cos θ = 2sqrt(2)/3, tan θ = sqrt(2)/4, cot θ = 2sqrt(2), csc θ = 3, sec θ =3 sqrt(2)/4
Practice makes perfect
We want to find the values of the other five trigonometric functions of θ given that sin θ = 13 and 0 < θ < π2.
Finding cos θ
To find the value of cos θ, we will use one of the Pythagorean Identities.
sin ^2 θ + cos^2 θ = 1We will substitute 13 for sin θ in the above equation and solve it for cos θ. Let's do it!
Be aware that we are told that θ lies between 0 and π2.
Therefore, θ is in Quadrant I.
In this quadrant, the cosine of θ is positive. Therefore, we will only keep the positive solution.
cos θ = 2sqrt(2)/3
Finding the Other Trigonometric Ratios
Knowing that sin θ = 13 and cos θ = 2 sqrt(2)3, we are allowed to find the four remaining trigonometric ratios. Remember to simplify fractions and rationalize denominators, if needed.
Function
Substitute
Simplify
tan θ=sin θ/cos θ
tan θ=13/2sqrt(2)3
tan θ = sqrt(2)/4
cot θ=cos θ/sin θ
cot θ=2sqrt(2)3/13
cot θ=2sqrt(2)
csc θ=1/sin θ
csc θ=1/13
csc θ=3
sec θ=1/cos θ
sec θ=1/2sqrt(2)3
sec θ=3sqrt(2)/4
Showing Our Work
Simplifying Fractions and Rationalizing Denominators
Simplifying the above fractions requires different methods for each case. Let's start with cot θ.
