Trigonometry: an angle as a ratio of sides

Where we are on the map

We study ratios preserved across similar right triangles.

Begin with a familiar image

A 10% road grade is a rise-to-run ratio. The same angle keeps this ratio at every drawing size.

Precise meaning

sin=opposite/hypotenuse, cos=adjacent/hypotenuse, and tan=opposite/adjacent.

The lesson’s support signal

meaning → model → calculation → check → explain in your own words

mark angle → name sides relative to it → choose known/unknown ratio → equation → estimate

Worked example

In a 3–4–5 triangle, for the angle opposite 3: sin=3/5, cos=4/5, and tan=3/4.

Example with a fading prompt

Change the numbers in the example and repeat the solution without copying its steps. Estimate first, calculate exactly, and check against the original condition.

Recall without a prompt

1. Define the main idea in one sentence.
2. Rebuild the support signal from memory.
3. Solve the example with different numbers and explain why every step is valid.

Find and correct the mistake

Calling the hypotenuse the adjacent leg is wrong: “adjacent leg” excludes the hypotenuse.

Transfer to a new setting

Apply the same relationship to something you can measure yourself. Name the quantities, units, and the boundary within which the model is valid. Check the answer by a second representation or an inverse operation.

Exercise

Required. An 8 m ladder makes a 25° angle with the wall. Find height and base distance, then check with Pythagoras.

With your own data. Create one everyday example with the same structure and show it in words, a formula, and a check.

Optional. Seven days later, change the numbers and repeat without the support map.

Open perspective

Vectors combine magnitude and direction so displacements can be added.

Course contents