Question 965932: If sec(t)=-3/2 and since sec(t) is the same as 1/cos(t) therefore cos(t)=-2/3. By using the pythagean idenitty sin^2(t)+cos^2(t)=1
sin^2(t)+(-2/3)^2=1
sin(t) = sqrt(5)/3
Now one of my questions ask what is sin(-t) from that given information. Does that make it
-sqrt(5)/2?
Thanks
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i believe that, following your logic, the solution would be as follows:
sec(t) = -3/2 which makes cos(t) = -2/3.
that part is true.
with no other restrictions, your angle will be in the second or third quadrant because cosine is negative in second or third quadrant and positive in first or fourth quadrant.
in fact, without any other restrictions, the angle with a cosine of (-2/3) can be in the second quadrant and it canalso be in the third quadrant.
use your calculator to confirm.
find cos^-1(-2/3) and you will get an angle of 131.8103149.
that's in the second quadrant.
the comparable angle in the third quadrant would be equal to 228.1896851.
by comparable angle, this means that the cosine of that angle is the same as the cosine of 131.8103149.
note that decimal equivalent of -2/3 is equal to -.66666.....
cos(131.8103149) is equal to -.66666.....
cos(228.1896851) is equal to -.66666....
the cosine is the same so the angles are equivalent as far as their trigonometric cosine functions are concerned.
back to your logic.
you state that sin^2(t) + cos^2(t) = 1
this is true.
you also state that cos(t) = -2/3.
this is also true.
you get sin^2(t) + cos^2(t) = 1 becomes sin^2(t) + (-2/3)^2 = 1 which becomes sin^2(t) + 4/9 = 1 which becomes sin^2(t) = 1 - 4/9 which becomes sin^2(t) = 5/9.
this is also true.
you are left with sin^2(t)= 5/9
this leads to sin(t) = plus or minus sqrt(5/9) which becomes sin(t) = plus or minus sqrt(5)/3.
your solution should be sin(t) = plus or minus sqrt(5)/3.
you stated that sin(t) = + sqrt(5)/3 only.
that would be an error, since sin(t) is equal to plus or minus sqrt(5)/3.
i'm not exactly sure where you got sin(-t) from, or why you even asked that question.
however, let's see what happens when we replace t with -t in this specific problem.
first of all, we know that t is in the second quadrant or the third quadrant.
it is not in the first quadrant or the fourth quadrant because the cosine is negative and we know that cosine is negative in second or third quadrant only.
let's assume that t is in the second quadrant.
we can actually solve for t to get t = 131.8103149.
if t = 131.8103149, then -t must be equal to -131.8103149 degrees.
that's a negative angle which we can just add 360 to until it becomes positive.
-131.8103149 + 360 = 228.1896851.
those angles are the same angles.
one is expressed as a positive angle and the other is expressed as a negative angle.
they occupy the same position on the unit circle.
we already determined earlier, that the angle in the third quadrant with the same cosine function was 228.1896851, so we have the same equivalent angle as far as the cosine trigonometric function is concerned.
in quadrant 2, the sine is positive.
in quadrant 3, the sine is negative.
looks like you would be correct.
in the second quadrant the sine is + sqrt(5)/3.
in the third quadrant the sine is - sqrt(5)/3.
in the second quadrant, the angle with a cosine of (-2/3) is 131.8103149.
the negative of that angle would be -131.8103149.
the equivalent positive angle of the negative angle would be -131.... + 360 = 228.1896851
in the third quadrant, the angle with a cosine of (-2/3) is 228.1896851.
the negative of that angle is -228.1896851.
the equivalent positive angle of the negative angle would be -228.... + 360 = 131.8103149.
you got it right although i probably would not have expressed it that way.
never the less, you are correct.
if sin(t) = sqrt(5)/3, then sin(-5) = sqrt(5)/3.
now an angle with a sine of sqrt(5)/3 can also be in the first quadrant.
that angle would be 48.1896851.
the negative of that angle would be -48.1896851.
the sine of (-48.1896851) will also be -sqrt(5)/3.
that angle in the first quadrant, however, will not have a negative cosine.
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