Questions and Answers
What is the measure of a right angle in degrees?
In the circular system, how is an angle defined?
Which of the following statements is true regarding the sexagesimal system?
What is the symbol used to denote one degree?
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In which system is a right angle divided into 100 equal parts?
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How many seconds are there in one minute in the sexagesimal system?
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What does a positive angle indicate in terms of rotation direction?
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What is one minute represented by in the centesimal system?
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What is the equivalent of 1 radian in degrees?
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In the formula $θ = l/r$, what does 'l' represent?
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Which of the following is true regarding the range of the sine function?
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Which trigonometric function is equal to $1/x$ when $x eq 0$?
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Which of the following describes trigonometric identities?
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Which angle is allied to the angle θ?
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What is the value of 1° in radians?
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What is the definition of trigonometric ratios?
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What is the defining characteristic of a periodic function?
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What does the symbol T represent in the context of periodic functions?
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Which of the following trigonometric functions is considered periodic?
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What is true about the angles used in trigonometric ratios of compound angles?
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Which statement is NOT true regarding transformation formulae in trigonometry?
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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.
