Beginning Trigonometry

[Algebraneophyte]

Hi Guys.

I'm new to combining algebra and trigonometry. I solved the first question right (form quadratic with identity), but the next question eludes me or... my calculator setting.



If you could please assist, I truly appreciate your time and effort

 

[Deleted member 4993]

Hi Guys.

I'm new to combining algebra and trigonometry. I solved the first question right (form quadratic with identity), but the next question eludes me or... my calculator setting.

View attachment 27316

If you could please assist, I truly appreciate your time and effort

Form quadratic using

sin²(Θ) = 1 - cos²(Θ)

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem

 

[tkhunny]

Please put your calculator away.

[math]\sin^{2}(\theta) + \cos^{2}(\theta) = 1[/math]. That's ALL the trigonometry you need. The second is almost EXACTLY the same as the first.

 

[nasi112]

we have

[MATH]4\sin^{2} \theta + 3\cos \theta = 3[/MATH]
we know that [MATH]\sin^2 \theta = 1 - \cos^2 \theta[/MATH]
then the equation will be

[MATH]4(1 - \cos^2 \theta) + 3\cos \theta = 3[/MATH]
simplify the equation and make it equal to zero

[MATH]4 - 4\cos^2 \theta + 3\cos \theta = 3[/MATH]
[MATH]4 - 4\cos^2 \theta + 3\cos \theta - 3 = 0[/MATH]
[MATH]-4\cos^2 \theta+ 3\cos \theta - 1= 0[/MATH]
multiply everything by [MATH]-1[/MATH]
[MATH]4\cos^2 \theta - 3\cos x + 1= 0[/MATH]
now, you know [MATH]x = \cos \theta[/MATH]
then, you have

[MATH]4x^2 - 3x + 1= 0[/MATH]
Can you find the [MATH]x[/MATH] values?

 

[Algebraneophyte]

we have

[MATH]4\sin^{2} \theta + 3\cos \theta = 3[/MATH]
we know that [MATH]\sin^2 \theta = 1 - \cos^2 \theta[/MATH]
then the equation will be

[MATH]4(1 - \cos^2 \theta) + 3\cos \theta = 3[/MATH]
simplify the equation and make it equal to zero

[MATH]4 - 4\cos^2 \theta + 3\cos \theta = 3[/MATH]
[MATH]4 - 4\cos^2 \theta + 3\cos \theta - 3 = 0[/MATH]
[MATH]-4\cos^2 \theta+ 3\cos \theta - 1= 0[/MATH]
multiply everything by [MATH]-1[/MATH]
[MATH]4\cos^2 \theta - 3\cos x + 1= 0[/MATH]
now, you know [MATH]x = \cos \theta[/MATH]
then, you have

[MATH]4x^2 - 3x + 1= 0[/MATH]
Can you find the [MATH]x[/MATH] values?

Thank you. I was along the way with your working before posting the question and now i am hindered by true culprit (imaginary numbers lol). I think one of the solutions is theta 1 =0. I will learn imaginary number solutions to quadratics soon.

 

[nasi112]

we have

[MATH]4\sin^{2} \theta + 3\cos \theta = 3[/MATH]
we know that [MATH]\sin^2 \theta = 1 - \cos^2 \theta[/MATH]
then the equation will be

[MATH]4(1 - \cos^2 \theta) + 3\cos \theta = 3[/MATH]
simplify the equation and make it equal to zero

[MATH]4 - 4\cos^2 \theta + 3\cos \theta = 3[/MATH]
[MATH]4 - 4\cos^2 \theta + 3\cos \theta - 3 = 0[/MATH]
[MATH]-4\cos^2 \theta+ 3\cos \theta - 1= 0[/MATH]
multiply everything by [MATH]-1[/MATH]
[MATH]4\cos^2 \theta - 3\cos x + 1= 0[/MATH]
now, you know [MATH]x = \cos \theta[/MATH]
then, you have

[MATH]4x^2 - 3x + 1= 0[/MATH]
Can you find the [MATH]x[/MATH] values?

lol
by speed i missed a negative sign

the equation must be

[MATH]4x^2 - 3x - 1= 0[/MATH]

 

[nasi112]

Thank you. I was along the way with your working before posting the question and now i am hindered by true culprit (imaginary numbers lol). I think one of the solutions is theta 1 =0. I will learn imaginary number solutions to quadratics soon.

You should not get imaginary numbers. All numbers here are real.

My mistake was in this step
[MATH]4 - 4\cos^2 \theta + 3\cos \theta - 3 = 0[/MATH]
[MATH]-4\cos^2 \theta+ 3\cos \theta - 1= 0[/MATH] -------> HERE

it should be

[MATH]-4\cos^2 \theta+ 3\cos \theta + 1= 0[/MATH]

 

[Steven G]

lol
by speed i missed a negative sign

the equation must be

[MATH]4x^2 - 3x - 1= 0[/MATH]

Have you heard of the forum's unofficial corner time rule? Whenever you make a mistake you need to go sit in the corner for a bit of time to think about your error. Welcome to the forum!

 

[nasi112]

Have you heard of the forum's unofficial corner time rule? Whenever you make a mistake you need to go sit in the corner for a bit of time to think about your error. Welcome to the forum!

lol
The Dark corner. I might have heard of it.

 

[tkhunny]

Rule #1 - Do NOT forget your Algebra. Hold it close and strengthen it as you advance.
Rule #2 - Do NOT forget your Geometry. Hold it close and strengthen it as you advance.
Rule #3 - Do NOT forget your Trigonometry. Hold it close and strengthen it as you advance.
Rule #4 - Do NOT get sloppy. Write what you can follow, reproduce, reverse, check, and explain.

 

[topsquark]

lol
The Dark corner. I might have heard of it.

I've been there a lot. Bring some cookies for the roaches. They're nice.

-Dan

 

[topsquark]

Rule #1 - Do NOT forget your Algebra. Hold it close and strengthen it as you advance.
Rule #2 - Do NOT forget your Geometry. Hold it close and strengthen it as you advance.
Rule #3 - Do NOT forget your Trigonometry. Hold it close and strengthen it as you advance.
Rule #4 - Do NOT get sloppy. Write what you can follow, reproduce, reverse, check, and explain.

Rule #5 - Do NOT pass Go. Do not collect your \$200.

-Dan

 

[tkhunny]

Rule #5 - Do NOT pass Go. Do not collect your \$200.

-Dan

Of course. One must PAY to follow this path.