# Trig

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If csc(theta)= -2 and theta lies in Quadrant III, find tan(theta).

• Trig -

cscØ = -2
sinØ = -1/2, Ø in III
Ø = 210°
then tan 210° = 1/√3

(I sketched the right-angled triangle in quad III with sides
1 - √3 - 2, for which you should know the ratios.
from the CAST rule I knew that the tangent in III is +

• Trig -

sin ( theta ) , cos ( theta ) , sec ( theta ) and csc( theta ) are negative.

tan ( theta ) = sin ( theta ) / cos ( theta )

are positive.

tan ( theta ) = + OR -1 / sqrt [ csc ( theta ) ^ 2 - 1 ]

tan ( theta ) = + OR -1 / sqrt [ ( -2 ) ^ 2 - 1 ]

tan ( theta ) = + OR -1 / sqrt ( 4 - 1 )

tan ( theta ) = + OR -1 / sqrt ( 3 )

tan ( theta ) are positive so:

tan ( theta ) = 1 / sqrt ( 3 )

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