Solving Trig Equation

**Peleus** · May 26th 2009, 11:39 PM

Hi all,

To start with - the problem.

$3\sin^2\theta + 7\cos\theta = 5$ solve for $0 <= \theta <= 2\pi$

The working I've done so far.

1. $3(1 - \cos^2\theta) + 7\cos\theta = 5$

2. $3 - 3\cos^2\theta + 7\cos\theta = 5$

3. $\cos\theta (7 - 3\cos\theta) = 2$

4. $\cos\theta = 2$ or $(7 - 3\cos\theta) = 2$

5. $7 - 3\cos\theta = 2$

6. $-3\cos\theta = -5$

7. $\cos\theta = \frac{3}{5}$

Therefore, $\theta = 53.13 deg$ or $\theta = 306.87 deg$

Which means that $\theta = \frac{53.13\pi}{180}$ or $\theta = \frac{306.87\pi}{180}$

I'm just not sure if step 4 is correct. Can I simply ignore one of the options because $\cos\theta = 2$ is invalid?

Cheers.

**alexmahone** · May 26th 2009, 11:43 PM

Step 4 is correct.

**Peleus** · May 26th 2009, 11:48 PM

Thanks

How about the rest of it lol.

**alexmahone** · May 26th 2009, 11:52 PM

I'm sorry; step 4 is incorrect!

How does step 4 follow from step 3.

3. $\cos\theta (7 - 3\cos\theta) = 2$

How did you get to step 4?

4. $\cos\theta = 2$ or $(7 - 3\cos\theta) = 2$

This conclusion can be made only if the RHS=0. I think you're missing something.

**Peleus** · May 26th 2009, 11:58 PM

I was thinking along the lines of factorising.

I.e.

$(x^2 + 3x) = 0$
$x (x + 3) = 0$
$x = 0$ or $(x + 3) = 0$
$x = 0$ or $x = -3$

Edit - Posted before your addition, I can only do that when x = 0 ? Learnt something new (although I should have known that before I guess)

Any further idea's of where to look?

**alexmahone** · May 27th 2009, 12:03 AM

$3cos^2 \theta - 7cos \theta+2=0$
$(3cos^2 \theta-6cos \theta - cos \theta+2)=0$
$3cos \theta (cos\theta-2) - (cos \theta-2)=0$
$(cos \theta-2) (3cos \theta-1)=0$
$(cos \theta-2)=0$ or $(3cos \theta-1)=0$

**scoobydoo4** · May 27th 2009, 02:24 PM

Originally Posted by Peleus

Hi all,

To start with - the problem.

$3\sin^2\theta + 7\cos\theta = 5$ solve for $0 <= \theta <= 2\pi$

The working I've done so far.

1. $3(1 - \cos^2\theta) + 7\cos\theta = 5$

2. $3 - 3\cos^2\theta + 7\cos\theta = 5$

3. $\cos\theta (7 - 3\cos\theta) = 2$

4. $\cos\theta = 2$ or $(7 - 3\cos\theta) = 2$

5. $7 - 3\cos\theta = 2$

6. $-3\cos\theta = -5$

7. $\cos\theta = \frac{3}{5}$

Therefore, $\theta = 53.13 deg$ or $\theta = 306.87 deg$

Which means that $\theta = \frac{53.13\pi}{180}$ or $\theta = \frac{306.87\pi}{180}$

I'm just not sure if step 4 is correct. Can I simply ignore one of the options because $\cos\theta = 2$ is invalid?

Cheers.

you can't do that on step three you need to do the following:

3-3cos^2x+7cosx=5

take everything to the right hand side
3cos^2x-7cosx+5-3=0
3cos^2x-7cosx+2=0

now you have to factorise
i would replace the cos with y first
3y^2-7y+2=0
(3y-1)(y-2)=0

so
3cosx=1
cosx=1/3

or cosx=2
x=error

so x=70.53 or 289.47 (2dp)

and i have used x instead of theta

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