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Right-Triangle Trigonometry

Students use sine, cosine, and tangent ratios to find missing sides and angles in right triangles and solve applied problems.

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Beginner 15–20 minutes Triangles & Transformations
Triangles & Transformations Page 8 of 8
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At a Glance

Right-Triangle Trigonometry
Binder SectionTriangles & Transformations
DifficultyBeginner
Estimated Time15–20 minutes
PrerequisitesNone
Essential Question

How do side ratios connect an acute angle to every similar right triangle?

Build Understanding

Right-triangle trigonometry connects acute angles with ratios of side lengths. Relative to angle theta, identify the opposite leg, adjacent leg, and hypotenuse.

SOH-CAH-TOA summarizes sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, and tangent = opposite/adjacent. Choose the ratio containing the known and unknown sides.

To find a side, write an equation and solve. To find an angle, use an inverse operation: theta = sin inverse(opposite/hypotenuse), for example. Ensure the calculator is in degree mode.

Angles of elevation and depression model sight lines. Draw and label a right triangle, choose a ratio, keep full calculator precision, round at the end, and include units. Sine and cosine of complementary acute angles are connected because opposite and adjacent sides switch roles.

Visual Model
4Thousands 3Hundreds 2Tens 7Ones

4,327 = 4,000 + 300 + 20 + 7

Goals

Learning Targets

  • Label opposite, adjacent, and hypotenuse relative to an angle.
  • Use sine, cosine, and tangent.
  • Select a ratio from the given information.
  • Use inverse trig to find an angle.
  • Solve elevation and depression problems.
  • Check calculator mode and reasonableness.
Click each word

Interactive Vocabulary

Reveal one step at a time

Worked Examples

Example 1

Angle 35° and hypotenuse 12; find opposite side

1

sin35°=x/12.

Example 2

Opposite 9 and adjacent 12; find angle

1

tanθ=9/12.

Ask and explain

Common Questions

Do opposite and adjacent stay fixed? No; they depend on the chosen angle.
Is the hypotenuse ever adjacent? It is not called the adjacent leg.
When do I use inverse trig? When an angle is unknown.
Why degree mode? Geometry problems usually measure angles in degrees.
Notice and correct

Common Misconceptions

Labeling sides without reference to the chosen angle

Circle the angle first.

Using degrees while calculator is in radians

Confirm degree mode.

Rounding during intermediate steps

Keep full precision until the final answer.

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Use the chapter markers, key vocabulary, and reflection prompts to stay actively engaged while you watch.

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Right-Triangle Trigonometry lesson and examples

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Warm-Up Guided Independent Challenge
1
Quick Start

Warm-Up

Warm-up 1

Directions: Solve the problem. Show your thinking, then enter only your final answer.

Relative to angle θ, which side is always longest?

Picture the lengthUse the picture to organize what you know before you solve.
0123456789107 equal units

Choose how you want to work or answer.

2
We Do

Guided Practice

3
You Do

Independent Practice

Independent 1

Directions: Solve independently. Show your work, check it, then enter only your final answer.

Find opposite side when θ=40° and adjacent=11.

Picture the coinsUse the picture to organize what you know before you solve.
25¢10¢5¢1¢

Choose how you want to work or answer.

Independent 2

Directions: Solve independently. Show your work, check it, then enter only your final answer.

Find θ when opposite=5 and hypotenuse=13.

Picture the triangleUse the picture to organize what you know before you solve.
leg aleg bc = ?right angle

Choose how you want to work or answer.

Independent 3

Directions: Solve independently. Show your work, check it, then enter only your final answer.

Find adjacent when θ=28° and hypotenuse=20.

Picture the coinsUse the picture to organize what you know before you solve.
25¢10¢5¢1¢

Choose how you want to work or answer.

4
Stretch Your Thinking

Challenge & Real-World Practice

🧩 Challenge

Plan an indirect height measurement for a flagpole. Choose a realistic horizontal distance and angle of elevation, calculate height, and describe sources of measurement error.

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Khan Academy

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Success Criteria

Before you begin, I can…

  • I label sides relative to the selected angle.
  • I choose the ratio containing the known and unknown quantities.
  • I use degree mode, units, and sensible rounding.
1
Question 1

Which statement shows the key idea in this lesson?

2
Question 2

Which habit best supports accuracy?

3
Question 3

What should a student do after solving?

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Question 4

Which explanation is strongest?

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Question 5

How can an error be found?

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