Geometry · 40 minutes
Suggested grades 9–11 · Unit 2: Similarity and right-triangle measures · Lesson 2 of 2
Right-Triangle Trigonometry in a Diagram
Choose a ratio from a labeled reference angle and check the result.
Mr. S Teaches · Original starter lesson · HS-GEO-04 · Grade range is planning guidance.Watch, notice, discuss
Video choices for this lesson
Preview a video, then choose what fits your learners. “Show to class” opens a larger view; use the player’s play and full-screen controls when you’re ready.
Teacher preview needed. Check the language, level and length before showing. YouTube may show ads or related videos.
Example: Trig to solve the sides and angles of a right triangle | Trigonometry | Khan Academy
By Khan Academy
Practice choosing sine, cosine or tangent and interpreting the resulting measurement.
Open on YouTube: Example: Trig to solve the sides and angles of a right triangle | Trigonometry | Khan Academy (opens a new tab) ↗Trigonometric Ratios
By The Organic Chemistry Tutor
Review sine, cosine and tangent ratios in a right triangle before choosing a ratio for the lesson’s unknown side or angle.
Open on YouTube: Trigonometric Ratios (opens a new tab) ↗How do you select a ratio using the known side, unknown quantity and reference angle?
An external player loads only after you choose Preview or Show to class. Playback connects to YouTube. Videos belong to their creators; availability and embedding can change.
Learning objective
Calculate an unknown side and angle in right-triangle models.
Before you begin
- Pythagorean theorem
- Ratios
- Degree-mode calculator
Gather your materials
- Paper and pencil
- Graph paper
- Calculator for checking results
Language to know
- sine
- Opposite-side length divided by hypotenuse in a right triangle.
- cosine
- Adjacent-leg length divided by hypotenuse.
- tangent
- Opposite-leg length divided by adjacent-leg length.
Understand the concept
Opposite and adjacent depend on the chosen acute angle. The hypotenuse is always opposite the right angle. Selecting the ratio follows the known and unknown sides, not a memorized picture orientation. A calculation must use the intended angle unit and a sensible domain; a hypotenuse shorter than a leg signals an error.
Teach it step by step
- Draw a right triangle with a 35° angle and adjacent leg 12 m; label the opposite leg h.
- Use tan 35°=h/12 and compute h with a degree-mode calculator.
- Find the hypotenuse using cosine and check approximately with the Pythagorean relationship.
- Reverse the task for opposite 5, adjacent 12, using inverse tangent for the acute angle.
Make the thinking visible
Worked example or model response
h=12tan35°≈8.40 m; hypotenuse=12/cos 35°≈14.65 m. For legs 5 and 12, the angle opposite 5 is arctan(5/12)≈22.62° and the hypotenuse is 13.
Guided practice
For an angle of 30° and hypotenuse 10 cm, find opposite and adjacent legs.
Independent practice
1. Find the opposite leg for 40° and adjacent 7. 2. Find the angle whose opposite and adjacent legs are 8 and 15.
Quick assessment
Why must the chosen reference angle be identified before naming a leg “opposite”?
Answer key & assessment criteria
Answers and teaching guidance
- Guided: Opposite 5 cm; adjacent 5√3≈8.66 cm.
- Independent 1: 7tan40°≈5.87. 2: arctan(8/15)≈28.07°.
- Quick check: Each acute angle has a different opposite leg; the labels switch when the reference angle changes.
Look for: Require the learner to explain the interpretation and check, not only state a result. Distinguish an exact value from a rounded approximation and attend to the stated domain.
Make it work for your learner
More support
Use a side-label checklist and provide calculator-mode reminders.
Extension challenge
Solve the same triangle from its complementary angle and explain the agreement.
Verified references · Selected contributions
Standards connections
Subject-content standards
These records identify the part of a subject standard practiced here. Course placement is instructional guidance; the source’s category or band remains authoritative.
Contributes to Common Core standard
HSG.SRT.C.8
Common Core · High school · Geometry
Regular high school mathematics standard
Use trigonometric ratios and a right-triangle check to solve applied diagrams.
Read the official standard ↗This lesson contributes selected skills; broader applications and full standard coverage require further instruction.
Verified 2026-11-04. Selected contribution, not full attainment.