MATH 130 · Module 10

Angles and Right Triangle Trigonometry

Concepts, worked examples, student practice, and synthesis

75 minutes · Offline capable

Today’s Learning Roadmap

  1. 1

    sketch angles in standard position

  2. 2

    find coterminal and reference angles in degrees and radians

  3. 3

    use radians to compute arc length

  4. 4

    use right-triangle trigonometric ratios

  5. 5

    solve special-triangle and right-triangle applications

Activate · 0.5 minSource: 1

Predict: sketch angles in standard position

Predict first

Where does a negative angle rotate, and what never changes about standard position?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

sketch angles in standard position

Concept
  • The initial side lies on the positive x-axis.
  • Positive rotation is counterclockwise; negative rotation is clockwise.

Explain · 2.5 minSource: 2

Model the Skill: sketch angles in standard position

Sketch $225^\circ$ and $-120^\circ$ and identify their quadrants.

  1. $225^\circ=180^\circ+45^\circ$, so it terminates in Quadrant III.
  2. $-120^\circ+360^\circ=240^\circ$, also in Quadrant III.
  3. Both begin on the positive x-axis; rotation direction distinguishes the sketches.

Model · 3.0 minSource: 3

You Try: sketch angles in standard position

Student practice

Sketch 310° and label initial and terminal sides.

Practice · 3.0 minSource: 4

Check and Diagnose: sketch angles in standard position

Error analysis
  • Do not measure from the y-axis.
  • Quadrant boundaries are not inside a quadrant.
Self-check

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

Feedback · 2.0 minSource: 5

Pacing checkpoint

Make It Stick: sketch angles in standard position

Describe standard position without drawing it.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: Authored

Predict: find coterminal and reference angles in degrees and radians

Predict first

How can infinitely many angles share one terminal side?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

find coterminal and reference angles in degrees and radians

Formula focus
  • Coterminal angles differ by $360^\circ k$ or $2\pi k$.
  • Reduce large magnitudes before locating the terminal side.

Explain · 2.5 minSource: 6

Model the Skill: find coterminal and reference angles in degrees and radians

1

$-97^\circ+360^\circ=263^\circ$ and $-97^\circ-360^\circ=-457^\circ$.

2

$9\pi/7+2\pi=23\pi/7$ and $9\pi/7-2\pi=-5\pi/7$.

3

Coterminal angles differ by full rotations.

Model · 3.0 minSource: 7

You Try: find coterminal and reference angles in degrees and radians

Student practice

Find a coterminal angle between 0° and 360° for 765°.

Practice · 3.0 minSource: Authored

Check and Diagnose: find coterminal and reference angles in degrees and radians

Error analysis
  • Add or subtract full rotations only.
  • Keep degree and radian units separate.
Self-check

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

Feedback · 2.0 minSource: 8

Pacing checkpoint

Make It Stick: find coterminal and reference angles in degrees and radians

Give the general coterminal-angle formulas.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: Authored

Predict: use radians to compute arc length

Predict first

What does one radian measure geometrically?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

use radians to compute arc length

Formula focus
  • Arc length is $s=r\theta$ when $\theta$ is in radians.
  • Radians connect central angle directly to distance.

Explain · 2.5 minSource: 9

Model the Skill: use radians to compute arc length

1

The angle is already in radians.

2

$s=r\theta=12(2\pi/3)=8\pi$ units.

3

$8\pi$ is about 25.133 units, less than the full circumference $24\pi$.

Model · 3.0 minSource: Authored

You Try: use radians to compute arc length

Student practice

Find s for $r=8$ and $\theta=135^\circ$.

Practice · 3.0 minSource: 10

Check and Diagnose: use radians to compute arc length

Error analysis
  • Convert degrees before using $s=r\theta$.
  • Arc length has linear units.
Self-check

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

Feedback · 2.0 minSource: Authored

Pacing checkpoint

Make It Stick: use radians to compute arc length

Explain why radians make $s=r\theta$ dimensionally natural.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: Authored

Predict: use right-triangle trigonometric ratios

Predict first

Which two sides define sine, cosine, and tangent relative to an angle?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

use right-triangle trigonometric ratios

Concept
  • SOH-CAH-TOA selects the ratio.
  • Reciprocal functions invert sine, cosine, and tangent.

Explain · 2.5 minSource: 11, 12

Model the Skill: use right-triangle trigonometric ratios

A right triangle has opposite leg 40, adjacent leg 9, and hypotenuse 41. Find all six ratios.

  1. $\sin\theta=40/41$, $\cos\theta=9/41$, and $\tan\theta=40/9$.
  2. $\csc\theta=41/40$, $\sec\theta=41/9$, and $\cot\theta=9/40$.
  3. $9^2+40^2=41^2$ verifies the side lengths.

Model · 3.0 minSource: 13, 14

You Try: use right-triangle trigonometric ratios

Student practice

Given $\cos\theta=9/41$, find $\sin\theta$ and $\tan\theta$.

Practice · 3.0 minSource: 15

Check and Diagnose: use right-triangle trigonometric ratios

Error analysis
  • Opposite and adjacent depend on the chosen reference angle.
  • Keep the calculator in the requested angle mode.
Self-check

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

Feedback · 2.0 minSource: 16, 17

Pacing checkpoint

Make It Stick: use right-triangle trigonometric ratios

Write the ratio-selection rule you will use under time pressure.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: 18

Predict: solve special-triangle and right-triangle applications

Predict first

When is an exact radical answer better than a decimal approximation?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

solve special-triangle and right-triangle applications

Concept
  • Special triangles produce exact values.
  • Word problems require a labeled diagram before choosing a ratio.

Explain · 2.5 minSource: 19, 20

Model the Skill: solve special-triangle and right-triangle applications

A 25 ft ladder makes a $70^\circ$ angle with the ground. Find its vertical reach.

  1. $\sin70^\circ=h/25$.
  2. $h=25\sin70^\circ\approx23.49$ ft.
  3. The height is less than the 25 ft hypotenuse.

Model · 3.0 minSource: 21

You Try: solve special-triangle and right-triangle applications

Student practice

A 25 ft ladder makes a 70° angle with the ground. Find the height.

Practice · 3.0 minSource: 22, 23

Check and Diagnose: solve special-triangle and right-triangle applications

Error analysis
  • Exact-value questions should retain radicals.
  • Check that a leg is shorter than the hypotenuse.
Self-check

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

Feedback · 2.0 minSource: 24

Pacing checkpoint

Make It Stick: solve special-triangle and right-triangle applications

List the diagram, ratio, solve, and reasonableness-check steps.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: 25

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: 26