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trigonometric function

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2 Answers

given (a,b) = (2/5,5/8)
 
c = SQRT[(2/5)2+ (5/8)2]
 
Sin = (5/8)/SQRT[4/25+25/64]
 
Cos = (2/5)/SQRT[4/25+25/64]
 
Tan = (5/8)/(2/5)= 25/16
 
Csc = 1/Sin = SQRT[4/25+25/64]/(5/8)
 
Sec = 1/Cos = SQRT[4/25+25/64]/(2/5)
 
Cot = 1/Tan = 16/25
You have a right triangle whose adjacent side relative to θ is 2/5 and whose opposite side is 5/8. According to the Pythagorean theorem, the hypotenuse is
√( (2/5)²+(5/8)² ) = √(881) /40.
 
Now use SOH-CAH-TOA:
sin(θ) = (5/8)/(√(881) /40) = 25 /√(881) (rationalize the denominator if you like)
cos(θ) =(2/5)/(√(881) /40) = 16 /√(881) 
tan(θ) = (5/8)/(2/5) = 25/16
 
The other three trig functions are the reciprocals of these:
sec(θ) = 1/cos(θ) = √(881) /16
csc(θ) = 1/sin(θ) = √(881) /25
cot(θ) = 1/tan(θ) = 16/25