Math Problem Statement
Find the reference number for each value of t.
(a)
t =
4π
7
(b)
t = β
7π
9
(c)
t = β3
(d)
t = 5
Solution
The reference number (also called the reference angle) is the positive acute angle that makes with the x-axis in the unit circle. It is always between and .
Finding the reference number for each given value of :
(a)
- Determine the quadrant:
- Since is between and , it is in Quadrant II because .
- Find the reference number:
(b)
- Determine the quadrant:
- Since is negative, we move clockwise.
- is between and , so it is in Quadrant III because .
- Find the reference number:
(c)
- Convert to radians:
- , so is close to .
- Since is in Quadrant III (because ), we compute the reference number:
- Find the reference number: Approximating ,
(d)
- Convert to radians:
- Since , we note that , meaning it is in Quadrant IV.
- The closest multiple of (which is about 6.28) is greater than 5, so it remains in the first rotation.
- Find the reference number: Approximating ,
Final Answers:
(a)
(b)
(c)
(d)
Do you want a step-by-step breakdown for any part?
Related Questions:
- What is the general formula for finding reference numbers in different quadrants?
- How do reference numbers help in trigonometry?
- Can reference numbers be negative?
- How do reference numbers relate to periodic functions like sine and cosine?
- What is the significance of the unit circle in calculating reference numbers?
Tip: If the angle is negative, always move clockwise to find its quadrant.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Trigonometry
Unit Circle
Reference Angle
Radians
Formulas
Reference Number = |Ο - t| (if in Quadrant II or III)
Reference Number = |t - Ο| (if in Quadrant IV)
Reference Number = |Ο - 3| (for non-exact radian values)
Reference Number = |5 - Ο| (for non-exact radian values)
Theorems
Reference Number Theorem
Quadrant Identification in Trigonometry
Unit Circle Properties
Suitable Grade Level
Grades 10-12