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Questions and Answers
Which of the following statements is true about the domain of trigonometric functions?
Which of the following statements is true about the domain of trigonometric functions?
What is the relationship between an angle of 360 degrees and an arc of length 2π?
What is the relationship between an angle of 360 degrees and an arc of length 2π?
Which of the following is NOT true about the domain of tangent of x?
Which of the following is NOT true about the domain of tangent of x?
What is the conclusion about the domain of trigonometric functions based on the relationship between an angle of 360 degrees and an arc of length 2π?
What is the conclusion about the domain of trigonometric functions based on the relationship between an angle of 360 degrees and an arc of length 2π?
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What is the context in which it makes sense to say that trigonometric functions are continuous except where they're not?
What is the context in which it makes sense to say that trigonometric functions are continuous except where they're not?
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Study Notes
Domain of Trigonometric Functions
- Trigonometric functions such as sine and cosine have a domain of all real numbers, while tangent has a restricted domain due to vertical asymptotes.
- The function tangent is undefined for angles where cosine equals zero, specifically at odd multiples of 90 degrees (±90°, ±270°, etc.).
Relationship Between Degrees and Arc Length
- An angle of 360 degrees corresponds to a full rotation around a circle, which is equivalent to an arc length of 2π when measured in radians.
- This relationship establishes that a complete circular path of radius 1 covers the arc length of 2π.
Properties of Tangent Function
- The statement "tangent of x has a domain of all real numbers" is false; it omits the points where the function is undefined.
- Identifying vertical asymptotes is crucial for understanding the discontinuous nature of tangent.
Conclusion on Trigonometric Functions
- The domain of trigonometric functions is determined by their periodic nature and the definition of their respective angles and arc lengths.
- The relationship between 360 degrees and 2π reinforces the periodicity, showing that trigonometric functions repeat every 2π radians.
Continuity of Trigonometric Functions
- Trigonometric functions are generally considered continuous; they only experience discontinuity at specific points like those in tangent and cotangent due to undefined values.
- The concept of continuity emphasizes that aside from these points, trigonometric functions demonstrate unbroken behavior across their domains.
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Description
Test your knowledge on the limits of trigonometric functions with this quiz. Explore the continuity of trigonometric functions and discover key concepts explained in the Khan Academy video.
