# Using the Double Angle Formula to Solve for Trigonometric Functions

• teme92
In summary, the cosine function can be found using the double angle formula. This equation allows for the determination of cos(2x) and cos(x). When working with sin(x), one must first find sin2x and then use the first equation to find sin(x). Finally, for sin(12), one must use sin2x=1-cos2x and then solve for sinx.
teme92

## Homework Statement

Given that cos($\pi$/6) =$\sqrt{}3$/2, use the double angle formula for the cosine function to find cos($\pi$/12) and sin($\pi$/12) explicitly.

## Homework Equations

cos(2x)=cos2x - sin2x
cos2x + sin2x = 1

## The Attempt at a Solution

So it wants me to find cos($\pi$/12) which is half the angle of cos($\pi$/6). So I called these cosx and cos 2x.

I then said $\sqrt{}3$/2 = cos2x - sin2x

I used cos2x + sin2x = 1 and got sin2x on its own and subbed into the first formula and then got cosx on its own.

For sin($\pi$/12) I subbed in sin2x = 1- cos2x and got sinx on its own.

Is this the correct method for finding the answers?

The inverse of cosx and sinx were $\pi$/12 so I assume I am but not sure. I'd be thankful to anyone who could clear this up.

Yes, $cos(2x)= cos^2(x)- sin^2(x)$ so that $cos(2x)= cos^2(x)- (1- cos^2(x))= 2cos^2(x)- 1$ and $cos(2x)= (1- sin^2(x))- sin^2(x)= 1- 2sin^2(x)$. Set cos(2x) equal to $\sqrt{3}/2$ and solve for sin(x) and cos(x). Of course, you take the positive root.

I am not clear why you would question your reasoning.

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1 person
teme92 said:

## Homework Statement

Given that cos($\pi$/6) =$\sqrt{}3$/2, use the double angle formula for the cosine function to find cos($\pi$/12) and sin($\pi$/12) explicitly.

## Homework Equations

cos(2x)=cos2x - sin2x
cos2x + sin2x = 1

## The Attempt at a Solution

So it wants me to find cos($\pi$/12) which is half the angle of cos($\pi$/6). So I called these cosx and cos 2x.

I then said $\sqrt{3}$/2 = cos2x - sin2x

I used cos2x + sin2x = 1 and got sin2x on its own and subbed into the first formula and then got cosx on its own.

For sin($\pi$/12) I subbed in sin2x = 1- cos2x and got sinx on its own.

Is this the correct method for finding the answers?

The inverse of cosx and sinx were $\pi$/12 so I assume I am but not sure. I'd be thankful to anyone who could clear this up.

That looks correct. Did you get exact radical values? You have$$\cos(2x) = \cos^2x -\sin^2 x = 2\cos^2x - 1 = 1-2\sin^2 x$$You are just using the last two equations to solve for ##\cos x## and ##\sin x## in terms of ##\cos(2x)##.

1 person
Thanks guys! HallsofIvy its just sometimes when I do these questions I think I'm right and then I only get it partly correct or not correct at all. I was just wanting to be sure as these type of questions may come up in my finals.

@teme92. You have to get into the habit of not doubting your reasoning. Yes, self-criticism will help you improve your abilities at problem solving, but when it gets in the way of you being confident in your answers it can be a problem.

Hey xiavatar, when it comes down to exams I will go with my instincts unquestionably but I just want to be safe in the run up to them. As you said, self-criticism has improved my understanding of a lot of topics in mathematics so I'd prefer to be safe than sorry in this instant :)

## What is the Double Angle Formula?

The Double Angle Formula is a trigonometric identity that allows us to express the trigonometric functions of a double angle in terms of the trigonometric functions of the original angle.

## What is the general formula for the Double Angle Formula?

The general formula for the Double Angle Formula is: sin(2x) = 2sin(x)cos(x), cos(2x) = cos^2(x) - sin^2(x), tan(2x) = 2tan(x)/1-tan^2(x)

## How do we use the Double Angle Formula to solve problems?

The Double Angle Formula is useful for solving trigonometric equations involving double angles. We can use it to simplify expressions, find exact values of trigonometric functions, and prove trigonometric identities.

## What are some common mistakes when using the Double Angle Formula?

Common mistakes when using the Double Angle Formula include forgetting to distribute the 2 in front of the trigonometric functions, incorrectly substituting values for x, and not simplifying the final expression.

## Can the Double Angle Formula be used for any angle?

Yes, the Double Angle Formula can be used for any angle, as long as the angle is doubled. However, it is most commonly used for angles that are multiples of 30, 45, and 60 degrees, as these can be easily simplified.

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