For an angle θ with the point (16, -30) on its terminating side, what is the value of cosine?

Solution:

The problem statement can be represented pictorially below:

The point on the circumference of the circle represented as (16, -30) implies that x = 16 and y = -30

The angle θ subtends the arc AB on the circumference. By dropping a perpendicular line from point B onto OA meeting at A', we get a right-angled triangle ∠OA'B. As the coordinates of point B are (12,-5) it implies:

A'B = 30 and OA' = 16

The hypotenuse OB can be obtained by applying the pythagorean theorem by using the relationship:

OB² = OA'² + A'B²

OB² = (30)² + (16)²

OB² = 900 + 256 = 1156

OB = √1156

OB = 34

Now Cosine of the angle θ is defined as :

Cos (θ) = base/hypotenuse

OA'/OB = 16/34 = 8/17

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For an angle θ with the point (16, -30) on its terminating side, what is the value of cosine?

Summary:

For an angle θ with the point (16, -30) on its terminating side, the cosine of the angle θ is defined as the base over hypotenuse which OA’/OB is = 8/17, which is the desired solution of the problem.