Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
![Title: Example Problem Using Right Triangle with Known Angle
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## Example Problem: Use the Picture Below
### Problem Statement:
Refer to the picture below to solve the following problem:
### Diagram:
The image shows a right triangle with:
- One angle marked as \( 30^\circ \)
- The side opposite the \( 30^\circ \) angle is labeled \( a \)
- The side adjacent to the \( 30^\circ \) angle is labeled \( b \)
- The hypotenuse is labeled \( 11 \)
### Explanation:
In the context of right triangles and trigonometric principles, this diagram may be used to illustrate how to apply sine, cosine, and tangent ratios to find missing lengths of sides or angle measures within the triangle.
1. **Using the Sine Function:**
\[
\sin(30^\circ) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{a}{11}
\]
Using the known value of \(\sin(30^\circ) = \frac{1}{2} \):
\[
\frac{1}{2} = \frac{a}{11} \implies a = \frac{11}{2} = 5.5
\]
2. **Using the Cosine Function:**
\[
\cos(30^\circ) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{b}{11}
\]
Using the known value of \(\cos(30^\circ) = \frac{\sqrt{3}}{2} \):
\[
\frac{\sqrt{3}}{2} = \frac{b}{11} \implies b = 11 \cdot \frac{\sqrt{3}}{2} = \frac{11\sqrt{3}}{2}
\]
3. **Pythagorean Theorem:**
To verify the sides, we can use the Pythagorean theorem:
\[
a^2 + b^2 = 11^2
\]
Substitute the values of \( a \) and \( b \):
\[
(5.5)^2 + \left( \frac{11\sqrt{3}}{2} \right)^2 = 121
\]
Calculate each](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F34bff1e1-af68-4229-b0ef-aa0aaa134e46%2F7e10c4e0-cd9c-437d-b9d3-2571b03f6b43%2F11u6fkh_processed.jpeg&w=3840&q=75)
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