Question

A circle centered at the origin has a radius of 10 units. The terminal side of an angle, Theta, intercepts the circle in Quadrant I I at point C. The y-coordinate of point C is 8. What is the value of cosine theta?

Answer 1

The value of cosine theta is -0.6, calculated using the y-coordinate of the point C(8), the radius of the circle (10), and the knowledge that in Quadrant II, x-coordinates are negative.

Step-by-step explanation:

The student is asking about finding the cosine value (cos Θ) of an angle whose terminal side intercepts a circle in Quadrant II and has a specific y-coordinate on a point.

To find the cosine of theta, we will apply the Pythagorean Theorem to a right triangle formed by the origin, point C, and the x-coordinate we seek to know. We have the y-coordinate (8 units), and the hypotenuse which is the radius of the circle (10 units).

Calculating Cosine Theta:

Using the Pythagorean Theorem:

• The square of the hypotenuse (radius r) is equal to the sum of the squares of the other two sides.
• So, r2 = x2 + y2.
• Plugging in the values r = 10, and y = 8 we get: 102 = x2 + 82.
• Simplifying, we find x2 = 100 - 64 = 36.
• Since we are in Quadrant II where the x-values are negative, x = -√36 = -6.

Finally, we use the definition of cosine for an angle Θ which is cos Θ = x/r. Here, cos Θ = -6/10 = -0.6.

Therefore, the value of cosine theta is -0.6.

Answer 2

Since the circle is centered at the origin, we can use the equations
x² + y² = r²
x = r cos θ
y = r sin θ

We know y = 8 and r = 10. Solving for x
x² + 8² = 10²
x = 6

From this equation:
x = r cos θ
6 = 10 cos θ
cos θ = 6/10 = 0.6