We want to find the values of the other five trigonometric functions of θ given that sin θ = - 710 and π < θ < 3π2.
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 - 710 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 π and 3π2.
Therefore, θ is in Quadrant III.
In this quadrant, the cosine of θ is negative. Therefore, we will only keep the negative solution.
cos θ = - sqrt(51)/10
Finding the Other Trigonometric Ratios
Knowing that sin θ = - 710 and cos θ = - sqrt(51)10, 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=- 710/- sqrt(51)10
tan θ = 7sqrt(51)/51
cot θ=cos θ/sin θ
cot θ=- sqrt(51)10/- 710
cot θ=sqrt(51)/7
csc θ=1/sin θ
csc θ=1/- 710
csc θ=- 10/7
sec θ=1/cos θ
sec θ=1/- sqrt(51)10
sec θ=- 10 sqrt(51)/51
Showing Our Work
Simplifying Fractions and Rationalizing Denominators
Simplifying the above fractions requires different methods for each case. Let's start with cot θ.
