Finding the Angle

nikkir

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Solve for each equation for 0≤ø≤2π

1) tanø-2cosøtanø=0
tanø-2[cosø(sinø/cosø)]=0
tanø-2sinø=0
sinø/cosø-2sinø=0
sinø-2sinøcosø/cosø=0
sinø=0
ø=0,π
I have this so far (in bold), but the answer is 0, π, π/3, 5π/3. What am I missing?

2) 2+cosø-2sin2ø=0
​I have no idea how to start this.
 
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Solve for each equation for 0≤ø≤2π

1) tanø-2cosøtanø=0
tanø-2[cosø(sinø/cosø)]=0
tanø-2sinø=0
sinø/cosø-2sinø=0
sinø-2sinøcosø/cosø=0
sinø=0

How did you get from the next-to-last line to the last line, above? What happened to the rest of the expression on the left-hand side?
2) 2+cosø-2sinø2ø=0
​I have no idea how to start this.
Hint: It can be converted into a quadratic in cosine. ;)
 
[/B]How did you get from the next-to-last line to the last line, above? What happened to the rest of the expression on the left-hand side?

I canceled cosø from the top and bottom.

Hint: It can be converted into a quadratic in cosine. ;)

What do you mean by quadratic?
 
Solve for each equation for 0≤ø≤2π

1) tanø-2cosøtanø=0
tanø-2[cosø(sinø/cosø)]=0
tanø-2sinø=0

tanΦ [1 - 2cosΦ] = 0

tanΦ = 0
Θ = 0,π

or

1-2cosΦ = 0
cosΦ = 1/2 Φ = π/3, 5π/3


sinø/cosø-2sinø=0
sinø-2sinøcosø/cosø=0
sinø=0
ø=0,π
I have this so far (in bold), but the answer is 0, π, π/3, 5π/3. What am I missing?

2) 2+cosø-2sin2ø=0
​I have no idea how to start this.
.
 
Solve for each equation for 0≤ø≤2π

1) tanø-2cosøtanø=0
tanø-2[cosø(sinø/cosø)]=0
tanø-2sinø=0
sinø/cosø-2sinø=0
sinø-2sinøcosø/cosø=0
sinø=0
ø=0,π
I have this so far (in bold), but the answer is 0, π, π/3, 5π/3. What am I missing?

2) 2+cosø-2sin2ø=0
​I have no idea how to start this.
sin2ϕ=1cos2ϕ\displaystyle sin^2\phi= 1- cos^2\phi so this can be written as 2+cosϕ2+2cos2ϕ=0\displaystyle 2+ cos\phi- 2+ 2cos^2\phi= 0. 2cos2ϕ+cosϕ=0\displaystyle 2cos^2\phi+cos\phi= 0 If you let x=cosϕ\displaystyle x= cos\phi, that becomes 2x2+x=0\displaystyle 2x^2+ x = 0. That's the "quadratic" staple was referring to.
 
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use:

sin2Φ = 1 - cos2Φ

Now you should have quadratic equation.


So now I should have:
2+cosø-2(1-cos2ø)=0
cosø+2cos2ø=0
cosø(1+2cosø)=0

cosø=0
ø=π/2, 3π/2

AND

1+2cosø=0
2cosø=-1
cosø=-1/2
ø=2π/3, 4π/3

CORRECT????
 
So now I should have:
2+cosø-2(1-cos2ø)=0
cosø+2cos2ø=0
cosø(1+2cosø)=0

cosø=0
ø=π/2, 3π/2

AND

1+2cosø=0
2cosø=-1
cosø=-1/2
ø=2π/3, 4π/3

CORRECT????

Put those value of Φ in the original equation - and perform a check of the answer.
 
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