Evaluate by using the Pythagorean identities
Find sin θ and cos θ if tan θ=1/6 and sin θ >0
tan = sin / cos ... cos = 6 sin
sin^2 + cos^2 = 1 ... 37 sin^2 = 1
tanθ and sinθ are both positive only in QI
Draw your standard triangle. You have
y = 1
x = 6
r = √37
tanθ = y/x
sinθ = y/r
cosθ = x/r
To evaluate sin θ and cos θ using the Pythagorean identities, we first need to find the value of tan^2 θ using the given information.
We are given that tan θ = 1/6. Recall that the Pythagorean identity for tangent is tan^2 θ = 1 + tan^2 θ.
Substituting the given value, we get:
(1/6)^2 = 1 + tan^2 θ
Simplifying, we have:
1/36 = 1 + tan^2 θ
Now, let's solve for tan^2 θ:
1 + tan^2 θ = 1/36
Subtracting 1 from both sides, we get:
tan^2 θ = 1/36 - 1
tan^2 θ = -35/36
Since tan^2 θ cannot be negative, we have an incorrect value for tan θ.
Therefore, the given information is not consistent, and we cannot determine the values of sin θ and cos θ using the Pythagorean identities.