Use similarity to compute trigonometric ratios when appropriate (SAT). | Right triangles and trigonometry | SAT Math

Similar right triangles keep the same trig ratios, so you can transfer a ratio from one copy to another.

Core Idea

Similar right triangles have the same acute angles, so their trig ratios are identical regardless of size. You can compute a trig ratio from any similar triangle and apply it to another.

Understanding

Rule: Trig ratios depend on angle, not size.

Similar right triangles have the same acute angles.
That means the same sine, cosine, and tangent values.
You can compute the ratio from the triangle with the clearest data.

If two triangles are similar, the ratio transfers exactly.

Step by Step

Confirm the triangles are similar (same angles).
Compute the trig ratio from the triangle that gives you the most information.
Apply that ratio to the other triangle to find the unknown side.
If needed, scale the known Pythagorean triple to match a given side length.

Misconceptions

Thinking trig ratios change when the triangle is scaled up or down.
Using sides from two different (non-similar) triangles in the same ratio.
Forgetting to verify similarity before transferring a trig ratio between triangles.

Question

Worked Example

Triangle $𝐴 𝐵 𝐶$ is similar to triangle $𝐷 𝐸 𝐹$ , and both are right triangles. In triangle $𝐴 𝐵 𝐶$ , $𝐴 𝐵 = 5$ , $𝐵 𝐶 = 12$ , and $𝐴 𝐶 = 13$ , with the right angle at $𝐵$ . In triangle $𝐷 𝐸 𝐹$ , $𝐷 𝐸 = 15$ . What is $s i n 𝐹$ ?

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