Trigonometric reasoning: mixed coordinate, inverse and threshold evidence
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 100 minutes, as an unvalidated planning estimate. This collection contains 6 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Attempt the public tasks before revealing guidance. Give drawings or coordinate descriptions and justify domain restrictions.
Use a calculator where requested and state its angle mode. Approximate answers are reviewed with reasoning, not exact-string checks.
This sample assessment requires human review and does not certify independent examination performance or full AP coverage.
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0 of 6 tasks have recorded responses. No overall mastery score is assigned.
A unit-circle point is (−7/25,24/25). Give the points for π+x, 2π−x and −x, then prove the general tangent identities behind those examples. Identify excluded angles.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Derive cosine subtraction from rotated unit directions, not from a quoted identity. Then evaluate cos(a−b) for acute a,b with coordinates (12/13,5/13) and (8/17,15/17). Explain why a positive answer does not force a−b to be positive.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Derive tan(a−b) and its domain. A learner substitutes a=b=π/2 and says the result is (∞−∞)/(1+∞²)=0. Repair the reasoning, and show how the original angle can still be evaluated.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Explain why arccos selects [0,π] using unit-circle monotonicity. Let β=arccos(−3/10). List all solutions to cos x=−3/10 on [−2π,2π] without decimal approximation.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A synthetic depth model is D(t)=6+4sin(π(t−2)/6), t in hours. Find every crossing D=7.5 and every interval D≥7.5 on [0,20], to 0.01 h. Explain the phase and endpoint choices.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For y(t)=2+3cos(2t), with t in seconds and trig inputs in radians, a report says: period π seconds; y=6 occurs at arccos(4/3)/2; all threshold solutions come from one inverse value. Audit every statement, then give complete formulas for y=3 and explain how a finite observation interval would be used.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
A unit-circle point is (−7/25,24/25). Give the points for π+x, 2π−x and −x, then prove the general tangent identities behind those examples. Identify excluded angles.
Derive cosine subtraction from rotated unit directions, not from a quoted identity. Then evaluate cos(a−b) for acute a,b with coordinates (12/13,5/13) and (8/17,15/17). Explain why a positive answer does not force a−b to be positive.
Derive tan(a−b) and its domain. A learner substitutes a=b=π/2 and says the result is (∞−∞)/(1+∞²)=0. Repair the reasoning, and show how the original angle can still be evaluated.
Explain why arccos selects [0,π] using unit-circle monotonicity. Let β=arccos(−3/10). List all solutions to cos x=−3/10 on [−2π,2π] without decimal approximation.
A synthetic depth model is D(t)=6+4sin(π(t−2)/6), t in hours. Find every crossing D=7.5 and every interval D≥7.5 on [0,20], to 0.01 h. Explain the phase and endpoint choices.
For y(t)=2+3cos(2t), with t in seconds and trig inputs in radians, a report says: period π seconds; y=6 occurs at arccos(4/3)/2; all threshold solutions come from one inverse value. Audit every statement, then give complete formulas for y=3 and explain how a finite observation interval would be used.
