SOLUTION: given that cos2a =2/5 and 0<2a<90, determine the exact values of sin a, cos a, tan a, csc a, sec a, and cot a.

Algebra ->  Trigonometry-basics -> SOLUTION: given that cos2a =2/5 and 0<2a<90, determine the exact values of sin a, cos a, tan a, csc a, sec a, and cot a.      Log On


   

Question 887545: given that cos2a =2/5 and 0<2a<90, determine the exact values of sin a, cos a, tan a, csc a, sec a, and cot a.
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
Since a is in the first quadrant, everything will be positive, so
when we take square roots we only need to take positive square
roots.

cos%282a%29+=+2%2F5

Use the identity cos%282a%29=2cos%5E2%28a%29-1

2%2F5=2cos%5E2%28a%29-1
2=10cos%5E2%28a%29-5
7=10cos%5E2%28a%29
7%2F10=cos%5E2%28a%29
sqrt%287%2F10%29=cos%28a%29, that's the cosine.
You can rationalize the denominator if you like.

Then use the identity sec%28a%29=1%2Fcos%28a%29

sec%28a%29=sqrt%2810%2F7%29

Use the identity: sin%5E2%28a%29%2Bcos%5E2%28a%29=1

sin%5E2%28a%29%2B7%2F10=1
sin%5E2%28a%29=3%2F10
sin%28a%29=sqrt%283%2F10%29, that's the sine.

Then use the identity csc%28a%29=1%2Fsin%28a%29

csc%28a%29=sqrt%2810%2F3%29, that's the cosecant.

Use the identity tan%28a%29=sin%28a%29%2Fcos%28a%29 
  

tan%28a%29=sqrt%283%2F7%29, that's the tangent

Use the identity cot%28a%29=1%2Ftan%28a%29 
  
cot%28a%29=sqrt%287%2F3%29, that's the cotangent.

So that's all six.

Edwin