Question

Find the exact values of the six trigonometric functions of the angle θ shown in the figure.

sin(θ)	=
cos(θ)	=
tan(θ)	=
csc(θ)	=
sec(θ)	=
cot(θ)	=

Answer 1

Consider a unit circle

Applying Pythagoras theorem

( hypoteneous ) = sqrt [ ( side 1 )^2 + ( side 2 )^2 ]

= sqrt [ 1^2 + 1^2 ]

= sqrt ( 2 ) = √ 2

sin( θ ) = opposite side / hypoteneous = 1 / √ 2

csc( θ ) = 1 / sin( θ ) = 1 / ( 1 / √ 2 ) = √ 2

cos( θ ) = adjacent side / hypoteneous = 1 / √ 2

sec( θ ) = 1 / cos( θ ) = 1 / ( 1 / √ 2 ) = √ 2

tan( θ ) = sin( θ ) / cos( θ ) = [ 1 / √ 2 ] / [ 1 / √ 2 ] = 1

cot( θ ) = 1 /  tan( θ ) = 1 / 1 = 1