What it does
Trigonometry starts with one idea: in a right triangle, the ratios of the sides depend only on the angle, not on how big the triangle is. This explorer lets you drag that angle and watch the three sides and the six ratios change together:
- sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent,
- and their upside-down partners cosec, sec and cot.
It covers the Class 10 chapter introduction to trigonometry (ratios, special angles, identities, complementary angles), and with the Class 11 level it takes the point all the way round the unit circle, as in the Class 11 chapter on trigonometric functions: angles beyond 90°, negative angles, radians and the sign of each ratio in each quadrant.
Start from the theorem behind the identities in the pythagoras proof playground playground, and plot y = sin x and friends as curves on the graphing calculator.
How to use it
- Drag the blue point along the circle, or use the θ slider, or type an angle. The opposite side is drawn in green, the adjacent side in blue and the hypotenuse in black, each with its length.
- Change r, the hypotenuse. The triangle grows or shrinks but every ratio stays the same.
- Tap a special angle (0°, 30°, 45°, 60°, 90°). Its column lights up in the table of exact values.
- The identities card puts your angle into sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ and the complementary-angle rules, so you can see them hold.
- Switch the Level to Class 11 to use the full circle. The page says which quadrant the angle is in, which ratios are positive there, and the angle in radians.
A worked example
At the start the angle is 30° and the hypotenuse is 5.
- Opposite = 5 × sin 30° = 2.5 and adjacent = 5 × cos 30° ≈ 4.33.
- sin 30° = 2.5 / 5 = 0.5, cos 30° = 4.33 / 5 ≈ 0.866, tan 30° = 2.5 / 4.33 ≈ 0.5774.
- These match the exact values 1/2, √3/2 and 1/√3.
- Identity: sin²30° + cos²30° = 0.25 + 0.75 = 1.
- Complement: sin 60° = 0.866 = cos 30°.
Now drag r to 10. Opposite becomes 5 and adjacent 8.66, but sin 30° is still 5 / 10 = 0.5.
Tips
- At 0° the opposite side has length 0, so cot and cosec are not defined. At 90° the adjacent side is 0, so tan and sec are not defined. The table says “not defined” there.
- A handy way to remember the special values: sin 0°, 30°, 45°, 60°, 90° are √0/2, √1/2, √2/2, √3/2, √4/2. Cos runs the same list backwards.
- In the Class 11 view, “All Students Take Coffee” names the positive ratios in quadrants I to IV: all, sin, tan, cos.
Check yourself
Four fresh questions each time: a ratio from a triangle’s sides, an expression with exact values, a complementary angle and a missing side from an angle. Type the answer and press Check; open the worked answer if you get stuck. Nothing is saved.
FAQ
Why is sin θ never more than 1? The opposite side can never be longer than the hypotenuse.
Degrees or radians? The page uses degrees, like the Class 10 book. The Class 11 level also gives the angle in radians (180° = π).
Is this part of NCERT? No. It follows the same topics, but the pictures and questions are fibub’s own.
Where the numbers come from
- Standard trigonometry: the six ratios of a right triangle, exact values at the special angles, sin²θ + cos²θ = 1 and its two partners, complementary angles and the signs in each quadrant. Written for fibub.
- Exact-value table checked entry by entry against the computed ratios; identities checked at every whole degree; values checked against Python's math module.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.