SOLUTION: Given that sin(x)=2/3 and pi/2 < x < pi find the values of the other five trigonometric functions of x

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Question 1035837: Given that sin(x)=2/3 and pi/2 < x < pi find the values of the other five trigonometric functions of x
Found 2 solutions by ikleyn, MathTherapy:
Answer by ikleyn(52756) About Me  (Show Source):
You can put this solution on YOUR website!
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Given that sin(x)=2/3 and pi/2 < x < pi find the values of the other five trigonometric functions of x
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If sin(x) = 2%2F3  and  pi%2F2 < x < pi   then

cos(x) = -sqrt%281+-+sin%5E2%28x%29%29 = -sqrt%281+-+%282%2F3%29%5E2%29 = -sqrt%281-4%2F9%29 = -sqrt%285%2F9%29 = -sqrt%285%29%2F3.

The sign "-" was chosen at the square root since the cosine is negative in the second quadrant.

Then tan(x) = sin%28x%29%2Fcos%28x%29 = %28%282%2F3%29%29%2F%28%28-sqrt%285%29%2F3%29%29 = -2%2Fsqrt%285%29 = -%282%2Asqrt%285%29%29%2F5.

Having this you can easily calculate the other trigonometric functions.


Answer by MathTherapy(10549) About Me  (Show Source):
You can put this solution on YOUR website!
Given that sin(x)=2/3 and pi/2 < x < pi find the values of the other five trigonometric functions of x 
sin+%28x%29+=+2%2F3+=+O%2FH+=+y%2Fr

x%5E2+=+r%5E2+-+y%5E2

x%5E2+=+3%5E2+-+2%5E2

x%5E2+=+9+-+4 ======> x+=+sqrt%285%29

sin+%28x%29+=+2%2F3 ======> csc+%28x%29+=+3%2F2

cos+%28x%29+=+A%2FH+=+x%2Fr+=+-+sqrt%285%29%2F3 ======> sec+%28x%29+=+-+3sqrt%285%29%2F5 ------ cos (x) and sec (x) are < 0 in the 2nd quadrant

tan+%28x%29+=+O%2FA+=+y%2Fx+=+2%2F%28-+sqrt%285%29%29+=+-+2sqrt%285%29%2F5 ======> cot+%28x%29+=+-+sqrt%285%29%2F2 ------ tan (x) and cot (x) are < 0 in the 2nd quadrant