Unit 05 · Lesson 04 · Week 14 · Differentiation Rules & Higher Derivatives
Derivatives of Sine and Cosine
Trigonometric derivatives connect periodic position with periodic rates. Angle measure must be in radians for the standard derivative formulas to hold without an additional conversion factor. This lesson emphasizes derivatives of sine and cosine through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in derivatives of sine and cosine.
- Move among symbolic, graphical, numerical, and contextual representations.
- Show complete reasoning with correct notation, domain conditions, units, and intervals.
- Verify a conclusion using a second method or representation.
Essential question
How do derivative rules make complex rates of change efficient, interpretable, and verifiable?
Visual model
Trigonometric derivatives connect periodic position with periodic rates. Angle measure must be in radians for the standard derivative formulas to hold without an additional conversion factor.
(\sin x)'=\cos x(\cos x)'=-\sin x(\tan x)'=\sec^2 x
Worked reasoning
Misconception check: The derivative formulas assume radians, not degrees.
Three learning pathways
Foundation
Reconstruct definitions, annotate a worked example, and solve representative problems with guided prompts.
Core
Complete mixed representations, justify every major step, and verify exact and approximate results.
Extended
Generalize a result, analyze an error or edge case, and connect the lesson to modelling or proof-informed reasoning.
Communication and modelling
State what is known, identify the theorem or structure used, show symbolic work, interpret the result in context, and explain how the conclusion was checked. Calculator or CAD evidence must include settings, units, and a written interpretation.
Khaemenes Academy · Open-Age Calculus I
Derivatives of Sine and Cosine
Teacher Answer Key · Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and technology evidence when used.
Name
Date
Differentiate sin x.
Answer: cos x
Reasoning: This standard rule assumes radian measure.
Reasoning: This standard rule assumes radian measure.
Differentiate cos x.
Answer: −sin x
Reasoning: Cosine's rate is negative sine.
Reasoning: Cosine's rate is negative sine.
Differentiate tan x.
Answer: sec²x
Reasoning: The tangent derivative is secant squared.
Reasoning: The tangent derivative is secant squared.
Differentiate 3sin x−2cos x.
Answer: 3cos x+2sin x
Reasoning: Apply linearity and the trig derivative rules.
Reasoning: Apply linearity and the trig derivative rules.
At x=π/2, find the slope of y=sin x.
Answer: 0
Reasoning: cos(π/2)=0.
Reasoning: cos(π/2)=0.
Why are radians required for the standard trig derivative rules?
Answer: Radians make the limiting arc-to-radius ratio equal 1.
Reasoning: Degree measure introduces an additional conversion factor.
Reasoning: Degree measure introduces an additional conversion factor.