How this calculator works
Reduce the input by full turns, then reflect it into the first quadrant. The table supplies exact fractions and radicals at 0°, 15°, 30°, 45°, 60°, 75° and 90°. Apply the coordinate signs for the original quadrant. Cotangent, secant and cosecant are obtained from the corresponding ratios and reciprocals with rationalised denominators. A zero denominator is Undefined. This focused tool accepts only multiples of 15°; it does not label a rounded decimal at another angle as exact.
Formula
sin 15° = sin(45°-30°) = (√6-√2)/4; cos 15° = (√6+√2)/4; tan 15° = 2-√3. tan θ = sin θ/cos θ; cot θ = cos θ/sin θ; sec θ = 1/cos θ; csc θ = 1/sin θ.
Worked example
At 135°, the reference angle is 45°. Sine is √2/2, cosine is -√2/2 and tangent is -1. Their reciprocals are cosecant √2, secant -√2 and cotangent -1. At 15°, cosecant rationalises to √6+√2.
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Frequently asked questions
Why are 15° and 75° included?
They follow from adding or subtracting 30° and 45°. Their radicals let you evaluate more angles than the usual 30° and 45° table.
Can I enter π/12?
Choose fraction-of-π mode with numerator 1 and denominator 12. Negative numerators and denominators are allowed, but the denominator cannot be zero.
Why is tan 90° undefined?
Its cosine denominator is exactly zero. An infinite decimal or a very large finite value would not be a defined real tangent.