How are sine and cosine slopes related?
You will be able to: Differentiate sine and cosine with radian inputs and correct signs.
How are sine and cosine slopes related?
A smooth oscillating signal rises fastest when it crosses its midline and flattens near its peaks. Sine and cosine encode this changing local slope.
A useful starting point: How do you differentiate a whole polynomial? →
Words and symbols before equations
- Radian
- Angle unit based on arc length divided by radius.
- Sine and cosine
- Unit-circle functions giving vertical and horizontal coordinates.
- Amplitude
- The scale of an oscillating output.
- Period
- Input interval over which a pattern repeats.
What this picture assumes
Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. Radian input. Curves use exact standard derivative rules; arbitrary compositions are deferred to Unit 3.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- sin at radian input 0: value=0, derivative=1. Blue: function; orange: tangent; second graph: derivative. Curves outside the finite vertical scale are clipped, not capped. Use radians and retain denominator exclusions.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
In radians, d(sin x)/dx=cos x and d(cos x)/dx=−sin x. At x=0, sine crosses upward with slope 1; cosine is at a peak with slope 0.
The sine difference quotient expands by the angle-addition identity to sin(x)(cos(h)−1)/h + cos(x)sin(h)/h. The two standard radian limits are 0 and 1, yielding cos(x).
The cosine angle-addition identity similarly yields −sin(x); the minus sign describes the downward local slope just to the right of its peak.
Degrees use a different input scale and therefore a π/180 rate factor. The simple derivative identities here require radians. Compositions such as sin(3x) require the later chain rule.
A worked example, step by step
Differentiate y=3sin x−2cos x and evaluate y′(0).
- Apply constant multiples and the difference rule.
- Derivative of 3sin x is 3cos x.
- Derivative of −2cos x is +2sin x.
- Thus y′=3cos x+2sin x and y′(0)=3.
Keep the minus sign in the cosine derivative and use radian mode for numerical checks.
Why is the derivative of cos x negative near x=0 on the right?
Compare with an explanation
Cosine decreases there; −sin x is negative for small positive radian inputs.
Predict. Change one thing. Explain.
Select sine or cosine and move the radian input. Find a zero function value with nonzero slope, then a peak with zero slope.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
sin at radian input 0: value=0, derivative=1. Blue: function; orange: tangent; second graph: derivative. Curves outside the finite vertical scale are clipped, not capped. Use radians and retain denominator exclusions.
Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. Radian input. Curves use exact standard derivative rules; arbitrary compositions are deferred to Unit 3.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using a difference quotient, tangent slope, or derivative rule with its domain conditions. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionFind the derivative of 4cos x+sin x at x=π and explain the angle convention.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Use radians.
- 1 point: The derivative is −4sin x+cos x.
- 1 point: At π, sin π=0 and cos π=−1.
- 1 point: The derivative is −1 per radian; degree inputs would require a scale factor.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1Derivative of sin x?
cos x, in radians.
RECALL 2Derivative of cos x?
−sin x, in radians.
RECALL 3Why mention the angle unit?
Changing the input scale changes rate values.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How are sine and cosine slopes related?
- (sin x)′=cos x; (cos x)′=−sin x, radians.
- Use constant and sum rules for linear combinations.
Remember: Keep the minus sign in the cosine derivative and use radian mode for numerical checks.
Conditions: Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. Radian input. Curves use exact standard derivative rules; arbitrary compositions are deferred to Unit 3.
Refresh Kid · AP Calculus AB Unit 2 · Objectives FUN-3.A, LIM-3.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 2.7, FUN-3.A, LIM-3.A. CED effective Fall 2020, with Fall 2026 clarifications, checked September 17, 2026. Unit 2 has 10 official topics. Focused lesson titles and questions are original Refresh Kid teaching material.
Derivative definitions require finite two-sided limits at interior inputs. Graphs and finite tables supply evidence and estimates, not universal proofs. Continuity is necessary but insufficient for differentiability. Rules retain their original domain restrictions, and trigonometric inputs use radians. Chain-rule compositions, implicit differentiation, inverse-function differentiation and L’Hôpital’s rule are deferred to later units. Piecewise joins are checked for continuity before their side slopes are compared.
The Organic Chemistry Tutor video titles, creators and descriptions were checked; full videos were not reviewed. Khan Academy’s unit destination was checked; its JavaScript lesson content was not fully readable by the research tool. OpenStax Sections 3.1, 3.3 and 3.5 were consulted for conceptual cross-checking. No creator scripts, questions, diagrams or artwork were copied. Refresh Kid is not endorsed by these providers.
GitHub’s 3D website collection and its camera-control example informed optional spatial inspection. Graph geometry is generated from the stated original equations; self-hosted Three.js retains its MIT license. The optional 3D model uses separate planes for f and f′. Depth distinguishes panels, not an additional variable; camera rotation does not change their values. Use the complete labeled 2D graphs and text to read precise coordinates and slopes.
Independent teacher review and observation of students remain pending. Checks do not certify scientific accuracy, accessibility or learning effectiveness. This is a review edition.
Optional official resource: Released AP Calculus AB questions and scoring guides. The archive spans multiple units; no entire exam question is assigned to this lesson.
Learn → Explore → Practice → Review is informed by the IES learning guide; this exact implementation has not been evaluated with learners.
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