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PREPITUS
GeometryAdvanced

Right Triangles and Trigonometry

Move fluently between the Pythagorean theorem, SOH-CAH-TOA, and the two special triangles the SAT reuses constantly.

⏱ 9 min📍 Learn first: Lines, Angles, and Triangles
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13 free resources
00:00 / 01:42

Right Triangle Ratios and Trigonometry Laws

A short spoken refresher on this concept.

00:00 / 22:41

Spatial cheat codes for SAT trigonometry

The full lesson, worked through in depth.

Concept Snapshot

Definition
In a right triangle: a² + b² = c², and the trig ratios sin = opp/hyp, cos = adj/hyp, tan = opp/adj relate angles to sides.
What the SAT tests
Finding sides with trig ratios, special right triangles (45-45-90, 30-60-90), and the sin/cos complement identity.
Key rule
45-45-90 sides: x, x, x√2. 30-60-90 sides: x, x√3, 2x. And sin(x°) = cos(90° − x°).
Most common mistake
Labeling opposite and adjacent from the wrong angle's point of view.
Prerequisite
Lines, Angles, and Triangles

Visual Explanation

θadjacentoppositehypotenuse
Geometry questions live on labeled figures. Put your finger on the angle in play — opposite is the side it doesn't touch, adjacent is the leg it does. The relationships follow from the labels, not the picture's orientation.

Concept Mind Map

DefinitionIn a right triangle: a² + b² = c², and the trig ratios sin = opp/hyp, cos = adj/hyp, tan = opp/adj relate angles to sides.
Key rule45-45-90 sides: x, x, x√2. 30-60-90 sides: x, x√3, 2x. And sin(x°) = cos(90° − x°).

Right Triangles and Trigonometry

Geometry

Ratio SolveOne angle + one side → use the right ratio for the target side.
Special TrianglesSpot 45° or 30/60° and use the template — no calculator trig needed.
Complement Identitysin A = cos B implies A + B = 90°.

How the SAT Tests It

Pattern

Ratio Solve

One angle + one side → use the right ratio for the target side.

Clue: An angle and a side given
Pattern

Special Triangles

Spot 45° or 30/60° and use the template — no calculator trig needed.

Clue: 45°, 30°, 60°, or x√2, x√3 in choices
Pattern

Complement Identity

sin A = cos B implies A + B = 90°.

Clue: "sin(x) = cos(…)"

Worked Example

A 24-ft ladder leans against a wall at a 60° angle with the ground. How high on the wall does it reach?

Right triangle: ladder = hypotenuse (24), target = side OPPOSITE the 60° angle.
Opposite and hypotenuse → sine: sin 60° = h/24.
sin 60° = √3/2, so h = 24 × √3/2 = 12√3.
12√3 ≈ 20.8 ft — less than the 24-ft ladder, as a height must be ✓.

Try It Yourself

In right triangle ABC, the right angle is at C, and sin A = 5/13. What is cos B?

Common Traps

What students do

Fixing "opposite" and "adjacent" to the triangle instead of the angle.

Why it fails

The same leg is opposite one acute angle and adjacent to the other — the labels travel with the angle you're using.

Do this instead

Before writing any ratio, put your finger on the angle in question. Opposite = the side your finger doesn't touch.

Interactive Flashcards

Review deckCard 1 of 8

Four-Question Knowledge Check

4 quick questions drawn from this concept's own snapshot, patterns and traps.

Question 1 of 4. Which statement is the key rule for this concept?

Audio Lesson

Spatial cheat codes for SAT trigonometry

SOH-CAH-TOA discipline plus the 45-45-90 and 30-60-90 templates.

00:00 / 22:41

22:41 · the full lesson

What to notice

45-45-90 sides: x, x, x√2. 30-60-90 sides: x, x√3, 2x. And sin(x°) = cos(90° − x°).

What the lesson covers

  • 0:00 Ratio setup
  • 1:40 Special triangles
  • 3:20 The complement identity
  • 5:00 Practice

Study Guide and Transcript

One-page study guide

The snapshot, the key rule, the patterns and the traps, on one page.

Source-Grounded AI Tutor

Ask the Right Triangles and Trigonometry notebook

A tutor grounded in this concept's own sources — ask for a simpler explanation, another example, or why a particular answer is wrong. It is in development and will appear here once its sources pass academic review.

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