Confused on what should be negative when finding with half angle identities

In summary, the author is trying to find sin2x, cos2x, and tan2x from the given information. They are able to find sin2x and cos2x, but are unable to find tan2x. They use the given information to solve for the unknown side and then use the half angle identities to plug in the numbers. However, they are still confused about what should be negative and positive. The author suggests using the Pythagorean theorem to solve for the unknown side and then write down the six trig functions.
  • #1
kieth89
31
0

Homework Statement


The question is to find [itex]sin 2x, cos 2x, tan 2x[/itex] from the given information: [itex]sin x = -\frac{3}{5}[/itex], x in Quadrant III


Homework Equations


Half Angle Identities
[itex]cos2x = cos^{2}x - sin^{2}x[/itex]

[itex]sin2x = 2sinxcosx[/itex]

[itex]tan2x = \frac{2tanx}{2-tan^{2}x}[/itex]


The Attempt at a Solution


I can find the solution for the most part, the only thing I can't figure out are the signs. What I do is use the given sin[itex](-\frac{3}{5})[/itex] to make a right triangle and solve for the unknown side. I then use that triangle to set up the 3 half angle identities and just plug in the numbers. I can do all that fine, but I can't figure out what should be negative and positive. I thought that since it is in Quadrant 3 both sin2x and cos2x should end up negative. However the back of the book answers say that they are all positive. Why?
 
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  • #2
When you draw the triangle in quadrant 3, you should see that the opposite side is negative and the adjacent side (all to x) is negative as well. As the hypotenuse is always positive, you can understand why sin2x is positive.

If you don't understand, draw the triangle within the 3rd quadrant, put in the appropriate signs and then write down what sinx and cosx are. Post back if you are still confused.
 
  • #3
rock.freak667 said:
When you draw the triangle in quadrant 3, you should see that the opposite side is negative and the adjacent side (all to x) is negative as well. As the hypotenuse is always positive, you can understand why sin2x is positive.

If you don't understand, draw the triangle within the 3rd quadrant, put in the appropriate signs and then write down what sinx and cosx are. Post back if you are still confused.

I don't understand why the hypotenuse is always positive..or really why we can say any of these distances are negative. I must be missing something. I tried to convey my thoughts/procedure through this (ugly photoshop) image:
In7vk.jpg


Basically I'm just setting up a triangle. But why couldn't the hypotenuse be negative and the two sides positive?
 
  • #4
kieth89 said:
I can find the solution for the most part, the only thing I can't figure out are the signs. What I do is use the given sin[itex](-\frac{3}{5})[/itex] to make a right triangle and solve for the unknown side. I then use that triangle to set up the 3 half angle identities and just plug in the numbers. I can do all that fine, but I can't figure out what should be negative and positive. I thought that since it is in Quadrant 3 both sin2x and cos2x should end up negative. However the back of the book answers say that they are all positive. Why?

You don't have to use identities, there's and easier way. Sin is -3/5, so you know the opp side is 3 and the hyp is 5. Think of the numbers as lengths, and you cannot have negative length, but the SIN and COS can the negative if in the quadrants like you said. Use the Pythagorean theorem to solve for the unknown side, then just jot down the six trug functions.

Sin is negative when it is moving down (below the x-axis, so quadrant 3 & 4).
Cosine is negative when moving to the left (quadrant 2 & 3)

Tangent = sin/cos
So -sin/cos = -tan
sin / -cos = -tan
sin / cos = tan
-sin / - cos = tan

"The question is to find sin2x,cos2x,tan2x from the given information: sinx=−35, x in Quadrant III"

Do you mean Sin2(x)? Because anything real squared is positive, in which case the book is right.
 
Last edited:
  • #5
If you double all the angles in 3rd. quadrant, the answers will be all in 1st. quadrant(all positive)
 
  • #6
kieth89 said:
[b

Homework Equations


Half Angle Identities
[itex]cos2x = cos^{2}x - sin^{2}x[/itex]

[itex]sin2x = 2sinxcosx[/itex]

[itex]tan2x = \frac{2tanx}{2-tan^{2}x}[/itex]

However the back of the book answers say that they are all positive. Why?

By substituting you can see all are positive.
If you double the angle, approximately 217° to 434°, all will be in first quadrant.
 
  • #7
e^(i Pi)+1=0 said:
[SNIP]
Sin is negative when it is moving down (below the x-axis, so quadrant 3 & 4).
Cosine is negative when moving to the left (quadrant 2 & 3)[/SNIP]

azizlwl said:
By substituting you can see all are positive.
If you double the angle, approximately 217° to 434°, all will be in first quadrant.

I think I see what I messed up now. I left off a negative sign on the horizontal side's measurement. I'm going to practice some more of these today and hopefully will become more comfortable with them. (although most of it is just knowing the formulas) Thanks for all the help.
 

What are half angle identities?

Half angle identities are trigonometric identities that allow us to find the values of sine, cosine, and tangent for half of a given angle. These identities are useful in solving equations and simplifying expressions.

Why is it important to know which values should be negative when using half angle identities?

It is important to know which values should be negative when using half angle identities because it affects the quadrant in which the angle lies and the sign of the trigonometric functions. This can make a significant difference in the final solution of a problem.

How do we know which values should be negative when using half angle identities?

The values that should be negative when using half angle identities depend on the quadrant in which the angle lies. If the angle lies in the second or third quadrant, the sine and tangent values should be negative, while the cosine value should be positive. If the angle lies in the fourth quadrant, all three values should be negative.

Can we use half angle identities for any angle?

Yes, half angle identities can be used for any angle, whether it is acute, obtuse, or even a reference angle. However, it is important to remember to adjust the values accordingly based on the quadrant in which the angle lies.

What are some common mistakes when using half angle identities?

Some common mistakes when using half angle identities include forgetting to adjust the values for the correct quadrant, using the wrong half angle formula, and not simplifying the final solution. It is important to double check all steps and use the correct formula for the given problem.

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