Question 866819: Hi, any help with the following question would be greatly appreciated. Using an analytical method (compound angle formula) solve the following trigonometric equation for angles in the range: sinθ=3sin(θ-Pi/6). Substitute your answers into the original expression to check that they are valid.
Thanks in advance.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i'll use x instead of theta.
doesn't change the problem.
it's just easier to type.
start with:
sin(x) = 3sin(x-pi/6)
since sin(d-e) = sin(d)*cos(e) - cos(d)*sin(e), then if d = x and e = pi/6, you get:
sin(x) = 3 * [sin(x)*cos(pi/6) - cos(x)*sin(pi/6)]
simplify this to get:
sin(x) = 3*sin(x)*cos(pi/6) - 3*cos(x)*sin(pi/6)
you can re-write this to be:
sin(x) = (3*cos(pi/6))*sin(x) - (3*sin(pi/6))*cos(x)
if you let a = 3*cos(pi/6) and you let b = 3*sin(pi/6), then you get:
sin(x) = a*sin(x) - b*cos(x)
subtract sin(x) from both sides of this equation and add b*cos(x) to both sides of this equation to get:
b*cos(x) = a*sin(x) - sin(x)
factor out the sin(x) in this equation to get:
b*cos(x) = sin(x) * (a-1)
divide both sides of this equation by b and divide both sides of this equation by sin(x) to get:
cos(x) / sin(x) = (a-1) / b
since cot(x) = cos(x) / sin(x), this equation becomes:
cot(x) = (a-1) / b
the equation you are working with now is:
cot(x) = (a-1) / b
now you want to find the value of a and b and substitute those values for a and b in the equation.
a = 3 * cos(pi/6) which makes a = 2.598076211
b = 3 * sin(pi/6) which makes b = 1.5
replace a and b in the formula of cot(x) = (a-1) / b to get:
cot(x) = (2.598076211 - 1) / 1.5
simplify to get:
cot(x) = 1.598076211 / 1.5
simplify further to get:
cot(x) = 1.065384141
solve for x to get:
x = arc-cot(1.065384141) = .753751599
that should be your answer.
in the original equation, replace x with .753751599 to get:
sin(x) = 3sin(x-pi/6) becomes:
sin(.753751599) = 3sin(.753751599-pi/6) which becomes:
.684378960 = .684370960
you did good.
don't forget to set your calculator to radians in order to solve this problem.
|
|
|
