Name: double-angle and half-angle formulas
Uploaded: 2025-11-26T05:44:27.000Z
Duration: 1 min 28 s
Channel: JoVE

スケートパークで、ライダーが湾曲したスロープに近づきます。その急勾配の変化によりライダーの加速度が変化し、角運動によって形作られた経路をたどります。ライダーがランプを上下すると、パスの急勾配が変化し、垂直方向の上昇と水平方向の走行を比較する接線関数と似ています。正接は、水平から測定された任意の角度の正弦を余弦で割ったものとして定義されます。この定義から、ダブルアングル恒等式を導き出すことができます。角度の2倍の正弦と余弦を思い出してください。これらを接線の定義に代入すると、角度の2倍の接線という結果が得られます。ライダーがより高く発進し、角度が 2 倍になると、この恒等式は新しい傾斜を予測します。ライダーが緩やかなカーブに入るランプの基部近くの小さな角度の場合、ハーフアングル ID が役立ちます。正弦と余弦の半角恒等式を思い出して接線定義に適用すると、式が得られます。共役を取り、1プラスまたは1マイナスコサインを掛けて単純化すると、接線半角恒等式の2つの等価形式が生成されます。これらの恒等式は、スケートパークのダイナミクスにおける方向変化の分析と曲線運動のモデル化に役立ちます。

三角関数の恒等式 ii

Trigonometric Identities II

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三角関数の恒等式 II

Education

JoVE Core

Math Fundamentals

数学の基礎

Unit

ユニット

Trigonometry

三角法

KEY TERMS AND CONCEPTS

主要用語と概念

double-angle and half-angle formulas

Double-angle and half-angle trigonometric formulas connect angles that are multiplied or divided by two. They come from the sum and difference formulas, so they build on earlier trigonometry skills. These identities help simplify expressions, solve equations, and evaluate integrals.

The double-angle formulas come from substituting equal angles into the sum formulas. For sine and cosine, they link the value at 2θ to the value at θ. The tangent double-angle identity follows the same idea. These formulas are useful when simplifying higher powers of trigonometric expressions.

The half-angle formulas are found by solving the double-angle equations for sin 2θ and cos 2θ. They give trigonometric values for half of an angle instead of the whole angle. The sign in a half-angle formula depends on the quadrant where the half-angle lies.

These identities are important for finding trigonometric values for nonstandard angles. They also appear in calculus, especially in trigonometric substitution and in methods for evaluating integrals. Their uses also extend into mathematics, physics, and engineering, including Fourier analysis and wave mechanics.

Double-Angle and Half-Angle Formulas

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trigonometric identities ii

Double-angle and half-angle trigonometric formulas connect angles that are multiplied or divided by two. They come from the sum and difference formulas, so they build on earlier trigonometry skills. These identities help simplify expressions, solve equations, and evaluate integrals. The double-angle formulas come from substituting equal angles into the sum formulas. For sine and cosine, they link the value at 2θ to the value at θ. The tangent double-angle identity follows the same idea. These...

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key terms and concepts

Learn how double-angle and half-angle trig formulas come from sum and difference identities, and how they help simplify equations, powers, and integrals.

Double-Angle and Half-Angle Formulas | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Double-Angle and Half-Angle Formulas

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