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.
![### Solving for the Unknown Side Using Trigonometric Ratios
**Problem Statement:**
Use a trigonometric ratio to find the value of \( x \). Round your answer to the nearest tenth.
**Diagram:**
The problem includes a right triangle with the following features:
- One angle measures 40°
- The side adjacent to the 40° angle has a length of 10 units
- The hypotenuse is labeled as \( x \)
**Note:** The diagram is not drawn to scale.
**Options:**
1. 6.4
2. 11.9
3. 7.7
4. 8.4
**Detailed Explanation:**
To solve for \( x \), we use the cosine function because we are given the length of the adjacent side (10 units) and need to find the hypotenuse (\( x \)).
The cosine of an angle in a right triangle is defined as the ratio of the adjacent side to the hypotenuse:
\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \]
For this problem:
\[ \cos(40^\circ) = \frac{10}{x} \]
Solving for \( x \):
\[ x = \frac{10}{\cos(40^\circ)} \]
Using a calculator to find \( \cos(40^\circ) \approx 0.766 \):
\[ x \approx \frac{10}{0.766} \approx 13.1 \]
However, since the provided answer choices are different, let's re-evaluate:
### Correct Calculation:
Given:
\[ \cos(40^\circ) \approx 0.766 \]
\[ \frac{10}{0.766} \approx 13.1 \]
According to the choices given, none match 13.1 precisely. Let's verify the provided answer choices to find the nearest approximation.
- Choice 1: 6.4
- Choice 2: 11.9
- Choice 3: 7.7
- Choice 4: 8.4
Based on our calculation, it looks like another trigonometric ratio or angle might be involved, ensuring to recheck calculation steps, eventual rounding errors might solve discrepancies.
**Conclusion:**
Calculate the correct values using precise cosine calculations accurately to find close matching answer from given multiple choice. Ensure to check through all](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe59874f9-f705-461a-94a3-07d771601ab4%2F21e89d14-0246-4b8f-816c-a9958ff7a287%2Fkb1hsg4_processed.jpeg&w=3840&q=75)
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