Math  /  Trigonometry

QuestionChallenge Question 5 Explain how the unit circle definition of sin\sin and cos\cos is related to the equations sin(θ)=OH\sin (\theta)=\frac{O}{H} and cos(θ)=AH\cos (\theta)=\frac{A}{H}
Type your explanation in the box below, sketching on top of the diagram if necessary. (You can use the example of θ=55\theta=55^{\circ} or explain why it is true in general.)

Studdy Solution

STEP 1

1. The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane.
2. The angle θ\theta is measured from the positive x-axis.
3. The definitions of sine and cosine on the unit circle relate to right triangle trigonometry.

STEP 2

1. Define the unit circle and the coordinates of a point on it.
2. Relate the unit circle to right triangle trigonometry.
3. Explain the equations sin(θ)=OH\sin(\theta) = \frac{O}{H} and cos(θ)=AH\cos(\theta) = \frac{A}{H}.

STEP 3

On the unit circle, any angle θ\theta corresponds to a point (x,y)(x, y) on the circle, where the radius is 1.
The x-coordinate of this point is cos(θ)\cos(\theta) and the y-coordinate is sin(θ)\sin(\theta).

STEP 4

Consider a right triangle formed by dropping a perpendicular from the point (x,y)(x, y) to the x-axis. This creates a right triangle with: - Hypotenuse = 1 (radius of the unit circle) - Opposite side = yy (vertical leg) - Adjacent side = xx (horizontal leg)

STEP 5

In right triangle trigonometry: - sin(θ)=OppositeHypotenuse=y1=y\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{y}{1} = y - cos(θ)=AdjacentHypotenuse=x1=x\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{x}{1} = x
Thus, on the unit circle, sin(θ)\sin(\theta) and cos(θ)\cos(\theta) directly relate to the ratios OH\frac{O}{H} and AH\frac{A}{H} because the hypotenuse is 1.
The unit circle definition of sin\sin and cos\cos is consistent with the right triangle definitions because the unit circle's radius is 1, making the hypotenuse 1 in the triangle definitions.

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