SOLUTION: In each case, find sin(a), cos(a), csc(a), sec(a), and cot(a). Note: 'a' means alpha. 45. cos(2a)=3/5 and 0�<2a<90� I've been trying to figure out this problem for some t

Question 820437: In each case, find sin(a), cos(a), csc(a), sec(a), and cot(a). Note: 'a' means alpha.
45. cos(2a)=3/5 and 0�<2a<90�

I've been trying to figure out this problem for some time, but I need help.
Found 2 solutions by stanbon, lwsshak3:
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
In each case, find sin(a), cos(a), csc(a), sec(a), and cot(a). Note: 'a' means alpha.
45. cos(2a)=3/5 and 0�<2a<90�
------
cos = x/r
If cos(2a) = 3/5, x = 3 and r = 5
------
Then y = sqrt[5^2-3^2] = sqrt[16] = 4
------
And sin(2a) = y/r = 4/5
-------
Now use the half-angle formulas to get:
sin(2a/2) = sqrt[(1-cos(2a)/2] = sqrt[(1-(3/5))/2] = sqrt[1/5] = (1/5)sqrt(5)
cos(2a/2) = sqrt[(1+cos(2a)/2] = sqrt[(1+(3/5))/2] = sqrt(4/5) = (2/5)sqrt(5)
----
tan(2a/2) = sin(2a/2)/cos(2a/2) = 1/2
------
Note: csc(a), sec(a), cot(a) are the inverse of above.
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Cheers,
Stan H.
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Answer by lwsshak3(11628) (Show Source): You can put this solution on YOUR website!
In each case, find sin(a), cos(a), csc(a), sec(a), and cot(a). Note: 'a' means alpha.45. cos(2a)=3/5 and 0�<2a<90�
***
use cos half-angle formula:

=2/√5=2√5/5
sin(a)=√(1-cos^2(a))=√(1-4/5)=√(1/5)=1/√5=√5/5
..
sin(a)=√5/5
csc(a)=√5
cos(a)=2√5/5
sec(a)=√5/2
cot(a)=cos(a)/sin(a)=2/1=2
..
calculator check:
cos(2a)=3/5
2a≈53.13˚
a≈26.565˚
..
sin(a)=sin(26.565)≈0.4472..
exact value as calculated=√5/5≈0.4472..
..
cos(a)=cos(26.565)≈0.8944..
exact value as calculated=2√5/5≈0.8944..

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