Explain the central structure of graphs of sine and cosine using precise mathematical language.
Apply the governing relationship and verify results with an alternate representation or substitution.
State domain, unit, precision, and context restrictions whenever they affect the result.
Analyze a misconception and communicate a correction with evidence.
Essential question: How do angle measure, the unit circle, graphs, and periodic functions model rotation, waves, cycles, and repeated behavior?
Governing relationship
y=sin x and y=cos x have amplitude 1 and period 2π.
Sine and cosine graphs encode periodic change. Key points at quarter-period intervals support accurate sketches and interpretation.
Worked examples
Use radians on the horizontal axis. One full revolution is 2π, and the unit-circle coordinates are (cos x, sin x). Work each example before opening its verification.
Investigation A · Construct one complete sine cycle.
Mark x=0, π/2, π, 3π/2, and 2π. The unit-circle vertical coordinate is sin x.
Record the five y-values: 0, 1, 0, −1, 0.
Join smoothly, not with sharp corners: the curve peaks at (π/2,1), bottoms at (3π/2,−1), and crosses y=0 at 0, π, 2π.
Since a unit-circle vertical coordinate cannot exceed 1 in magnitude, the range is [−1,1]. Repeating after 2π proves the period is 2π.
Verification: sin(x+2π)=sin x, while sin(7π/2)=sin(3π/2)=−1. This agrees with the graph.
Investigation B · Derive the phase relationship with cosine.
The sine addition rule is sin(x+π/2)=sin x cos(π/2)+cos x sin(π/2).
Substitute cos(π/2)=0 and sin(π/2)=1 to obtain sin(x+π/2)=cos x.
Thus the cosine graph is the sine graph shifted left by π/2: cosine begins at its maximum 1 when x=0, where sine is 0.
Both graphs have amplitude 1, midline y=0, period 2π and range [−1,1]; their horizontal positions differ.
Verification: At x=0, sin(π/2)=1=cos 0; at x=π, sin(3π/2)=−1=cos π.
Visual model · one period of sine and cosine
One cycle, both on the same scale. Read the curves at quarter-period intervals; the blue sine maximum aligns with the amber cosine zero.
Quarter-cycle checkpoint table
x (radians)
0
π/2
π
3π/2
2π
sin x
0
1
0
−1
0
cos x
1
0
−1
0
1
Units: The quarter-cycle marks above are radians, not degrees. If degrees are required, π/2=90°, π=180°, and 2π=360°.
Guided and independent practice
Without using a graphing tool, plot the five sine and cosine landmarks from the table, then compare to the SVG.
Explain why sin x=0 at both x=0 and x=π but cos x differs at those points.
Use the phase identity to prove cos(x−π/2)=sin x. State whether that graph is shifted right or left relative to cosine.
Build a model T(x)=18+4sin x, where T is degrees Celsius and x is measured in radians. Identify midline, amplitude, minimum, maximum and repeating period.
Use Trigonometric Wave Studio to test one claim, recording a numerical and graphical verification in your lesson note.
Independent transfer · check only after solving
Evaluate sin(7π/2) and cos(−π) using periodicity.
Find the range, period and midline of y=2sin x−1.
Explain why the claim cos x=sin(x−π/2) is incorrect by evaluating both sides at x=0.
Reveal worked verification and answers
sin(7π/2)=sin(3π/2)=−1; cos(−π)=−1 because cosine is even and cos π=−1.
Amplitude 2, midline −1, range [−3,1], and unchanged period 2π.
At x=0, cos 0=1 but sin(−π/2)=−1. The correct identity is cos x=sin(x+π/2).
Differentiated pathways
Foundation
Review prerequisite vocabulary, annotate each worked step, complete the first four prompts, and use a table or graph to support symbolic work.
Core
Complete every prompt, explain domain and verification, correct errors, and submit the applied task with clear mathematical communication.
Extended
Generalize the relationship, compare methods, use technology strategically, analyze sensitivity, and create a proof, counterexample, or advanced model.
Application laboratory
Connect a periodic position or temperature cycle to sine or cosine.
Required evidence: defined quantities and units; at least two representations; transparent calculations; verification; one limitation; and a defensible conclusion.
Misconception and correction
Common misconception: Using degrees on an axis labeled in radians without conversion.
Write the first invalid step, the governing rule that corrects it, and one counterexample or verification that demonstrates the correction.
Mastery evidence
Accurate symbolic work with preserved equivalence.
Correct domain, interval, unit, and precision statements.
At least one independent verification.
A correction explaining why an incorrect approach fails.
A clear application conclusion that distinguishes mathematical result from recommendation.
Reflection
What structure did you recognize? Which representation was most useful? What assumption or restriction mattered most? What will you do differently on the next problem?