More trigonometry help? :(
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More trigonometry help? :(

[From: ] [author: ] [Date: 12-04-14] [Hit: ]
4 (rounded). Now, what do I do with this? I know theres another angle within that domain, but how do I find it? :( Would it be different if cos or sin were in place of tan?......
Okay, my teacher went over this concept in class but I'm still extremely confused on it. Any help? :(

On this problem:

tan(x)=sqrt(7)/3

It asks for all the angles between 0 and 2pi radians. I know you want to find the inverse in your calculator, which is: 41.4 (rounded). Now, what do I do with this? I know there's another angle within that domain, but how do I find it? :( Would it be different if cos or sin were in place of tan? Thanks a TON in advance. :)

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tan^-1(sqrt(7)/3) and tan^-1(sqrt(7)/3) + pi, since the period for tan(x) is pi.
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If cos(x) = sqrt(7)/3, then x = cos^-1(sqrt(7)/3) and 2pi - cos^-1(sqrt(7)/3), since it is either in the first or in the fourth quadrant.

If sin(x) = sqrt(7)/3, then x = sin^-1(sqrt(7)/3) and pi - cos^-1(sqrt(7)/3), since it is either in the first or in the second quadrant.

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To explain the idea, simply remember this trigonometry positive rule:

- 0 ---> pi/2 ALL ( it means all trigonometric functions, i.e. sin, cos, tan ..etc are positive numbers)
- pi/2 --> pi Sine (Only the sine is positive, the corrsponding angle to x is pi -x in this quarter ).
- pi -->3*pi/2 Tangent ( only tan is positive, the corrsponding angle to x is pi+x in this quarter )
- 3*pi/2 -->2*pi Cosine (only cosine is positive, the corrsponding angle to x is 2*pi -x in this quarter ).

Then if sqrt(7)/3 is positive, the corresponding angle can only be in the first and 3rd quarters. Since it is 41.4 degrees, then the two corresponding angles are :
- 41.4 degrees. (in the first quater).
- 180+ 41.4 = 221.4 degrees. ( in the 3rd quarter)

If sqrt(7)/3 is negative, since the tan will be negative on the 2nd and 4rth quarters, the corresponding angles will be:
- 180 - 41.4 = 138.6 degrees. (in the 2nd quater).
- 360 - 41.4 = 318.6 degrees. (in the 4th quater).

Notes:
-If you want the results in radian just use pi instead of 180 degree and 2*pi instead of 360 degrees and 0.72 radian instead of 41.4 degrees. Or simply convert from degree to radian or vice versa.

- These rules are covers all the solutions in the range 0-->2*pi.

Best Regards.
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