solutions, this equation has infinitely many solutions for x. The solution set can be represented by the general form above, where n can take any integer value.
To solve the equation sin(6x)cos(11x) - cos(6x)sin(11x) = -0.1 for all solutions, we can make use of the trigonometric identity for the sine of a difference of angles:
sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
By rewriting the equation using this identity, we get:
sin(6x - 11x) = -0.1
Simplifying the equation further:
sin(-5x) = -0.1
To find the solutions, we can take the inverse sine function (also called arcsin or sin^(-1)) of both sides:
-5x = arcsin(-0.1)
Now, we need to solve for x. To isolate x, divide both sides by -5:
x = (1/5) * arcsin(-0.1)
Using a calculator, we can find the approximate value of arcsin(-0.1) to be approximately -0.10016 radians.
Therefore, the solutions for x are:
x = (1/5) * -0.10016
Simplifying:
x = -0.020032
Hence, one possible solution for the equation sin(6x)cos(11x) - cos(6x)sin(11x) = -0.1 is x = -0.020032.