trigonometric function

Question

A point on the terminal side Θ is (2/5, 5/8) determine the exact value of the 6 trigonometric fictional of Θ

Answer 1

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

Answer 2

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