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...

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?

 

Asked on by moeenmoeen

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

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.`

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moeenmoeen | Student, Grade 11 | (Level 1) Honors

Posted on

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!

http://www.flickr.com/photos/82941247@N04/8243008603/in/photostream/lightbox/

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moeenmoeen | Student, Grade 11 | (Level 1) Honors

Posted on

sorry the graph did not post in the question so here is the graph:

 

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