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三角関数の恒等式の計算機：恒等式を確かめる

この計算機は、sin²x + cos²x = 1 のような等式が恒等式かどうかを確かめます。両辺を数十個の角で計算し、値が異なるものがあれば反例を示し、なければ恒等式であると確認します。下の公式は証明を書くときに役立ちます。

角には x を使います。sin、cos、tan、sec、csc、cot が使えます。sin^2(x) は (sin x)² の意味です。

結果

値を入力して「計算」を押すと結果が表示されます。

公式

sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1

The Pythagorean identity; dividing by cos²x gives 1 + tan²x = sec²x.

tan⁡x=sin⁡xcos⁡x\tan x = \frac{\sin x}{\cos x}

The quotient identity, for cos x ≠ 0.

sin⁡2x=2sin⁡xcos⁡x,cos⁡2x=cos⁡2x−sin⁡2x\sin 2x = 2\sin x\cos x, \quad \cos 2x = \cos^2 x - \sin^2 x

Double-angle identities.

この計算ツールの使い方

  1. Type the identity with an equals sign, using x for the angle — for example tan(x)*cos(x) = sin(x).
  2. Press Calculate. Both sides are evaluated at 48 angles between about −6 and 6 radians, skipping any where a side is undefined.
  3. If the sides ever differ, you get the angle where they differ — the identity is false. If they always agree, it is an identity, and you can prove it with the formulas below.

解き方付き例題

The Pythagorean identity

Both sides are evaluated at 48 angles; they agree every time.

Proof: on the unit circle a point is (cos x, sin x), and by Pythagoras cos²x + sin²x = 1.

Verify a quotient identity

tan x · cos x = (sin x / cos x) · cos x = sin x, wherever cos x ≠ 0.

The calculator confirms it at every angle it tested.

Using sec and tan

Divide sin²x + cos²x = 1 by cos²x: tan²x + 1 = sec²x, so sec²x − tan²x = 1.

A false “identity”

At x = −6.1: sin(−12.2) ≈ 0.3582 but 2 sin(−6.1) ≈ 0.3643, so the two sides differ.

The correct double-angle identity is sin 2x = 2 sin x cos x.

よくある間違い

  • Treating sin 2x as 2 sin x — the 2 is inside the function.
  • Writing sin x² when you mean (sin x)²; type sin(x)^2 or sin^2(x).
  • Working on both sides at once in a proof: transform one side until it matches the other.
  • Dividing by an expression such as cos x without noting where it is zero.

よくある質問

How does the trig identities calculator verify an identity?

It evaluates both sides at 48 different angles. One mismatch proves the equation is not an identity; agreement everywhere is strong evidence that it is — and then the formulas help you write a proof.

Is a numerical check a proof?

No. It can prove an identity false (with a counterexample) but only confirm a true one. Teachers expect an algebraic proof using identities such as sin²x + cos²x = 1.

Which trig identities should I learn first?

The Pythagorean identity sin²x + cos²x = 1, tan x = sin x / cos x, and the double-angle formulas for sin 2x and cos 2x.

Does it work in degrees?

The angle is in radians, but an identity is true in both units or in neither, so the verdict is the same.