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.
![### Using Trigonometric Ratios to Solve for Triangle Sides
In the image provided, we have a right triangle with the following components:
- One side labeled as 10 (this is the opposite side to the given angle).
- An angle marked as 45°.
- The hypotenuse is labeled as \( b \).
- The adjacent side to the given angle is labeled as \( a \).
- A right-angle symbol is present, indicating that it is a right triangle.
Given this triangle, we can use trigonometric ratios to find the lengths of the unknown sides \( a \) and \( b \).
#### Important Trigonometric Ratios
For a right triangle:
- Sine (\(\sin\)) of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
- Cosine (\(\cos\)) of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.
- Tangent (\(\tan\)) of an angle is the ratio of the length of the opposite side to the length of the adjacent side.
#### Using the 45° Angle
1. **Finding \( a \):**
\[ \tan(45^\circ) = \frac{\text{opposite}}{\text{adjacent}} \]
\[ 1 = \frac{10}{a} \]
\[ a = 10 \]
2. **Finding \( b \):**
\[ \sin(45^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} \]
\[ \frac{\sqrt{2}}{2} = \frac{10}{b} \]
\[ b = 10 \times \frac{\sqrt{2}}{2} \]
\[ b = 10\sqrt{2} \approx 14.14 \]
Therefore, the lengths of the sides \( a \) and \( b \) are 10 and \( 10\sqrt{2} \) (approximately 14.14), respectively.
This triangle demonstrates the usage of basic trigonometric principles to determine unknown side lengths when one side length and one angle (apart from the right angle) are known.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F34bff1e1-af68-4229-b0ef-aa0aaa134e46%2F2774bc6b-0834-4860-9560-cd917cacace5%2F8ad66z_processed.jpeg&w=3840&q=75)
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