If sin θ = 1 over 3 and tan θ < 0, what is the value of cos θ?

Solution:

It is given that

sin θ = 1/3, tan θ < 0.

Consider sin θ = 1/3

By squaring both sides we get,

sin² θ = 1/9 .... [equation 1]

As we know that cos² θ + sin² θ = 1

So, on substituting the value of sin² θ = 1 - cos² θ in equation 1, we get

1 - cos² θ = 1/9

cos² θ = 1 - 1/9

cos² θ = (9 -1)/9

cos² θ = 8 / 9

cos θ = 2 √2 / 3 or - 2√2/3

Now it is also given that tan θ < 0 hence, cos θ < 0 since sin θ > 0.

Therefore, the value of cos θ is  - 2√2 / 3.

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If sin θ = 1 over 3 and tan θ < 0, what is the value of cos θ?

Summary:

sin θ = 1 over 3 and tan θ < 0, the value of cos θ - 2√2/3.