Find cos θ if sin θ = -(5/13) and tan θ > 0.

Solution:

We know that in the first quadrant, All positive

Second quadrant, Sine positive

Third quadrant, tan positive

Fourth quadrant, cos positive

The statement of the problem is summarized in the two diagrams above. In the 3rd quadrant sinθ is -ve and tanθ > 0.

sinθ = -5/13 which implies:

A’B = 5

OB = 13

Using the Pythagorean Relationship (for the Right-angled triangle OA’B). The base of the triangle OA’B can be calculated as follows:

OB² = A'B² + OA'²

(13)² = (5)² + (OA')²

(OA')² = (13)² - (5)²

(OA')² = 169 - 25

(OA')² = 144

OA' = Base = 12

cos θ = Base/Hypotenuse = -5/12 (cos θ is negative in the 3rd quadrant)

cos θ = -5/12

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Find cos θ if sin θ = -(5/13) and tan θ > 0.

Summary:

If sin θ = -(5/13) and tan θ > 0 then cos θ = -5/12.