Geometry: Angles and Trigonometry Concepts

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Questions and Answers

What is the measure of a right angle in degrees?

90° (correct)

60°

45°

180°

In the circular system, how is an angle defined?

As the circumference of a circle divided by its diameter

As the area of a circle divided by its radius

As the angle subtended at the center of a circle by an arc equal to the radius (correct)

As the length of a circle divided by the radius

Which of the following statements is true regarding the sexagesimal system?

Each minute is divided into 10 seconds.

Each degree is divided into 60 minutes. (correct)

A degree is equivalent to 100 grades.

A right angle is divided into 100 equal parts.

What is the symbol used to denote one degree?

1° Signup and view all the answers

In which system is a right angle divided into 100 equal parts?

Centesimal System Signup and view all the answers

How many seconds are there in one minute in the sexagesimal system?

60 Signup and view all the answers

What does a positive angle indicate in terms of rotation direction?

Anti-clockwise direction Signup and view all the answers

What is one minute represented by in the centesimal system?

1′ Signup and view all the answers

What is the equivalent of 1 radian in degrees?

57°16’22” Signup and view all the answers

In the formula $θ = l/r$, what does 'l' represent?

Length of the arc Signup and view all the answers

Which of the following is true regarding the range of the sine function?

|sin θ| ≤ 1 Signup and view all the answers

Which trigonometric function is equal to $1/x$ when $x eq 0$?

sec θ Signup and view all the answers

Which of the following describes trigonometric identities?

Equations that hold true for all values Signup and view all the answers

Which angle is allied to the angle θ?

270° + θ Signup and view all the answers

What is the value of 1° in radians?

0.01746 rad Signup and view all the answers

What is the definition of trigonometric ratios?

Relationships between different sides and angles of a triangle Signup and view all the answers

What is the defining characteristic of a periodic function?

It repeats its values at regular intervals. Signup and view all the answers

What does the symbol T represent in the context of periodic functions?

The period of the function Signup and view all the answers

Which of the following trigonometric functions is considered periodic?

Trigonometric functions Signup and view all the answers

What is true about the angles used in trigonometric ratios of compound angles?

They can be formed by summing multiple angles. Signup and view all the answers

Which statement is NOT true regarding transformation formulae in trigonometry?

They create new trigonometric functions. Signup and view all the answers

Study Notes

Angles

An angle is formed by a ray OA rotating about point O to position OB, denoted as ∠AOB.
Positive angles result from counterclockwise rotation, while negative angles arise from clockwise rotation.

Measurement of Angles

Sexagesimal System (Degrees)

A right angle is divided into 90 degrees, symbolized as 1°.
Each degree has 60 minutes (1′) and each minute has 60 seconds (1″).
Relationships: 1 right angle = 90°, 1° = 60′, 1′ = 60″.

Centesimal System (Grades)

A right angle is divided into 100 grades, with 1g = 100′ and 1′ = 100″.

Circular System (Radians)

One radian is the angle subtended at the center of a circle by an arc equal to the circle's radius.
Relationships: π radians = 180°, 1° = (π/180) rad.

Trigonometric Ratios

Defined as the relationship between the sides and angles of a right triangle.
Crucial for solving various mathematical problems and understanding circular functions.

Trigonometric Functions on the Unit Circle

Defined on the coordinate axes, using unit radius:
- cos(θ) = x
- sin(θ) = y
- sec(θ) = 1/x (where x ≠ 0)
- cosec(θ) = 1/y (where y ≠ 0)
- tan(θ) = y/x (where x ≠ 0)
- cot(θ) = x/y (where y ≠ 0)

Range of Trigonometric Functions

|sin(θ)| ≤ 1
|cos(θ)| ≤ 1
|sec(θ)| ≥ 1
|cosec(θ)| ≥ 1 when defined.

Trigonometric Identities

Equations involving trigonometric functions, valid for all angles where the functions are defined.

Trigonometric Ratios of Standard Angles

Special angles have determined trigonometric ratios aiding in calculations.

Allied Angles

Angles are allied if their sum or difference yields zero or is a multiple of 90°.
Examples include angles like θ, 90° ± θ, 180° ± θ, and others depending on θ.

Periodic Functions

A function f(x) is periodic if there exists a T > 0 such that f(x + T) = f(x) for all x.
All trigonometric functions are classified as periodic.

Maximum and Minimum Values

Trigonometric expressions can reach specific maximum and minimum values related to their periodicity.

Compound Angles

The sum of two or more angles is called a compound angle, relevant in trigonometric calculations.

Transformation Formulae

These help simplify and transform trigonometric expressions for ease of use in calculations.

Ratios of Multiple Angles

Specific formulas exist to handle trigonometric ratios when dealing with multiples of angles.

Useful Angles

Certain angles, due to their easy relationships, are often used for quick calculations in trigonometry.

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Description

This quiz covers fundamental concepts related to angles, their measurement in different systems, and trigonometric ratios. Test your understanding of degrees, radians, and the relationships between angles and their respective measurements. Perfect for students looking to strengthen their knowledge in geometry and trigonometry.