Question about sin and cos

[elcatracho]

The question is... find exact solutions for 0<x< 2pi (less than or equal/ greater than or equal)

2sin^2 (theta)=1-sin(theta)


2tan(theta)-sec^2(theta)=0


sin2(theta)+sin(theta)=0

I would really be thankful if someone could help me figure out how to solve these problems and get the exact solutions, thanks, Luis
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[stapel]

I will use "@" to stand for "theta".

1) 2sin²(@) = 1 - sin(@)

This is a quadratic in sine:

. . . . .2sin²(@) + sin(@) - 1 = 0

Factor:

. . . . .(2sin(@) - 1)(sin(@) + 1) = 0

...and solve for the values of sin(@):

. . . . .2sin(@) - 1 = 0, so sin(@) = 1/2

. . . . .sin(@) + 1 = 0, so sin(@) = -1

Then solve these two equations for the angle values.

2) 2tan(@) - sec²(@) = 0

Convert this, using "sec²(@) = 1 + tan²(@)", into a quadratic in tangent. Then solve as in (1).

3) sin(2@) + sin(@) = 0

Convert using "sin(2@) = 2sin(@)cos(@)". Then factor and solve.

Eliz.

Edit: Mis-read an equation.

 

[Gene]

heta = T

1) cos²(u)= 1-sin²(u)
(If it is easier substitute
v=sin(u), factor the quadratic and un-sub.)

2) 2sin(T)/cos(T) - (1/cos²(T)) = 0
Multiply by cos²(T)
Use identity 2sin(T)cos(T) = sin(2T)

3) sin2(T)+sin(T)=0
Same identity (reversed) then factor.

Different reading on 3). Take the one that agrees with the book.
PS Nevermind.

 

[tkhunny]

. . . . .2sin(@) - 1 = 0, so sin(@) = 1/2

. . . . .sin(@) + 1 = 0, so sin(@) = -1

Then solve these two equations for the angle values.

Do NOT forget that there may be more than one value from each factor. Tell us what you get so we can see if you find them all.

 

[elcatracho]

I'm a bit confused on how to do the last two. For the first question I got the answers to be 3pi/2 5pi/6 and pi/6. The other two questions i'm confused on still. I'm not sure how to factor them out.. can someone take me step by step through the factoring process, that way i understand how to factor them.. Thanks, Luis

 

[Gene]

I had in mind
theta = T

2) 2sin(T)/cos(T) - (1/cos²(T)) = 0
Multiply by cos²(T)
2sin(T)cos(t) - 1 = 0
Use identity 2sin(T)cos(T) = sin(2T)
sin(2T) = 1

Eliz liked
2) 2tan(@) - sec^2(@) = 0
Convert this, using "sec^2(@) = 1 + tan^2(@)", into a quadratic in tangent. Then solve as in (1).
2tan(@) -1 - tan^2(@) = 0
tan^2(@) - 2tan(@) + 1 = 0
(tan(@)-1)^2 = 0

3) sin2(T)+sin(T)=0
Same identity (reversed) then factor.
2sin(T)cos(T)+sin(T) = 0
sin(T)*(2cos(T)+1) = 0

From there on it's the same as the first (which you got right.)