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If sinθ =2/3, which of the following are possible?

o secθ = -3/2 and tanθ 2/√5
o cosθ = -√5/3 and tanθ 2/√5
o cosθ = √5/3 and tanθ 2/√5
o secθ = 3/√5 and tanθ 2/√5

User Farris
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1 Answer

4 votes

Final answer:

Given that sinθ = 2/3, the possible values for cosine and tangent are cosθ = -√5/3 and tanθ = 2/√5, as these adhere to the trigonometric functions' relationships in a right triangle. Option 2 is correct.

Step-by-step explanation:

When given that sinθ = 2/3, we can find the corresponding cosine and tangent values by using Pythagorean identities and the definitions of trigonometric functions in a right triangle.

Firstly, using the Pythagorean identity sin^2θ + cos^2θ = 1, we can solve for cosθ:

sin^2θ = (2/3)^2 = 4/9

cos^2θ = 1 - sin^2θ = 1 - 4/9 = 5/9

cosθ = ±√(5/9) = ±√5/3

Since cosθ can be either positive or negative, depending on the quadrant, we look at the possible answers:

cosθ = -√5/3 is possible if θ is in the second or third quadrant.

cosθ = √5/3 is only possible if θ is in the first quadrant, which is not the case here as secθ cannot be negative when cosθ is positive.

Now, let's determine tanθ using tanθ = sinθ / cosθ:

For cosθ = -√5/3, tanθ = (2/3) / (-√5/3) = -2/√5

As a result, the pair cosθ = -√5/3 and tanθ = 2/√5 is the correct answer since the signs of cosine and tangent must agree in the second and third quadrants.

User Juris
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9.1k points