Sin θ=√5/3, cos θ=2/3

3/2

2√5/5

√5/2

3√5/5

9514 1404 393

  cos θ = 2/3

Step-by-step explanation:

We presume you want the value of the cosine for

  sin θ = √5/3 ≈ 0.745356

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There are several possible approaches you can take to this problem.

1) Using the trig identity:

  cos = √(1 -sin²)

  cos θ = √(1 -(√5/3)²) = √(1 -5/9) = √(4/9)

  cos θ = 2/3

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2) Another approach you can take is to use a calculator to find the angle, then use the calculator to find the cosine of the angle.

  cos θ = cos(arcsin(√5/3)) = cos(arcsin(0.745356)) = cos(48.19°) ≈ 0.666667

  cos θ = 2/3

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3) Yet another approach is to check the answer choices for reasonableness.

You are given sin θ = 0.745356. This is more than 0.707107, the value where sine and cosine are equal, so the cosine value must be less than 0.707. The answer choices are ...

  2/3 ≈ 0.667

  3/2 . . . greater than 1; not a viable cosine value

  2√5/5 ≈ 0.894 . . . too large

  √5/2 . . . greater than 1; not a viable cosine value

  3√5/5 . . . greater than 1; not a viable cosine value

So, just looking at the values of the offered answer choices, you can choose the correct one. It is the only value with the appropriate magnitude.

  cos θ = 2/3