Math

QuestionFind the primary trigonometric ratios for θ=2\theta = 2 radians at the point (9,4)(-9,4) on the circle's circumference.

Studdy Solution

STEP 1

Assumptions1. The center of the circle is at (9,4)(-9,4). . The point chosen by Khalid on the circumference of the circle is also (9,4)(-9,4).
3. The angle θ\theta is given as radians.
4. We need to find the primary trigonometric ratios (sine, cosine, and tangent) for θ\theta.

STEP 2

The primary trigonometric ratios are defined in terms of the sides of a right triangle. However, in this case, the point chosen by Khalid is the same as the center of the circle. This means that the radius of the circle, which would form the hypotenuse of our right triangle, is0.

STEP 3

Since the radius is0, this means that the "triangle" we are dealing with is actually just a single point. Therefore, the lengths of the sides of our "triangle" are all0.

STEP 4

Now, let's calculate the sine of θ\theta. The sine of an angle in a right triangle is defined as the length of the opposite side divided by the length of the hypotenuse.
sin(θ)=OppositeHypotenuse\sin(\theta) = \frac{Opposite}{Hypotenuse}

STEP 5

Substitute the lengths of the opposite side and the hypotenuse into the formula.
sin(θ)=00\sin(\theta) = \frac{0}{0}

STEP 6

The division of0 by0 is undefined. Therefore, the sine of θ\theta is undefined.

STEP 7

Now, let's calculate the cosine of θ\theta. The cosine of an angle in a right triangle is defined as the length of the adjacent side divided by the length of the hypotenuse.
cos(θ)=AdjacentHypotenuse\cos(\theta) = \frac{Adjacent}{Hypotenuse}

STEP 8

Substitute the lengths of the adjacent side and the hypotenuse into the formula.
cos(θ)=00\cos(\theta) = \frac{0}{0}

STEP 9

The division of by is undefined. Therefore, the cosine of θ\theta is undefined.

STEP 10

Finally, let's calculate the tangent of θ\theta. The tangent of an angle in a right triangle is defined as the length of the opposite side divided by the length of the adjacent side.
tan(θ)=OppositeAdjacent\tan(\theta) = \frac{Opposite}{Adjacent}

STEP 11

Substitute the lengths of the opposite side and the adjacent side into the formula.
tan(θ)=00\tan(\theta) = \frac{0}{0}

STEP 12

The division of0 by0 is undefined. Therefore, the tangent of θ\theta is undefined.
The primary trigonometric ratios for θ\theta are all undefined.

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