Welch Teaching & Coaching
Trigonometry • College Preparatory Mathematics

Unit 2: Right Triangle Trigonometry

Use special triangles, inverse functions, reference angles, identities, and equations to solve right triangles and applications.

U2
Unit Road Map

Lesson Map

Use this table as your route through the unit. Lesson numbers jump directly to the matching explanation.

LessonTopicYou should be able to…
2.145-45-90 TrianglesUse the 1:1:√2 side ratio to determine exact lengths.
2.230-60-90 TrianglesUse the 1:√3:2 side ratio to determine exact lengths.
2.3Solving Right TrianglesUse trig ratios and inverse trig functions to find missing sides and angles.
2.4ApplicationsTranslate elevation, depression, distance, and height contexts into right triangles.
2.5Reference AnglesFind reference angles for positive, negative, and coterminal angles.
2.6Exact Unit-Circle ValuesUse reference angles and quadrant signs to find exact trig values.
2.7Cofunctions & Calculator ValuesUse cofunction relationships and calculators appropriately.
2.8Trigonometric EquationsUse identities and factoring to solve introductory trigonometric equations.
Navigation

How the Unit Is Built

Each resource below is part of the same learning cycle. Select a card to jump to it.

Lesson MapSee the sequence and jump directly to any lesson.Content SummaryRead the ideas, methods, examples, and checks.Video LibraryOpen a targeted Khan Academy or YouTube search for each lesson.VocabularyReview the language needed to read and explain the mathematics.Sample ProblemsPractice with expandable worked solutions.Equations & RulesUse a compact reference for formulas and decision rules.Review GuidePrepare with a skill checklist and fresh mixed practice.Performance TaskApply the unit in a modeling or measurement context.
Learn

Content Summary

Work through these lessons in order. Focus on the meaning of the mathematics, the decision that selects a method, and a check that confirms the result.

Lesson 2.1

45-45-90 Triangles

Use the 1:1:√2 side ratio to determine exact lengths.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.
Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.
Lesson 2.2

30-60-90 Triangles

Use the 1:√3:2 side ratio to determine exact lengths.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.
Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.
Lesson 2.3

Solving Right Triangles

Use trig ratios and inverse trig functions to find missing sides and angles.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.
Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.
Lesson 2.4

Applications

Translate elevation, depression, distance, and height contexts into right triangles.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.
Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.
Lesson 2.5

Reference Angles

Find reference angles for positive, negative, and coterminal angles.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.
Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.
Lesson 2.6

Exact Unit-Circle Values

Use reference angles and quadrant signs to find exact trig values.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.
Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.
Lesson 2.7

Cofunctions & Calculator Values

Use cofunction relationships and calculators appropriately.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.
Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.
Lesson 2.8

Trigonometric Equations

Use identities and factoring to solve introductory trigonometric equations.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.
Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.
Watch & Reinforce

Video Library

These links open focused video searches by lesson topic. Use them as reinforcement rather than as a substitute for showing your own reasoning.

Lesson 2.1

45-45-90 Triangles

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →
Lesson 2.2

30-60-90 Triangles

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →
Lesson 2.3

Solving Right Triangles

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →
Lesson 2.4

Applications

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →
Lesson 2.5

Reference Angles

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →
Lesson 2.6

Exact Unit-Circle Values

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →
Lesson 2.7

Cofunctions & Calculator Values

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →
Lesson 2.8

Trigonometric Equations

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →
Language

Vocabulary

These terms should appear naturally in explanations, diagrams, and written reasoning.

special right triangle
Write a definition in your own words and add an example or sketch in your STRIDES notes.
inverse trigonometric function
Write a definition in your own words and add an example or sketch in your STRIDES notes.
angle of elevation
Write a definition in your own words and add an example or sketch in your STRIDES notes.
angle of depression
Write a definition in your own words and add an example or sketch in your STRIDES notes.
reference angle
Write a definition in your own words and add an example or sketch in your STRIDES notes.
exact value
Write a definition in your own words and add an example or sketch in your STRIDES notes.
cofunction
Write a definition in your own words and add an example or sketch in your STRIDES notes.
identity
Write a definition in your own words and add an example or sketch in your STRIDES notes.
trigonometric equation
Write a definition in your own words and add an example or sketch in your STRIDES notes.
Practice

Sample Problems

Try each problem before opening the solution. The examples are original practice aligned to the unit skills.

1

45-45-90

A square has diagonal 18 cm. Find its side length exactly.
Show solution
s√2=18, so s=18/√2=9√2 cm.
2

30-60-90

The short leg is 7. Find the long leg and hypotenuse.
Show solution
Long leg=7√3; hypotenuse=14.
3

Solve a triangle

A right triangle has adjacent side 12 and opposite side 5 relative to θ. Find θ.
Show solution
tan θ=5/12, so θ=tan⁻¹(5/12)≈22.6°.
4

Application

From 40 m away, the angle of elevation to the top of a structure is 32°. Find the height above eye level.
Show solution
tan32°=h/40, so h=40tan32°≈25.0 m.
5

Reference angle

Find the reference angle for 230°.
Show solution
230°−180°=50°.
6

Exact value

Find cos 150° exactly.
Show solution
Reference angle 30°; cosine is negative in QII, so cos150°=−√3/2.
Reference

Equations & Rules

Know what each relationship means and when it applies; do not treat the box as a list of disconnected formulas.

45-45-90
x, x, x√2
30-60-90
x, x√3, 2x
Inverse Sine
θ = sin⁻¹(opposite/hypotenuse)
Inverse Cosine
θ = cos⁻¹(adjacent/hypotenuse)
Inverse Tangent
θ = tan⁻¹(opposite/adjacent)
Cofunction
sin θ = cos(90°−θ)
Quadrant Signs
Use the terminal point or ASTC sign pattern; do not change the reference-angle magnitude.
Factoring Equations
Move to one side, factor, set factors equal to zero, then find angles in the requested interval.
Prepare

Review Guide

Skill Checklist

  • I can use the 1:1:√2 side ratio to determine exact lengths.
  • I can use the 1:√3:2 side ratio to determine exact lengths.
  • I can use trig ratios and inverse trig functions to find missing sides and angles.
  • I can translate elevation, depression, distance, and height contexts into right triangles.
  • I can find reference angles for positive, negative, and coterminal angles.
  • I can use reference angles and quadrant signs to find exact trig values.
  • I can use cofunction relationships and calculators appropriately.
  • I can use identities and factoring to solve introductory trigonometric equations.

How to review

  1. Complete the unit Skills Check packet without notes.
  2. Mark the exact skill behind each error.
  3. Return to that lesson and complete a fresh example.
  4. Explain the correction in words, then retry a similar problem.
  5. Finish with mixed practice so you must choose the method yourself.
Apply

Performance Task

Campus Trigonometry Field Study

Choose several objects or locations that can be modeled with right triangles. Collect measurements, solve each triangle, and compare direct measurements with trigonometric estimates.

Evidence to include

A labeled mathematical model, organized calculations, appropriate units and precision, a written interpretation, and a check or discussion of limitations.