MATH 130 · Module 12

Right Triangle Applications

Concepts, worked examples, student practice, and synthesis

55 minutes · Offline capable

Today’s Learning Roadmap

  1. 1

    model one-triangle applications with SOH-CAH-TOA

  2. 2

    apply elevation, depression, and motion components

  3. 3

    solve complete and two-triangle application problems

Activate · 0.5 minSource: 1

Predict: model one-triangle applications with SOH-CAH-TOA

Predict first

What must be labeled before selecting a trig ratio?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

model one-triangle applications with SOH-CAH-TOA

Concept
  • Read, draw, label, choose a ratio, solve, and check.
  • Inverse trig finds an angle from a ratio.

Explain · 2.5 minSource: 2, 3

Model the Skill: model one-triangle applications with SOH-CAH-TOA

Annotated diagram
  • A 20 ft ladder makes a $65^\circ$ angle with the ground. Find the height.
  • $\sin65^\circ=h/20$.
  • $h=20\sin65^\circ\approx18.13$ ft.
  • The result agrees with the corrected diagram and is less than 20 ft.
Right triangle ladder diagram with a 20 foot ladder and 65 degree ground angle
Right triangle ladder diagram with a 20 foot ladder and 65 degree ground angle

Model · 3.0 minSource: 4

You Try: model one-triangle applications with SOH-CAH-TOA

Student practice

A surveyor stands 80 ft from a building at 42° elevation. Find the height.

Practice · 3.0 minSource: 5

Check and Diagnose: model one-triangle applications with SOH-CAH-TOA

Error analysis
  • Calculator mode must be degrees.
  • A leg must be shorter than the hypotenuse.
Self-check

What evidence confirms the setup, sign, unit, or magnitude?

Feedback · 2.0 minSource: 6

Pacing checkpoint

Make It Stick: model one-triangle applications with SOH-CAH-TOA

State the six-step word-problem strategy.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: 7

Predict: apply elevation, depression, and motion components

Predict first

Why are an angle of depression and its matching angle of elevation equal?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

apply elevation, depression, and motion components

Formula focus
  • Both are measured from parallel horizontal lines.
  • Resolve motion with $d_x=d\cos\theta$ and $d_y=d\sin\theta$.

Explain · 2.5 minSource: 8

Model the Skill: apply elevation, depression, and motion components

1

East component: $300\cos25^\circ\approx271.89$ mi.

2

North component: $300\sin25^\circ\approx126.79$ mi.

3

$\sqrt{271.89^2+126.79^2}\approx300$ verifies the components.

Model · 3.0 minSource: 9

You Try: apply elevation, depression, and motion components

Student practice

A vehicle travels 300 miles at 25° north of east. Find east and north components.

Practice · 3.0 minSource: 10

Check and Diagnose: apply elevation, depression, and motion components

Error analysis
  • Angles are measured from horizontal unless stated otherwise.
  • Use total distance before resolving components.
Self-check

What evidence confirms the setup, sign, unit, or magnitude?

Feedback · 2.0 minSource: 11

Pacing checkpoint

Make It Stick: apply elevation, depression, and motion components

Explain how a horizontal reference line organizes both elevation and component problems.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: Authored

Predict: solve complete and two-triangle application problems

Predict first

What information is shared by two right triangles aimed at the same height?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

solve complete and two-triangle application problems

Concept
  • To solve a right triangle, find all missing sides and angles.
  • Two-triangle problems share a height and use related horizontal distances.

Explain · 2.5 minSource: 12, 13

Model the Skill: solve complete and two-triangle application problems

Two tower observations are 50 ft apart with elevation angles $53^\circ$ and $31^\circ$. Find the closer distance d and height h.

  1. $h=d\tan53^\circ=(d+50)\tan31^\circ$.
  2. $d\approx41.37$ ft and $h\approx54.90$ ft.
  3. Using the farther distance $91.37$ ft reproduces the same height.

Model · 3.0 minSource: 14

You Try: solve complete and two-triangle application problems

Student practice

Set up, but do not immediately solve, a two-observation tower problem with a 50 ft gap.

Practice · 3.0 minSource: 15, 16

Check and Diagnose: solve complete and two-triangle application problems

Error analysis
  • Keep d and d+50 attached to the correct observation points.
  • Show the algebraic distribution step before dividing.
Self-check

What evidence confirms the setup, sign, unit, or magnitude?

Feedback · 2.0 minSource: 17

Pacing checkpoint

Make It Stick: solve complete and two-triangle application problems

Write the shared-height equation that connects the two triangles.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: 18

Session Summary and Exit Ticket

  1. State the most important idea from today without looking at your notes.
  2. Complete one representative setup and identify the step most likely to cause an error.
  3. Write one question that should be answered before the next assessment.
Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 1.5 minSource: 19