Rankwise

Value of tan 22.5°

tan 22.5° = √2 − 1 ≈ 0.4142

√2 − 1Exact value
0.414214Decimal
π/822.5° in radians
cot 67.5°Same value (co-function)

How to derive it

  1. Use tan(θ/2) = (1 − cos θ)/sin θ with θ = 45°.
  2. = (1 − 1/√2)/(1/√2) = √2 − 1.
  3. So tan 22.5° = √2 − 1.

All trig ratios of 22.5°

RatioExactDecimal
sin 22.5°√(2 − √2)/20.3827
cos 22.5°√(2 + √2)/20.9239
tan 22.5°√2 − 10.4142
cosec 22.5°1 ÷ sin 22.5°2.6131
sec 22.5°1 ÷ cos 22.5°1.0824
cot 22.5°1 ÷ tan 22.5°2.4142

Exact values of special angles

Anglesincostan
15°(√6 − √2)/4(√6 + √2)/42 − √3
18°(√5 − 1)/4√(10 + 2√5)/4√(25 − 10√5)/5
22.5°√(2 − √2)/2√(2 + √2)/2√2 − 1
30°1/2√3/21/√3
36°√(10 − 2√5)/4(√5 + 1)/4√(5 − 2√5)
45°1/√21/√21
54°(√5 + 1)/4√(10 − 2√5)/4√(25 + 10√5)/5
60°√3/21/2√3
67.5°√(2 + √2)/2√(2 − √2)/2√2 + 1
72°√(10 + 2√5)/4(√5 − 1)/4√(5 + 2√5)
75°(√6 + √2)/4(√6 − √2)/42 + √3

For whole-degree values from 0° to 90°, see the trigonometry table.

Questions students ask

What is the value of tan 22.5°?

√2 − 1, which is about 0.4142.

What is tan 22.5° in radians?

22.5° is π/8 radians, so tan(π/8) = √2 − 1.

Is tan 22.5° equal to cot 67.5°?

Yes. tan θ = cot(90° − θ), so tan 22.5° = cot 67.5°.

How do you find tan 22.5° without a calculator?

Use tan(θ/2) = (1 − cos θ)/sin θ with θ = 45°. = (1 − 1/√2)/(1/√2) = √2 − 1. So tan 22.5° = √2 − 1.

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Trigonometry tableSin, cos, tan and more for standard angles, 0° to 360°.Maths formula sheetAlgebra to integrals, the formulas JEE Main uses most.Log tableFour-figure log table with mean differences, and a log calculator.Squares, cubes and rootsSquares, cubes, square roots and cube roots from 1 to 100.