- #1

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## Homework Statement

## Homework Equations

cos 2theta = costheta^2 - sintheta^2

## The Attempt at a Solution

cos2theta = 1

2theta = 0, 2phi

but i get wrong answer.. how is it?

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- Thread starter Helly123
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- #1

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cos 2theta = costheta^2 - sintheta^2

cos2theta = 1

2theta = 0, 2phi

but i get wrong answer.. how is it?

- #2

Ray Vickson

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## Homework Statement

View attachment 205960

## Homework Equations

cos 2theta = costheta^2 - sintheta^2

## The Attempt at a Solution

cos2theta = 1

2theta = 0, 2phi

but i get wrong answer.. how is it?

Well, given your values for ##\sin \theta## and ##\cos \theta## you should NOT get ##\cos 2 \theta = 1##. Check your algebra.

- #3

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oh.. cos 2 theta = root 2 / 2Well, given your values for ##\sin \theta## and ##\cos \theta## you should NOT get ##\cos 2 \theta = 1##. Check your algebra.

2theta = 45degrees

the sin is negative, theta must on quadrant 2 or 3

225 or 315 degrees.

- #4

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oops. never mind

- #5

mjc123

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No, with cos +ve and sin -ve, the angle must be in the 4th quadrant. All silly tom cats. (A calculator will give you different angles.)You get the "wrong" answer because the problem posits an impossible condition. Just use arccos and arcsin and you'll see they are talking about two different angles so it can't be right to call them both the same. For that matter, you don't even have to do any math; the signs alone tell you the angles are different.

θ = 2π - π/8 = 15π/8, and 2θ = 15π/4 (note the question asked for the answer as a multiple of π, not in degrees).

- #6

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Yes, I had already deleted my post before you posted thisNo, with cos +ve and sin -ve, the angle must be in the 4th quadrant. All silly tom cats. (A calculator will give you different angles.)

θ = 2π - π/8 = 15π/8, and 2θ = 15π/4 (note the question asked for the answer as a multiple of π, not in degrees).

- #7

Ray Vickson

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π

No, with cos +ve and sin -ve, the angle must be in the 4th quadrant. All silly tom cats. (A calculator will give you different angles.)

θ = 2π - π/8 = 15π/8, and 2θ = 15π/4 (note the question asked for the answer as a multiple of π, not in degrees).

Or, the angle could be ##-\pi/4##, assuming negative angles are allowed.

- #8

mjc123

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The question said 0 ≤ θ < 2π.

- #9

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But the answer also 15/4 pi.. Which is 337.5 degree

- #10

Ray Vickson

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But the answer also 15/4 pi.. Which is 337.5 degree

No, it isn't. ##2 \pi \leftrightarrow 360^o##, and ##15/4> 3> 2##, so ##15/4 \pi ## is a lot bigger than ##360^0##.

- #11

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the key answer it is. the 2 theta allowed to be bigger than 2 pi. just theta < 2piNo, it isn't. ##2 \pi \leftrightarrow 360^o##, and ##15/4> 3> 2##, so ##15/4 \pi ## is a lot bigger than ##360^0##.

- #12

SammyS

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no

the key answer it is. the 2 theta allowed to be bigger than 2 pi. just theta < 2pi

Yes, but you said:

As Ray said, that's not true. After all, (15/4)π ≠ 337.5° .But the answer also 15/4 pi.. Which is 337.5 degree

Perhaps you meant that

one of the answers is 2θ = (15/4)π , which means that θ = 337.5° ,

so, of course, θ = (15/8)π < 2π .- #13

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Hence, 2theta=-pie/4.

From our restriction, this -pie/4 should be equal to 7pie/4 (sin is negative and cos is positive/ IV Quadrant).

Therefore 2theta=7pie/4

theta=7pie/8

- #14

Mark44

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You do know that there is a difference between "pie" and "pi" with the latter being the name of a Greek letter, right?

Hence, 2theta=-pie/4.

