Last updated October 10, 2026 · Report an error
Trigonometry has six ratios in total — sine, cosine, tangent, and their three reciprocals cotangent, secant, and cosecant — but they're never truly independent of one another. A small set of fundamental identities ties them all together, meaning that knowing just one ratio (plus which quadrant the angle sits in) is enough information to derive all six. This calculator does exactly that: given either an angle directly, or just one known trig ratio of an unknown angle, it derives every other ratio using the fundamental identities. This is a genuinely practical shortcut in trigonometry problem-solving — rather than looking up or recomputing each ratio independently, you can start from whichever single piece of information you actually have (an angle, or one ratio from a word problem) and derive the rest algebraically, seeing exactly which identity produced each value along the way.
How it works
Choose whether you know the angle directly, or just one ratio of an unknown angle. If you enter an angle directly, all six ratios are computed from it in the standard way. If instead you know just one ratio (like sinθ = 0.6), you also need to specify which quadrant the angle falls in, since a single ratio value is mathematically consistent with two different angles in two different quadrants that have opposite signs for some of the other ratios — the quadrant tells the calculator which sign to use when resolving the rest. From there, the Pythagorean identity (sin²θ + cos²θ = 1) finds the paired ratio, the quotient identities (tanθ = sinθ/cosθ, cotθ = cosθ/sinθ) derive tangent and cotangent, and the reciprocal identities (secθ = 1/cosθ, cscθ = 1/sinθ) derive the remaining two.
- Enter what do you know?.
- Enter angle θ (if known).
- Enter angle unit.
- Enter known ratio (if angle unknown).
- Enter ratio value.
- Enter quadrant of θ.
- Click Calculate to see your results.
Examples
sin(θ) = 0.6, quadrant II
cos(θ) = −0.8 (from the Pythagorean identity: cos = ±√(1 − 0.6²) = ±0.8, negative because cosine is negative in quadrant II), then tan(θ) = 0.6 ÷ (−0.8) = −0.75, cot(θ) = −1.3333, sec(θ) = −1.25, and csc(θ) = 1.6667, all following from the quotient and reciprocal identities.
cos(θ) = 0.5, quadrant I
sin(θ) = √(1 − 0.5²) ≈ 0.8660 (positive, since sine is positive in quadrant I) — this actually corresponds to the well-known 60° angle, and tan(θ) ≈ 1.7321, cot(θ) ≈ 0.5774, sec(θ) = 2, csc(θ) ≈ 1.1547 follow from there.
Starting from a known angle directly
Entering θ = 30° directly gives the standard reference values: sin = 0.5, cos ≈ 0.8660, tan ≈ 0.5774, along with their reciprocal ratios — no quadrant ambiguity to resolve since the angle itself is already fully known.
Who should use it
- Solving trigonometry word problems that give one ratio and a quadrant description.
- Verifying a manually derived set of trig ratios against the fundamental identities.
- Trigonometry and precalculus coursework.
Industry applications
- Mathematics education
- Engineering and physics coursework involving trigonometric relationships
Advantages
- Derives all six trig ratios from just one known ratio and a quadrant.
- Shows exactly which identity produced each derived value.
- Works whether you start from an angle or from a single ratio.
Limitations
- Requires knowing the correct quadrant to resolve sign ambiguity — an incorrect quadrant produces an incorrect (though internally consistent) set of ratios.
Common mistakes to avoid
- Forgetting to set the quadrant when starting from a single ratio, which can silently assume the wrong sign.
- Mixing up which ratios are reciprocals of which — secant is the reciprocal of cosine (not sine), and cosecant is the reciprocal of sine (not cosine).
- Assuming the Pythagorean identity always gives a positive square root, when the correct sign depends entirely on the quadrant.
Best practices
- Always double-check the quadrant matches the actual context of your problem (e.g. a word problem describing an obtuse angle implies quadrant II).
- Verify your result using the Pythagorean identity as a sanity check: sin²θ + cos²θ should always equal 1.
- Keep the sign conventions for each quadrant (ASTC: All positive in QI, Sine in QII, Tangent in QIII, Cosine in QIV) handy as a quick memory aid.
Tips
- Remember the ASTC mnemonic (All, Sine, Tangent, Cosine) for which ratios are positive in each quadrant.
- Use the Pythagorean identity as your go-to cross-check whenever deriving sine or cosine from the other.