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Homework Statement
I have no idea how to do these type of problems.
-------Problem--------
Solve each equation on the interval 0 less than or equal to theta less than 2 pi
42. SQRT(3) sin theta + cos theta = 1
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There is an example prior to the excersises that attempts to explain how to do these types of problems. I do not understand it...
Homework Equations
-------EXAMPLE---------
Solve the equation on the interval 0 less than or equal to theta less than 2 pi
a sin theta + b cos theta = c (2)
where a, b, and c are constants and either a does not equal 0 or b does not equal 0.
We divide each side each side of equation (2) by SQRT(a^2 + b^2). Then
a sin theta /SQRT(a^2 + b^2) + b cos theta/SQRT(a^2 + b^2) = c/SQRT(a^2 + b^2) (3)
There is a unique angle phi, 0 less than or equal to phi less than 2 pi, for which
cos phi = a/SQRT(a^2 + b^2) and sin phi = b/SQRT(a^2 + b^2) (4)
Figure 36
http://img269.imageshack.us/img269/3002/capturenmk.jpg
See Figure 36. Equation (3) may be written as sin theta cos phi + cos theta sin phi = c/SQRT(a^2 + b^2)
or, equivalentley,
sin(theta + phi) = c/SQRT(a^2 + b^2) (5)
where phi satisfies equation (4).
If |c| > SQRT(a^2 + b^2), then sin(theta + phi) > 1 or sin(theta + phi) < -1, and equation (5) has no solution.
If |c| less than or equal to SQRT(a^2 + b^2), then the solutions of equation (5) are
theta + phi =sin^-1 c/SQRT(a^2 + b^2) or theta + phi = pi - sin^-1 c/SQRT(a^2 + b^2)
Because the angle phi is determined by equations (4), these are the solutions to equation (2).
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Well I didn't understand but here is my attempt at solving the PROBLEM which is clearly wrong
The Attempt at a Solution
-----ATEMPT AT A SOLUTION-------
Solve each equation on the interval 0 less than or equal to theta less than 2 pi
42. SQRT(3) sin theta + cos theta = 1
I just plugged and chugged into what the book claims to be the solutions into the formula
theta + phi =sin^-1 c/SQRT(a^2 + b^2) or theta + phi = pi - sin^-1 c/SQRT(a^2 + b^2)
theta + phi = sin^-1 1/SQRT(3 +1) = sin^-1 1/2 = pi/6
or
theta + phi = pi - pi/6 = (5 pi)/6
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Well I'm lost so if you could guide me in some way to solve these type of problems that would be extremely appreciated... TAHNKS!
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