Mathematics
Grade 11
15 min
Which customary unit is appropriate?
Which customary unit is appropriate?
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1
Introduction & Learning Objectives
Learning Objectives
Differentiate between degree and radian measures and identify the appropriate context for each.
Select the most logical customary unit of length (e.g., inches, feet, miles) for the solution to a real-world trigonometry problem.
Analyze the parameters of a sinusoidal function and determine the appropriate units for amplitude and period.
Correctly apply the arc length formula by ensuring the angle is measured in radians.
Convert between degrees and radians to solve problems that involve mixed angular units.
Justify their choice of units when modeling periodic phenomena like circular motion or wave behavior.
If you use trigonometry to find the height of Mount Everest, should your answer be in inches? 🤔 Choosing the right unit is as important as the calculat...
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Key Concepts & Vocabulary
TermDefinitionExample
Degree (°)A unit of angular measure, where one full rotation is divided into 360 degrees. It is the most 'customary' and commonly used unit in introductory geometry and real-world contexts like construction and navigation.A right angle is 90°. The angle of elevation to the top of a flagpole from a certain distance might be 35°.
Radian (rad)A unit of angular measure based on the radius of a circle. One radian is the angle created when the arc length equals the radius. It is the standard unit for calculus and many physics formulas because it simplifies calculations.A full circle is 2π radians. The arc length formula, s = rθ, only works when θ is in radians.
Linear UnitA standard unit used to measure distance or length, such as inches, feet, yards, or miles. T...
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Core Formulas
Degree to Radian Conversion
\text{radians} = \text{degrees} \times \frac{\pi}{180}
Use this formula to convert an angle from the customary unit (degrees) to radians. This is essential before using formulas like arc length (s = rθ) or in calculus.
Radian to Degree Conversion
\text{degrees} = \text{radians} \times \frac{180}{\pi}
Use this formula to convert an angle from radians back to degrees. This is useful for interpreting an answer in a more familiar, real-world context.
Arc Length Formula
s = r\theta
Calculates the length of an arc (s) of a circle with radius (r) subtended by a central angle (θ). A critical rule is that θ MUST be in radians for this formula to be valid. The unit for 's' will be the same as the unit for 'r'.
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Challenging
A student is modeling the horizontal position, x(t), of a swinging pendulum of length L. The model is x(t) = A sin(ωt). The maximum angle the pendulum reaches is θ_max. For small angles, the amplitude A is approximately L * θ_max. For this approximation to be dimensionally consistent and correct, what must be the unit of θ_max?
A.Degrees, because the angle is small.
B.It must be a dimensionless ratio, so radians are used.
C.It does not matter, as the units will cancel out.
D.Meters per second, to match the velocity.
Challenging
A ship's navigator uses a bearing of 45° (North-East) for direction. A physicist on board is modeling the ship's motion using vector components, which requires trigonometric functions for calculations. Why is it more appropriate for the physicist to convert the bearing to π/4 radians for their model?
A.Because π/4 is a smaller number than 45, simplifying arithmetic.
B.Because navigational tools are more accurate in radians.
C.Because vector calculus and physics formulas for motion (like derivatives for velocity) are defined and simplified using radians.
D.Because the ship is traveling on a curved surface (the Earth), which is better measured in radians.
Challenging
A function modeling daily temperature is T(t) = 20sin(Ï€/12 * (t - 9)) + 70, where T is in degrees Fahrenheit and t is hours after midnight. A student is asked to find the rate of change of temperature at t=15. The derivative is T'(t) = 20(Ï€/12)cos(Ï€/12 * (t - 9)). What are the appropriate units for the final calculated rate of change?
A.Degrees Fahrenheit
B.Hours per degree Fahrenheit
C.Degrees Fahrenheit per hour
D.Radians per hour
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