Mark the statements that are true.
A. An angle that measures $\frac{\pi}{6}$ radians also measures $30^{\circ}$.
B. An angle that measures $180^{\circ}$ also measures $\frac{\pi}{2}$ radians.
C. An angle that measures $\frac{\pi}{3}$ radians also measures $60^{\circ}$.
D. An angle that measures $30^{\circ}$ also measures $\frac{\pi}{3}$ radians.
Answer (1)
Convert radians to degrees by multiplying by π 18 0 ∘ .
Convert degrees to radians by multiplying by 18 0 ∘ π .
Statement A: 6 π radians = 3 0 ∘ (True).
Statement B: 18 0 ∘ = π radians (False).
Statement C: 3 π radians = 6 0 ∘ (True).
Statement D: 3 0 ∘ = 6 π radians (False).
The true statements are: A , C .
Explanation
Analyze the statements Let's analyze each statement to determine if it's true. We'll use the conversion factor π radians = 18 0 ∘ .
Check statement A Statement A: An angle that measures 6 π radians also measures 3 0 ∘ .
To convert 6 π radians to degrees, we multiply by π 18 0 ∘ :
6 π ⋅ π 18 0 ∘ = 6 18 0 ∘ = 3 0 ∘ So, statement A is true.
Check statement B Statement B: An angle that measures 18 0 ∘ also measures 2 π radians. To convert 18 0 ∘ to radians, we multiply by 18 0 ∘ π :
18 0 ∘ ⋅ 18 0 ∘ π = π radians So, 18 0 ∘ = π radians, not 2 π radians. Therefore, statement B is false.
Check statement C Statement C: An angle that measures 3 π radians also measures 6 0 ∘ .
To convert 3 π radians to degrees, we multiply by π 18 0 ∘ :
3 π ⋅ π 18 0 ∘ = 3 18 0 ∘ = 6 0 ∘ So, statement C is true.
Check statement D Statement D: An angle that measures 3 0 ∘ also measures 3 π radians. To convert 3 0 ∘ to radians, we multiply by 18 0 ∘ π :
3 0 ∘ ⋅ 18 0 ∘ π = 6 π radians So, 3 0 ∘ = 6 π radians, not 3 π radians. Therefore, statement D is false.
Conclusion Based on our analysis, statements A and C are true, while statements B and D are false.
Examples
Understanding radian and degree conversions is crucial in fields like engineering and physics, especially when dealing with circular motion or wave phenomena. For example, when calculating the angular velocity of a rotating object, you need to ensure that the angle is measured in radians for the formulas to work correctly. Similarly, in signal processing, the frequency of a wave is often expressed in radians per second.
