Trig Ratios — High School Math Quiz
Master sine, cosine, and tangent — the three foundational trigonometric ratios. 45 problems across three tiers: Geometry (SOHCAHTOA definitions, 3-4-5 triangles, special angles 30/45/60), Pre-Calculus (reciprocal functions csc/sec/cot, radian conversion, co-function identities), and Advanced (inverse trig, second-quadrant values, Pythagorean identities, real-world triangle applications).
How to Play Trig Ratios
- 1
Study the Triangle Reference
The lobby shows a labeled right triangle and the three SOH-CAH-TOA cards. Memorize which side is 'opposite' (across from θ), which is 'adjacent' (next to θ), and which is the 'hypotenuse' (longest side, across from the right angle). This spatial understanding is the foundation of everything else.
- 2
Choose a Difficulty
Geometry focuses on the six definitions and exact values at 30°, 45°, 60°, 90°. Pre-Calculus adds reciprocal functions, radians, and identities. Advanced introduces inverse trig, angles beyond 90°, and application problems — all of which appear on SAT and ACT math sections.
- 3
Identify the Ratio
For EVALUATE questions: decide which ratio applies (sin, cos, or tan), then plug in the sides or use a memorized special-angle value. For IDENTIFY questions: recall the definition or identity directly. For INVERSE questions: ask 'what angle produces this ratio?' For APPLY questions: label the triangle sides relative to the given angle, then choose sin, cos, or tan.
- 4
Use Hints Strategically
Each hint states the key formula or identity and walks through the exact calculation. The hint for EVALUATE problems shows which triangle and side lengths to use. INVERSE hints name the special triangle. IDENTITY hints show the algebraic derivation step by step.
Key Features
45 Problems Across 3 Tiers
Geometry drills SOHCAHTOA definitions, the 3-4-5 right triangle, and the six exact values for the special angles 30°, 45°, and 60°. Pre-Calculus introduces the three reciprocal functions (csc, sec, cot), degree-to-radian conversion, the co-function identity sin θ = cos(90°−θ), and evaluating trig functions at 0° and 90°. Advanced covers inverse trig (sin⁻¹, cos⁻¹, tan⁻¹), second-quadrant values (sin 120°, cos 135°), the fundamental identities sin²θ + cos²θ = 1 and tan²θ + 1 = sec²θ, and real-world ladder and right-triangle problems.
Labeled Right-Triangle Lobby
The start screen features a custom SVG right triangle with sides labeled 'opposite', 'adjacent', and 'hypotenuse' alongside the angle θ. Three color-coded SOH / CAH / TOA cards define each ratio in symbolic form, giving you the complete reference before you begin.
4 Problem Types — Color-Coded
EVALUATE (cyan) asks for the exact value of a trig expression. IDENTIFY (teal) tests your knowledge of definitions, reciprocal functions, conversions, and identities. INVERSE (sky) gives you a ratio and asks for the angle. APPLY (indigo) presents a real-world or geometric scenario requiring trig to find a missing side or angle.
Dark Starry-Sky Theme
The game uses a deep navy slate background with cyan, teal, and sky-blue accents — evoking the astronomical origins of trigonometry. Answer tiles glow on hover, and correct answers pulse green against the dark backdrop.
Frequently Asked Questions
What does SOHCAHTOA mean?
It's a mnemonic for the three main trig ratios: SOH = Sin is Opposite over Hypotenuse, CAH = Cos is Adjacent over Hypotenuse, TOA = Tan is Opposite over Adjacent. The 'opposite' side is across from the angle θ, 'adjacent' is next to it, and 'hypotenuse' is always the longest side (across from the 90° angle).
What are the special angle values I need to memorize?
sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3. sin 45° = cos 45° = √2/2, tan 45° = 1. sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3. Also: sin 0° = 0, cos 0° = 1, sin 90° = 1, cos 90° = 0, tan 90° = undefined. These come from the 30-60-90 and 45-45-90 special right triangles.
What are csc, sec, and cot?
They are the three reciprocal functions: csc θ = 1/sin θ (cosecant), sec θ = 1/cos θ (secant), cot θ = 1/tan θ = cos θ/sin θ (cotangent). They are less common in early coursework but appear in Pre-Calculus, AP Calculus, and the SAT.
What are the Pythagorean trig identities?
The three Pythagorean identities are: (1) sin²θ + cos²θ = 1 (fundamental — comes directly from the Pythagorean theorem on the unit circle), (2) tan²θ + 1 = sec²θ (derived by dividing identity 1 by cos²θ), (3) 1 + cot²θ = csc²θ (derived by dividing identity 1 by sin²θ). The first identity is the most important to memorize.
How do I find trig values for angles greater than 90°?
Use the reference angle and the quadrant sign rules. In Q2 (90° to 180°): sin is positive, cos and tan are negative. sin 120° = sin(180°−120°) = sin 60° = √3/2. cos 135° = −cos(180°−135°) = −cos 45° = −√2/2. A useful memory aid is ASTC (All Students Take Calculus): All positive in Q1, Sin positive in Q2, Tan positive in Q3, Cos positive in Q4.
How does scoring work?
Correct answers earn 10 pts (Geometry), 15 pts (Pre-Calculus), or 20 pts (Advanced). Consecutive correct answers add a 5-point streak bonus per answer after the first. A wrong answer resets the streak to zero.