Trigonometric Identities: How to Find Cos x and Tan 2 x given Sin x = 2/√13

In summary, the conversation discusses finding the values of cos x and tan 2x given that sin x = 2/√13 for the given range of x. The speaker used a triangle to find that cos x = 3/√13 and added a negative sign to adjust for the given range. The speaker then asks about finding tan 2x and the expert suggests using the double angle identity for tangent. The speaker also mentions not having the tan identity in their formula booklet. The expert clarifies that the speaker's initial approach could have worked as well.
  • #1
Peter G.
442
0
Hi,

I am given that, for π/2 < x < π, sin x = 2/√13

a) Find Cos x
b) Find tan 2 x

So, what I did was: I drew a triangle and found that the missing side was equal to 3. From then, I deduced that cos x was equal to 3/√13

The problem was however that the angle must lie between the values given above. What I did was I simply added a negative sign. Is that right?

For part b I did sin 2 x / cos 2 x = tan 2 x and solved. Is that right? I got a negative answer too, which makes sense in terms of the unit circle.

Thanks,
Peter
 
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  • #2
Peter G. said:
Hi,

I am given that, for π/2 < x < π, sin x = 2/√13

a) Find Cos x
b) Find tan 2 x

So, what I did was: I drew a triangle and found that the missing side was equal to 3. From then, I deduced that cos x was equal to 3/√13

The problem was however that the angle must lie between the values given above. What I did was I simply added a negative sign. Is that right?
Yes.
Peter G. said:
For part b I did sin 2 x / cos 2 x = tan 2 x and solved. Is that right?
I don't think so. You know sin(x) and you have found cos(x), but you don't know sin(2x) or cos(2x).

Use the double angle identity for tangent: tan(2x) = 2tanx/(1 - tan2x).
Peter G. said:
I got a negative answer too, which makes sense in terms of the unit circle.

Thanks,
Peter
 
  • #3
Ah ok. I did the sin 2(x)/cos2(x) because I hadn't learned the tan identity and therefore didn't have it in my formula booklet. Maybe I had to know it and I didn't :redface:

Thanks!
 
  • #4
Actually, what you started to do would have worked. Since you know both sin(x) and cos(x) you could have used them to get sin(2x) and cos(2x), and then evaluated sin(2x)/cos(2x). What I suggested is just more direct.
 

1. What are trigonometric identities?

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. They are used to simplify and solve equations involving trigonometric functions.

2. What are the most commonly used trigonometric identities?

The most commonly used trigonometric identities include the Pythagorean identities, sum and difference identities, double angle identities, and half angle identities.

3. How are trigonometric identities used in real life?

Trigonometric identities are used in various fields such as engineering, physics, and astronomy to solve problems involving angles and triangles. They are also used in navigation and surveying.

4. How can one prove a trigonometric identity?

To prove a trigonometric identity, one must manipulate the given equation using algebraic techniques and known trigonometric identities to show that both sides of the equation are equal. This can be done by starting with one side of the equation and manipulating it until it matches the other side.

5. Can trigonometric identities be used to solve any trigonometric equation?

No, not all trigonometric equations can be solved using trigonometric identities. Some equations may require the use of other techniques such as the unit circle or trigonometric substitution.

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