Math Grade 11 Identeties and equations
The graphs of f(θ) = 2sin(θ) - 1 (blue) , and g(θ) = 3cos(θ) + 2 (red) are shown above.
What equation would have the intersection points of the graphs as its solutions?
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You need to find the points of intersection between the graphs of `f(theta)` and `g(theta), ` hence, should solve the following equation such that:
`f(theta) = g(theta) => 2 sin theta - 1 = 3 cos theta + 2`
`2 sin theta - 3 costheta = 3`
You should use the following trigonometric identities such that:
`sin theta = (2tan(theta/2))/(1 + tan^2(theta/2))`
`cos theta = (1 - tan^(theta/2))/(1 + tan^2(theta/2))`
Substituting t for `tan(theta/2) ` yields:
`2*(2t)/(1 + t^2) - 3*(1 - t^2)/(1 + t^2) = 3`
You need to bring the terms to a common denominator such that:
`4t - 3 + 3t^2 - 3 - 3t^2 = 0`
Reducing like terms yields:
`4t - 6 = 0 => t = 6/4 => t = 3/2`
You need to solve for x the equation `tan(theta/2) = 3/2 => theta = 2arctan(3/2) + npi` .
Hence, the equation that has as solutions the points of intersection of graphs is `2 sin theta - 3 cos theta - 3 = 0` and the general solution to this equation is `theta = 2arctan(3/2) + npi.`
sorry again, the graph will not post on this website so i have this link, and once u go here, the graph is right there. thanks!
sorry the graph did not post in the question so here is the graph:
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