How Do You Solve Trigonometric Equations Using Graphs?

[Trail_Builder]

just encountered this question and kinda confused at how to solve it since I havn't bin told and havn't worked it out for myself. hope you can help.

Homework Statement

Use the graphs (shows 2 graphs) to find the values of x in the range $$0 /leq x /leq 720$$ when 2sinx = cosx -1

Homework Equations

The Attempt at a Solution

I found from the graph that x could equal 0, 360, and 720. but the 2 lines cross at another point before x=360 so there should be 2 more values of x to satisfy the question.

however, i can't easily read off an accurate result. i am only 16 so do you rekon they allow for error reading off the graph instead of working it out a very accurate way?

thnx

 

[Gib Z]

Yea theyll allow for an error even if your not 16. The question says use the graphs to read off the result, so its the best you can do. If you want to be cautious, say this agrees with an algebraic result:
http://en.wikipedia.org/wiki/Trigonometric_identities#Linear_combinations
there you can find your solutions.

 

[HallsofIvy]

Graphing calculators will allow you to "zoom" in on a point so if you have one that should give you very accurate solutions. It is, however, possible to get "exact" solutions. $cos x= \sqrt{1- sin^2 x}$ so you can rewrite the equation as $2 sin x= \sqrt{1- sin^2 x}- 1$. Now, just to simplify the writing, let y= sin x. The equation is $2 y= \sqrt{1- y^2}- 1$. Add 1 to both sides: $2y+ 1= \sqrt{1- y^2}$ and, finally, squaring, $4y^2+ 4y+ 1= 1- y^2$ or $5y^2+ 4y= y(5y+4)= 0$. One solution to that is y= sin x= 0 (and we must also have cos x= 2sin x+ 1= 1). that gives you multiples of 360 as solutions. If y is not 0 then we must have 5y+ 4= 0 so y= sin x= -4/5 (and cos x= 2sin x+ 1= -8/5+1= 3/5. x= Arcsin(-4/5)+ multiples of 360.