Trigonometry and representations: mixed assessment
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 110 minutes, as an unvalidated planning estimate. This collection contains 10 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.
Choose methods without opening the worked solutions first. Use paper for diagrams and calculations; write enough reasoning to justify assumptions and every requested conclusion.
The suggested working time is guidance, not a timer. Written and symbolic responses require human review; a recorded response is not a proof-quality grade.
This public mixed practice samples the published unit. It is not a secure examination, complete AP paper or guarantee of course readiness. Saved first responses and known answer exposure persist across retries and imports.
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0 of 10 tasks have recorded responses. No overall mastery score is assigned.
A wheel of radius 4 cm rotates counterclockwise through 750°. Express the signed angle in radians, find its endpoint coordinates on the unit circle, and calculate the distance traveled by a point on its rim. Explain why a 30° rotation gives the same endpoint but a different distance.
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 temperature reaches its minimum −2 at t=1 hour and its next maximum 10 at t=7. Assume a sinusoidal model with no extra extrema between these times. Construct a formula, give its period and range, and predict the value at t=4. Explain what observations would test the assumption.
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.
Simplify (1−cos(2x))/sin(2x) using double-angle identities. State the original domain and the domain of the simplified tangent expression. Give an input where the simplified expression exists but the original does not.
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.
Evaluate arcsin(sin(7π/6)) and arccos(cos(7π/6)), explaining the principal ranges. Is arctan(tan(π/2)) meaningful over the reals? Explain before applying any inverse identity.
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.
Solve cos(2x)=sin x for 0≤x<2π. Show that every solution is included and check all candidates in the original equation.
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 triangle data A=30°, a=7, b=10, determine how many nondegenerate triangles are possible, with a opposite A. Separately, a triangle has adjacent sides7 and10 with included angle60°; find its third side and area exactly. Explain why the two data sets require different methods.
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.
Let u=(2,−1,2) and v=(1,2,2). Find u·v, both lengths, the cosine of their angle and the projection of u onto v. Verify the residual is perpendicular to v.
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.
Convert r=6cosθ to a Cartesian circle and explain why 0≤θ<2π traces the circle twice. At θ=π/3 give r, Cartesian coordinates and a negative-radius representation of the same point. Give a parameter interval that traces the entire circle once without repeating an endpoint.
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.
Find all complex solutions of z³=8i in polar and Cartesian forms. Explain why taking only the principal cube root is incomplete.
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 x=2+3cos t, y=−1+2sin t on 0≤t<2π, give the Cartesian equation, center, semiaxes, starting point and tracing direction. Compare with replacing t by2t on the same parameter interval.
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 wheel of radius 4 cm rotates counterclockwise through 750°. Express the signed angle in radians, find its endpoint coordinates on the unit circle, and calculate the distance traveled by a point on its rim. Explain why a 30° rotation gives the same endpoint but a different distance.
A synthetic temperature reaches its minimum −2 at t=1 hour and its next maximum 10 at t=7. Assume a sinusoidal model with no extra extrema between these times. Construct a formula, give its period and range, and predict the value at t=4. Explain what observations would test the assumption.
Simplify (1−cos(2x))/sin(2x) using double-angle identities. State the original domain and the domain of the simplified tangent expression. Give an input where the simplified expression exists but the original does not.
Evaluate arcsin(sin(7π/6)) and arccos(cos(7π/6)), explaining the principal ranges. Is arctan(tan(π/2)) meaningful over the reals? Explain before applying any inverse identity.
For triangle data A=30°, a=7, b=10, determine how many nondegenerate triangles are possible, with a opposite A. Separately, a triangle has adjacent sides7 and10 with included angle60°; find its third side and area exactly. Explain why the two data sets require different methods.
Let u=(2,−1,2) and v=(1,2,2). Find u·v, both lengths, the cosine of their angle and the projection of u onto v. Verify the residual is perpendicular to v.
Convert r=6cosθ to a Cartesian circle and explain why 0≤θ<2π traces the circle twice. At θ=π/3 give r, Cartesian coordinates and a negative-radius representation of the same point. Give a parameter interval that traces the entire circle once without repeating an endpoint.
For x=2+3cos t, y=−1+2sin t on 0≤t<2π, give the Cartesian equation, center, semiaxes, starting point and tracing direction. Compare with replacing t by2t on the same parameter interval.
