Find all six trig functions for angle 0, given that sec e V5, and 0 is in quadrant IV. Remember to rationalize denominators where required.

Answer 1

Given:

sec θ = √( 5 )

θ is in Quadrant IV

Find:

cos θ = ?

tan θ = ?

sin θ = ?

csc θ = ?

cot θ = ?

Solution:

Because θ is in Quadrant IV, this means:

sec θ > 0 (secant will be POSITIVE, and already is from the givens)

tan θ < 0 (tangent will be negative)

cot θ < 0 (cotangent will be negative)

sin θ < 0 (sine will be negative)

cos θ > 0 (cosine will be POSITIVE)

csc θ < 0 (cosecant will negative)

This helps in choosing between the + and - in the ±

tangent-secant Identity:

1 + tan² θ = sec² θ

1 + tan² θ = 5

tan² θ = 4

tan θ = -2

cot θ = 1 / tan θ

cot θ = -1/2

cos θ = 1 / sec θ = 1 / √( 5 )

cos θ = √( 5 ) / 5

sin θ / cos θ = tan θ

sin θ = tan θ * cos θ

sin θ = ( -2 ) * ( √( 5 ) / 5 )

sin θ = - 2 * √( 5 ) / 5

csc θ = 1 / sin θ = 1 / ( -2 / √( 5 ) )

csc θ = - √( 5 ) / 2

I hope this helps! Message me in the comments with any questions, comments, or concerns on what I did above!