From our restriction, this -pie/4 should be equal to 7pie/4 (sin is negative and cos is positive/ IV Quadrant).

Therefore 2theta=7pie/4

theta=7pie/8

- #15

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You do know that there is a difference between "pie" and "pi" with the latter being the name of a Greek letter, right?

typing with a new tablet... with auto correct...

You do know that making a post within a topic, that is not relevant to the discussion is considered trolling? That is a violation of PF rules of conduct.

- #16

mjc123

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The restriction applies to θ, not 2θ. It is also θ that must be in the fourth quadrant, therefore θ = 7π/8 is not acceptable

Hence, 2theta=-pie/4.

From our restriction, this -pie/4 should be equal to 7pie/4 (sin is negative and cos is positive/ IV Quadrant).

Therefore 2theta=7pie/4

theta=7pie/8

- #17

ehild

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The restriction applies to theta, but cos(2θ) is positive and sin(2θ) is negative, therefore 2θ is in the fourth quadrant or 2pi more. And the problem asks 2θ.The restriction applies to θ, not 2θ. It is also θ that must be in the fourth quadrant, therefore θ = 7π/8 is not acceptable

- #18

mjc123

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- #19

- #20

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@mjc123 2 θ = is 45 degrees, theta = 22.5 degreesπ

No, with cos +ve and sin -ve, the angle must be in the 4th quadrant. All silly tom cats. (A calculator will give you different angles.)

θ = 2π - π/8 = 15π/8, and 2θ = 15π/4 (note the question asked for the answer as a multiple of π, not in degrees).

So quadrant 4 is 360-22.5 = 337.5

So 2theta = 337.5 x 2 = 675 degrees (15/4)pi

Why can't i use directly 45degrees, so theta in quadrant 4 is 360-45 = 315 degrees

Since 315 and 675 have same cos value, why the answer is 675?

- #21

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Never mind i get it. Thanks for all help

- #22

mjc123

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From the magnitudes of cosθ and sinθ, the basic angle is 22.5° (π/8). (If you have to do this without a calculator, from the magnitude of cos 2θ, 2θ = 45°.) However, since cosθ is positive and sinθ is negative, θ must be in the 4th quadrant, i.e. between 3π/2 and 2π.@mjc123 why θ = 2π - π/8 = 15π/8, and 2θ = 15π/4 ?

Why you substract π/8 from 2π?

I assume you mean "2 theta in quadrant 4 is 360-45". You can't use this because (for the millionth time)Why can't i use directly 45degrees, so theta in quadrant 4 is 360-45 = 315 degrees

Since 315 and 675 have same cos value, why the answer is 675?

- #23

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Haha yes. ThanksFrom the magnitudes of cosθ and sinθ, the basic angle is 22.5° (π/8). (If you have to do this without a calculator, from the magnitude of cos 2θ, 2θ = 45°.) However, since cosθ is positive and sinθ is negative, θ must be in the 4th quadrant, i.e. between 3π/2 and 2π.

I assume you mean "2 theta in quadrant 4 is 360-45". You can't use this because (for the millionth time)θ must be in the 4th quadrant.So 2θ must be between 3π and 4π.

- #24

Mark44

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At PF we endeavor to maintain a high-quality site for the pursuit of science and mathematics, one aspect of which is to not conflate the number ##\pi## (or pi) with the dessert whose name is pronounced the same. I am aware that you have a strong knowledge of mathematics, but because you wrote "pie" six times in your post, I believed that this deserved comment and was relevant to the discussion.You do know that making a post within a topic, that is not relevant to the discussion is considered trolling?

As a mentor, I am very familiar with the forum rules. Hijacking a thread is a violation, but comments that relate directly to posts in the thread are neither off-topic nor hijacks.MidgetDwarf said:That is a violation of PF rules of conduct.

- #25

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The restriction applies to θ, not 2θ. It is also θ that must be in the fourth quadrant, therefore θ = 7π/8 is not acceptable

Thank you. I now fully understand. So silly to make that common mistake.

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