If we know that sin(t) = 5/7 , how to find other identies like cos(t) and tan(t) ?
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We know that sin t = 5/7. We use the relation (sin x)^2 + (cos x)^2 = 1, to find cos t
(cos t)^2 + (5/ 7)^2 = 1
=> (cos t)^2 = 1 - 25 / 49
=> (cos t)^2 = 24 / 49
=> cos t = (sqrt 24) / 7 or -(sqrt 24) / 7
tan t = sin t / cos t
=> (5/7)/ [(sqrt 24)/7] or (5/7)/ [-(sqrt 24)/7]
=> 5/ (sqrt 24) or -5/ (sqrt 24)
The required value of...
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We'll start from the identity:
1 + (cot t)^2 = 1/(sin t)^2
(cot t)^2 = 1/(sin t)^2 - 1
(cot t)^2 = 49/25 - 1
(cot t)^2 = (49-25)/25
(cot t)^2 = 24/25
cot t = 2sqrt6/5 or cot t = -2sqrt6/5
We know that tan t = 1/cot t
tan t = 5/2sqrt6 => tan t = 5sqrt6/12 or tan t = -5sqrt6/12
We also know that tan t = sin t/cos t
cos t = sin t/tan t
cos t = 5/7/5sqrt6/12 => cos t = 12/7sqrt6
cos t = 12sqrt6/42 = 6sqrt6/21
cos t = -6sqrt6/21
The values for cos t and tan t are: cos t = +6sqrt6/21; cos t = - 6sqrt6/21 ; tan t = 5sqrt6/12 ; tan t = -5sqrt6/12.
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