find the exact values for all 6 trigonometric functions with the provided information sec(x)=2, sin(x)>0

Question

find the exact values for all 6 trigonometric functions with the provided information sec(x)=2, sin(x)>0

sin

 

cos

 

tan

 

csc

 

sec

 

cot

 

 

Answer 1

Assuming you know the inter-relationship between the trigonometric ratios

If sin (x) is positive and sec (x) is also positive, then all 6 ratios are positive

Since sec(x) = 2, cos (x) = 1/2

sin²(x) = 1 - cos²(x) = 1 - (1/2)² = 3/4, therefore sin (x) = sqrt(3)/2

csc (x) = 2/sqrt(3)

tan (x) = sin (x)/cos (x) = sqrt(3)

cot (x) = 1/sqrt(3)

Answer 2

This problem involves one of the special triangles, namely the 30-60-90 triangles. Since sin x > 0, and sec Θ = 2, then theta lies in quadrant I. For what value of theta does sec θ = 2. This is true for the angle 60°.  Now, find the value of the six trig functions for this angle theta. All of them will be positive since both sine and cosine are positive in quadrant I. 

 

sin 60° = √3/2

cos 60° = 1/2

tan 60° = √3

csc 60° = 2/√3

cot 60° = 1/√3