sin(6x)cos(11x)+cos(6x)sin(11x)=-0.1
solve for all solutions and provide steps so I can follow.. thank you
sin a cos b + cos a sin b = sin (a+b)
so what we really have is
sin(17x) = -.1
well there are an infinite number of solutions but just between 0 and 360 we have solutions in quadrants 3 and 4
sin of what = .1?
what = 5.8 degrees.
so5.8/17 = .34 degrees below + and - x axes
so x = 180.34 deg and x = - .34 deg
since the period of sin(17x) is 360/17, there are 17 more pairs of solutions for x in [0,360) such that
17x = 180.34 + n*360 for n=0..17
17x = 359.66 + n*360
To solve the equation sin(6x)cos(11x) + cos(6x)sin(11x) = -0.1, we can use the product-to-sum identities for sine and cosine functions.
First, rewrite the left side of the equation using the product-to-sum identities:
sin(6x)cos(11x) + cos(6x)sin(11x) = sin(6x + 11x) = sin(17x)
Now the equation becomes sin(17x) = -0.1.
To solve sin(17x) = -0.1, we need to find the values of x that satisfy this equation. Here's how you can find the solutions step by step:
Step 1: Solve the equation sin(17x) = -0.1 for the principal solution.
- Find the principal solution using the inverse sine function (also known as arcsin) on both sides of the equation:
arcsin(sin(17x)) = arcsin(-0.1)
- Since arcsin(sin(θ)) = θ only when -π/2 ≤ θ ≤ π/2, we need to find the principal solution within this range.
- Apply the arcsin function to both sides of the equation:
17x = arcsin(-0.1)
- Solve for x by dividing both sides by 17:
x = arcsin(-0.1)/17
Step 2: Find the general solutions.
- To find all the solutions to the equation, we need to consider both the principal solution and the periodicity of the sine function.
- The sine function is periodic with a period of 2π, which means that sin(x + 2π) = sin(x) for any value of x.
- We can use this periodicity to find the general solutions of the equation by adding integer multiples of the period (2π) to the principal solution.
- The general solutions can be expressed as:
x = arcsin(-0.1)/17 + 2πn/17, where n is an integer.
These are the steps to solve the equation sin(6x)cos(11x) + cos(6x)sin(11x) = -0.1 and find all the solutions.