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. :)
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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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.
- 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.