Radians Flashcards
What is a radian?
A radian is the angle subtended at the center of a circle by an arc equal in length to the radius.
How many radians are in a full circle?
2π radians.
Convert 180° to radians.
π radians.
Convert 1 radian to degrees.
Approximately 57.3°.
What is the formula for arc length (s)?
s = rθ, where r is radius and θ is in radians.
What is the formula for the area of a sector?
A = 1/2 * r² * θ, where θ is in radians.
What is the formula for the area of a segment?
A = 1/2 * r²(θ - sinθ), where θ is in radians.
Why must θ be in radians for the arc length formula?
Because the formula s = rθ is only valid when θ is in radians.
What is the small angle approximation for sin(θ)?
sin(θ) ≈ θ when θ is small and in radians.
What is the small angle approximation for tan(θ)?
tan(θ) ≈ θ when θ is small and in radians.
What is the small angle approximation for cos(θ)?
cos(θ) ≈ 1 - θ²/2 when θ is small and in radians.
In which units must you work when using small angle approximations?
Radians.
What is the range of values for which small angle approximations are accurate?
Accurate for small values of θ (typically < 0.1 radians).
How do you convert degrees to radians?
Multiply degrees by π/180.
How do you convert radians to degrees?
Multiply radians by 180/π.
How do you express π/3 radians in degrees?
60°.
What is the unit of angle used in calculus (differentiation/integration of trig)?
Radians.
Why are radians preferred in advanced mathematics?
Because they simplify formulas and derivatives of trigonometric functions.
How do you find the length of a chord given r and θ?
Chord length = 2r * sin(θ/2)
What happens to arc length and area formulas if θ is in degrees?
They must be adjusted with a conversion factor; otherwise, results are incorrect.
