Why does $\sin 35^\circ$ have the same value in every right triangle containing a $35^\circ$ angle? The answer is similarity. Any two such triangles share a right angle and the chosen acute angle, so they are similar by AA. Their corresponding sides are proportional, and each scale factor cancels from the ratios.
This editable lesson begins with that argument, defines all six trigonometric ratios, proves their central identities, and derives every exact value at $30^\circ$, $45^\circ$, and $60^\circ$ from geometry. It then develops unknown-side and unknown-angle problems, angles of elevation and depression, a two-observation height model, and thirty sequenced questions. A separate teacher key contains full equations, proof checkpoints, a scoring guide, and misconception notes.
Exact values are derived, not announced
An isosceles right triangle with legs $1$ has two $45^\circ$ angles. Pythagoras gives its hypotenuse as $\sqrt2$, so
$$\sin45^\circ=\cos45^\circ=\frac{1}{\sqrt2}=\frac{\sqrt2}{2},\qquad \tan45^\circ=1.$$
For $30^\circ$ and $60^\circ$, bisect an equilateral triangle of side $2$ with an altitude. Congruence shows that the altitude bisects both the base and the top angle. Each half has hypotenuse $2$, short leg $1$, and other leg
$$\sqrt{2^2-1^2}=\sqrt3.$$
Thus the side ratio is $1:\sqrt3:2$. Relative to $30^\circ$ this gives
$$\sin30^\circ=\frac12,\qquad \cos30^\circ=\frac{\sqrt3}{2},\qquad \tan30^\circ=\frac{\sqrt3}{3},$$
and relative to $60^\circ$ it gives
$$\sin60^\circ=\frac{\sqrt3}{2},\qquad \cos60^\circ=\frac12,\qquad \tan60^\circ=\sqrt3.$$
The reciprocal ratios follow immediately. The editable project includes the complete proof, both geometric constructions, and a table of all eighteen values.
The identities have short reasons
If opposite, adjacent, and hypotenuse are $y,x,h$, then
$$\frac{\sin\theta}{\cos\theta}=\frac{y/h}{x/h}=\frac{y}{x}=\tan\theta.$$
Dividing $x^2+y^2=h^2$ by $h^2$ proves
$$\sin^2\theta+\cos^2\theta=1.$$
Dividing this identity by $\cos^2\theta$ or $\sin^2\theta$ proves
$$1+\tan^2\theta=\sec^2\theta,\qquad 1+\cot^2\theta=\csc^2\theta.$$
The two acute angles in a right triangle are complementary. A leg opposite one is adjacent to the other, so $\sin\theta=\cos(90^\circ-\theta)$ and $\tan\theta=\cot(90^\circ-\theta)$. The lesson presents these as consequences of the diagram rather than unrelated formulas.
A representative application
An observer stands $28.0$ m from a building on level ground. The observer’s eye is $1.65$ m high, and the angle of elevation to the roof is $41^\circ$. The rise above eye level is
$$r=28.0\tan41^\circ\approx24.34\text{ m}.$$
The building height is therefore approximately $24.34+1.65=25.99$ m, or $26.0$ m to the nearest tenth. Separating the right-triangle rise from observer height prevents a common modeling error.
What students practice
The thirty questions move through four connected strands:
- naming sides relative to a reference angle and proving ratio invariance by similarity;
- reconstructing special triangles and proving exact values and identities;
- solving for sides and angles with inverse trigonometric functions;
- modeling flagpoles, ladders, ramps, cliffs, surveying, and error analysis.
Representative prompts ask students to prove why tripling every side leaves the ratios unchanged, derive $\cos30^\circ$ without a calculator, recover five exact ratios from $\sin\theta=5/13$, solve a two-observation tower problem, and diagnose radian-mode and reciprocal errors.
Adapt it for a class
The primary file and teacher key are separate. A teacher can shorten the packet to a similarity-and-ratios lesson, use the exact-value proof as a second lesson, or reserve two-observation problems for extension. Units, calculator policy, rounding, diagram labels, practice selection, and answer space are editable.
The included AI authoring guide uses small, ordinary prompts rather than one oversized request. One prompt asks for a similarity-based definition, another asks for geometric special-angle proofs, and later prompts add identities, worked examples, applications, and the separate key. Each result is meant to be checked before useful material is copied into the document.
Sources and boundaries
- Common Core State Standards for Mathematics, HSG-SRT.C.6-8, for similarity-based ratio definitions, complementary-angle relationships, and applied right-triangle problems.
- OpenStax Precalculus 2e, Section 5.4, for a modern open treatment of right-triangle ratios, exact values, cofunctions, and applications.
- S. L. Loney, Plane Trigonometry (1893), public domain in the United States, consulted for classical scope and proof emphasis.
- Hugh Blackburn, Elements of Plane Trigonometry, public domain in the United States, consulted as a second classical reference.
The wording, exercises, and vector diagrams are newly created for this package; the classical books are references rather than copied source text. This resource treats acute-angle right-triangle trigonometry only. Unit-circle definitions, general angles, radian measure, and non-right-triangle laws belong in later lessons. Independent mathematics-teacher review and classroom trial remain outstanding.