Find the values of the trigonometric ratios from the graph ### Understanding the Unit CircleThe image depicts a unit circle, which is a fundamental concept in trigonometry and is often used to define sine, cosine, and tangent functions for all real numbers.#### Components of the Diagram:- **Circle:** The circle shown has a radius of 1 unit and is centered at the origin of the coordinate system (0, 0).- **Axes:** The horizontal and vertical lines represent the x-axis and y-axis, respectively.- **Point:** The point on the circumference of the circle is marked as $(-0.4, 0.917)$.- **Arrows and Curve:** The curved arrow indicates the direction in which the angle is measured from the positive x-axis, typically counterclockwise in standard position.#### Significance of Each Component:1. **Unit Circle and Coordinates:** - The unit circle helps in visualizing the trigonometric functions. - Any point $(x, y)$ on the circle satisfies the equation $x^2 + y^2 = 1$. - In this graph, the point $(-0.4, 0.917)$ indicates a position on the unit circle. The x-coordinate is -0.4, and the y-coordinate is 0.917.2. **Trigonometric Functions:** - For the point $(x, y)$ on the unit circle, $x$ represents the cosine of the angle, and $y$ represents the sine of the angle. - Here, $\cos(\theta) = -0.4$ and $\sin(\theta) = 0.917$, where $\theta$ is the angle formed with the positive x-axis.3. **Direction of Measurement:** - The arrow illustrates the direction in which the angle increases. Angles are typically measured counterclockwise from the positive x-axis. #### Application:Understanding the unit circle is critical for solving problems in trigonometry, particularly those involving periodic functions, wave functions, and oscillations. It provides a geometric representation of the trigonometric functions, making it easier to understand their properties and relationships.By studying the coordinates of various points on the unit circle, students can grasp the values of sine and cosine functions for angles in all four quadrants.For more detailed study, students should practice finding coordinates of specific angles and using them to evaluate trigonometric functions. ### Trigonometric Function ValuesUse the dropdown menus to select the appropriate values for the following trigonometric functions:1. $ \sin x = $ ____2. $ \cos x = $ ____3. $ \tan x = $ ____#### Select the appropriate value from the following options:- a. $-0.4$- b. $0.4$- c. $-2.293$- d. $-0.917$- e. $0.917$Make sure to review your choices carefully based on the context provided.

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Trigonometry

Find the values of the trigonometric ratios from the graph

Trigonometry (11th Edition)

11th Edition

ISBN:9780134217437

Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Chapter1: Trigonometric Functions

Section: Chapter Questions

Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.

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Fundamentals of Trigonometric Identities

Trigonometric Identities are the equalities that involve trigonometry functions and hold for all the values of variables given in the equation.

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Find the values of the trigonometric ratios from the graph

### Understanding the Unit Circle

The image depicts a unit circle, which is a fundamental concept in trigonometry and is often used to define sine, cosine, and tangent functions for all real numbers.

#### Components of the Diagram:

- **Circle:** The circle shown has a radius of 1 unit and is centered at the origin of the coordinate system (0, 0).
- **Axes:** The horizontal and vertical lines represent the x-axis and y-axis, respectively.
- **Point:** The point on the circumference of the circle is marked as $(-0.4, 0.917)$.
- **Arrows and Curve:** The curved arrow indicates the direction in which the angle is measured from the positive x-axis, typically counterclockwise in standard position.

#### Significance of Each Component:

1. **Unit Circle and Coordinates:**
- The unit circle helps in visualizing the trigonometric functions.
- Any point $(x, y)$ on the circle satisfies the equation $x^2 + y^2 = 1$.
- In this graph, the point $(-0.4, 0.917)$ indicates a position on the unit circle. The x-coordinate is -0.4, and the y-coordinate is 0.917.

2. **Trigonometric Functions:**
- For the point $(x, y)$ on the unit circle, $x$ represents the cosine of the angle, and $y$ represents the sine of the angle.
- Here, $\cos(\theta) = -0.4$ and $\sin(\theta) = 0.917$, where $\theta$ is the angle formed with the positive x-axis.

3. **Direction of Measurement:**
- The arrow illustrates the direction in which the angle increases. Angles are typically measured counterclockwise from the positive x-axis.

#### Application:

Understanding the unit circle is critical for solving problems in trigonometry, particularly those involving periodic functions, wave functions, and oscillations. It provides a geometric representation of the trigonometric functions, making it easier to understand their properties and relationships.

By studying the coordinates of various points on the unit circle, students can grasp the values of sine and cosine functions for angles in all four quadrants.

For more detailed study, students should practice finding coordinates of specific angles and using them to evaluate trigonometric functions.

### Trigonometric Function Values

Use the dropdown menus to select the appropriate values for the following trigonometric functions:

1. $ \sin x = $ ____
2. $ \cos x = $ ____
3. $ \tan x = $ ____

#### Select the appropriate value from the following options:
- a. $-0.4$
- b. $0.4$
- c. $-2.293$
- d. $-0.917$
- e. $0.917$

Make sure to review your choices carefully based on the context provided.

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