Suppose a = 12 and b = 9. Find an exact value or give at least two decimal places: sin(A) = %3D

Calculus: Early Transcendentals
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Author:James Stewart
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Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Trigonometric Ratios in a Right Triangle

#### Overview:
This section covers how to find the trigonometric ratios for a given right triangle. 

#### Triangle Diagram:
- **Vertices**: The triangle has vertices labeled as A, B, and the right angle.
- **Sides**: The sides of the triangle are labeled as follows:
  - "a" is opposite A
  - "b" is adjacent to A
  - "c" is the hypotenuse

#### Note:
- The triangle may not be drawn to scale.

#### Given Values:
- \( a = 12 \)
- \( b = 9 \)

#### Task:
Find an exact value or give at least two decimal places for the following trigonometric ratios:

1. **Sin(A)**: \(\sin(A) = \)
   - This represents the ratio of the length of the side opposite angle A to the length of the hypotenuse.

2. **Cos(A)**: \(\cos(A) = \)
   - This represents the ratio of the length of the side adjacent to angle A to the length of the hypotenuse.

3. **Tan(A)**: \(\tan(A) = \)
   - This represents the ratio of the length of the side opposite angle A to the length of the side adjacent to angle A.

4. **Sec(A)**: \(\sec(A) = \)
   - This is the reciprocal of the cosine of angle A.

5. **Csc(A)**: \(\csc(A) = \)
   - This is the reciprocal of the sine of angle A.

6. **Cot(A)**: \(\cot(A) = \)
   - This is the reciprocal of the tangent of angle A.

#### Diagram Description:
- The triangle shown in the diagram is a right-angled triangle.
- Angle B is the right angle.
- Angle A is an acute angle between sides 'b' and 'c'.
- Side 'a' is opposite angle A.
- Side 'b' is adjacent to angle A.
- Side 'c' is the hypotenuse of the triangle.

#### Calculation Example:
To find the values, you will first need to find the length of the hypotenuse (c) using the Pythagorean theorem:
\[ c = \sqrt{a^2 + b^2} \]

Then use this length to calculate the required trigonometric ratios
Transcribed Image Text:### Trigonometric Ratios in a Right Triangle #### Overview: This section covers how to find the trigonometric ratios for a given right triangle. #### Triangle Diagram: - **Vertices**: The triangle has vertices labeled as A, B, and the right angle. - **Sides**: The sides of the triangle are labeled as follows: - "a" is opposite A - "b" is adjacent to A - "c" is the hypotenuse #### Note: - The triangle may not be drawn to scale. #### Given Values: - \( a = 12 \) - \( b = 9 \) #### Task: Find an exact value or give at least two decimal places for the following trigonometric ratios: 1. **Sin(A)**: \(\sin(A) = \) - This represents the ratio of the length of the side opposite angle A to the length of the hypotenuse. 2. **Cos(A)**: \(\cos(A) = \) - This represents the ratio of the length of the side adjacent to angle A to the length of the hypotenuse. 3. **Tan(A)**: \(\tan(A) = \) - This represents the ratio of the length of the side opposite angle A to the length of the side adjacent to angle A. 4. **Sec(A)**: \(\sec(A) = \) - This is the reciprocal of the cosine of angle A. 5. **Csc(A)**: \(\csc(A) = \) - This is the reciprocal of the sine of angle A. 6. **Cot(A)**: \(\cot(A) = \) - This is the reciprocal of the tangent of angle A. #### Diagram Description: - The triangle shown in the diagram is a right-angled triangle. - Angle B is the right angle. - Angle A is an acute angle between sides 'b' and 'c'. - Side 'a' is opposite angle A. - Side 'b' is adjacent to angle A. - Side 'c' is the hypotenuse of the triangle. #### Calculation Example: To find the values, you will first need to find the length of the hypotenuse (c) using the Pythagorean theorem: \[ c = \sqrt{a^2 + b^2} \] Then use this length to calculate the required trigonometric ratios
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