Unit 7
Trigonometry
Trigonometry here is not a side quest. It’s what happens when students name the side ratios that similarity guarantees and treat them as functions of an angle. Building on similarity, proportional reasoning, and right triangles from earlier units, students first discover and invert the basic trig ratios, then prove exact values for special angles, and finally extend trig beyond right triangles with the Laws of Sines and Cosines. Along the way, classroom stories, algebraic missteps, and the Everest problem keep the focus on real student thinking and on trig as a natural outgrowth of geometry, not a separate course. Start by exploring the Unit 7 Navigation Instrument and Unit Overview to see how the lessons connect, what students will build, and where the key moments live.
Unit 7 Lessons
Lesson 7.1 – Ratios That Refuse to Change
Students grow families of similar right triangles on paper and hunt for side ratios that stay the same. They name those invariant ratios sine, cosine, and tangent, connect them to opposite/adjacent/hypotenuse, and build a class trig table that will anchor the rest of the unit. The emphasis is on AA similarity, precision measurement, and seeing trig ratios as named geometric quantities, not button presses.
Lesson 7.2 – If We Can Measure the Angle, We Can Measure Anything
Students put the class trig table to work solving right triangles, always estimating from the table before using a calculator. They design and build simple clinometers, then take them into the field to measure the height of an inaccessible object and analyze percent error. The lesson builds fluency with sin, cos, and tan while treating trig as a practical measurement tool students own.
Lesson 7.3 – Choosing the Steepest Safe Path
An avalanche safety scenario pushes students to decide which slopes are safe using rise:run ratios and a danger zone of 30°–45°. Guess‑and‑check with the class table makes inverse trig feel necessary before it is named. Students then connect inverse functions to reading the table in reverse and apply them to an ADA ramp compliance investigation, strengthening both estimation and structural understanding of when to use trig vs. inverse trig
Lesson 7.4 – Triangles Worth Remembering
Students derive the 45‑45‑90 and 30‑60‑90 triangles from first principles, then use them to generate exact trig values for 30°, 45°, and 60°. Along the way they confront common algebra slips and prove the Pythagorean Identity geometrically. The goal is for students to see these values as logical consequences they can reconstruct, not facts to memorize.
Lesson 7.5 – The Shape of Sine (Optional)
Students plot y = sinθ and y = cosθ from 0° to 90° using exact values they can justify, then overlay the graph on the verbal, symbolic, and tabular representations they already know. A short Desmos animation of the unit circle traces the sine and cosine waves, giving a preview of the continuous, periodic functions they will study in precalculus. This lesson is a synthesis moment for classes with time and readiness; the unit remains coherent without it.
Lesson 7.6 – The Law of Sines
Starting from the area of a triangle given two sides and an included angle, students find the area of specific triangles three different ways and generalize their work to derive the Law of Sines. They then apply it to the Mount Everest problem, where right‑triangle trig alone is not enough. The lesson highlights the move from inductive pattern‑finding in specific diagrams to a deductive proof that works for every triangle.
Lesson 7.7 – The Law of Cosines
Students extend the Pythagorean theorem to any triangle by creating hidden right triangles, managing variables carefully, and working through the algebra that leads to the Law of Cosines. They then see the Pythagorean theorem emerge as a special case and tackle an ambiguous‑case problem that requires both Laws of Sines and Cosines plus earlier theorems. The unit’s through‑line becomes explicit: each result becomes a tool, and many of those tools converge in a single rich problem.
Unit 7 Assessment
I’m happy to share my Unit 7 assessments and grading rubrics for teacher use, but I don’t want them publicly available on the internet. If you’d like a copy, please email me from your school account at jay@thinkingthroughgeometry.net and let me know which school and state you teach in.