MATH 130 · Test 3
Test 3 Review
Concepts, worked examples, student practice, and synthesis
55 minutes · Offline capable
Today’s Learning Roadmap
- 1
solve angle, reference-angle, and exact-value problems
- 2
solve right-triangle, area, and two-triangle applications
- 3
perform vector operations and direction calculations
Activate · 0.5 minSource: 1
Predict: solve angle, reference-angle, and exact-value problems
Predict firstWhich problems need a sketch before any calculation?
Commit to a response before discussion.
Activate · 1.0 minSource: Authored
solve angle, reference-angle, and exact-value problems
ConceptUse coterminal reduction, reference angles, ASTC, and special triangles as one connected workflow.
Explain · 2.5 minSource: 2, 3
Model the Skill: solve angle, reference-angle, and exact-value problems
Find the remaining ratios when $\cos\theta=9/41$ in Quadrant I, then analyze $-97^\circ$.
- The missing leg is 40, so $\sin\theta=40/41$ and $\tan\theta=40/9$.
- $-97^\circ$ is coterminal with $263^\circ$ in Quadrant III.
- Its reference angle is $263^\circ-180^\circ=83^\circ$.
Model · 3.0 minSource: 4
You Try: solve angle, reference-angle, and exact-value problems
Student practiceFind $\sin225^\circ$. Then find a positive coterminal angle and reference angle for $-97^\circ$.
Feedback and solution- $\sin225^\circ=-\sqrt2/2$.
- Positive coterminal angle: $263^\circ$; reference angle: $83^\circ$.
Practice · 3.0 minSource: 5, 6
Check and Diagnose: solve angle, reference-angle, and exact-value problems
Error analysisCheck calculator mode and retain exact radicals when requested.
Self-checkWhat evidence confirms the setup, sign, unit, or magnitude?
Feedback · 2.0 minSource: 7
Pacing checkpointMake It Stick: solve angle, reference-angle, and exact-value problems
Write the order: sketch, reference angle, sign, exact value.
Exit responseAnswer in complete mathematical sentences and identify one remaining question.
Synthesize · 0.5 minSource: 8
Predict: solve right-triangle, area, and two-triangle applications
Predict firstWhat distinguishes a one-triangle problem from a shared-height two-triangle problem?
Commit to a response before discussion.
Activate · 1.0 minSource: Authored
solve right-triangle, area, and two-triangle applications
ConceptUse inverse trig to find angles, Pythagorean theorem for missing sides, and $A=\tfrac12bc\sin A$ for included-angle area.
Explain · 2.5 minSource: 9
Model the Skill: solve right-triangle, area, and two-triangle applications
Solve a right triangle with leg $a=11$, hypotenuse $c=35$, then find area for $A=76^\circ$, $b=34$, $c=21$.
- $b=\sqrt{35^2-11^2}=4\sqrt{69}\approx33.23$.
- $A\approx18.32^\circ$ and $B\approx71.68^\circ$.
- Included-angle area is $\tfrac12(34)(21)\sin76^\circ\approx346.40$ square units.
Model · 3.0 minSource: 10
You Try: solve right-triangle, area, and two-triangle applications
Student practiceObservation points 50 ft apart measure tower elevation angles $53^\circ$ and $31^\circ$. Find the tower height.
Feedback and solution- $d\tan53^\circ=(d+50)\tan31^\circ$, so $d\approx41.37$ ft.
- $h=d\tan53^\circ\approx54.90$ ft.
Practice · 3.0 minSource: Authored
Check and Diagnose: solve right-triangle, area, and two-triangle applications
Error analysis- Draw and label before substituting.
- Use square units for area.
Self-checkWhat evidence confirms the setup, sign, unit, or magnitude?
Feedback · 2.0 minSource: 11
Pacing checkpointMake It Stick: solve right-triangle, area, and two-triangle applications
State the setup pattern for each triangle problem type.
Exit responseAnswer in complete mathematical sentences and identify one remaining question.
Synthesize · 0.5 minSource: Authored
Predict: perform vector operations and direction calculations
Predict firstWhich vector tasks are component-wise and which require geometry?
Commit to a response before discussion.
Activate · 1.0 minSource: Authored
perform vector operations and direction calculations
ConceptOperations are component-wise; magnitude and direction use the Pythagorean theorem and inverse tangent.
Explain · 2.5 minSource: 12
Model the Skill: perform vector operations and direction calculations
Find the vector from $P=(3,-1)$ to $Q=(-2,4)$, then its magnitude and direction.
- $\vec v=\langle-2-3,4-(-1)\rangle=\langle-5,5\rangle$.
- $|v|=\sqrt{50}=5\sqrt2$.
- The Quadrant II direction angle is $135^\circ$.
Model · 3.0 minSource: Authored
You Try: perform vector operations and direction calculations
Student practiceFind magnitude and direction of $\langle-5,5\rangle$.
Feedback and solutionMagnitude $5\sqrt2$; direction 135°.
Practice · 3.0 minSource: Authored
Check and Diagnose: perform vector operations and direction calculations
Error analysis- Use terminal minus initial.
- Check quadrant before accepting inverse tangent.
Self-checkWhat evidence confirms the setup, sign, unit, or magnitude?
Feedback · 2.0 minSource: Authored
Pacing checkpointMake It Stick: perform vector operations and direction calculations
Write a vector problem checklist from components through direction.
Exit responseAnswer in complete mathematical sentences and identify one remaining question.
Synthesize · 0.5 minSource: Authored
Session Summary and Exit Ticket
- State the most important idea from today without looking at your notes.
- Complete one representative setup and identify the step most likely to cause an error.
- Write one question that should be answered before the next assessment.
Exit responseAnswer in complete mathematical sentences and identify one remaining question.
Feedback and solutionUse the objective roadmap to identify any skill that still needs deliberate practice.
Synthesize · 1.5 minSource: 13