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

At a skate park, a rider approaches a curved ramp. Its changing steepness alters the rider&#8217;s acceleration, tracing a path shaped by angular motion.As the rider moves up or down the ramp, the path&#8217;s steepness changes&#8212;with similarities to the tangent function, which compares vertical rise to horizontal run.Tangent is defined as sine divided by cosine for any angle measured from the horizontal.From this definition, the double-angle identity can be derived. Recall sine and cosine of twice the angle. Substituting these into the definition of tangent leads to the result: tangent of twice the angle. When the rider launches higher, and the angle doubles, this identity predicts the new slope.For smaller angles near the base of the ramp, where the rider enters a gentle curve, the half-angle identity becomes useful.Recalling and applying the sine and cosine half-angle identities to the tangent definition gives an expression. Simplifying it by taking its conjugate and multiplying by one plus or one minus cosine produces two equivalent forms of the tangent half-angle identity.These identities aid in analyzing directional changes and modeling curved motion in skate park dynamics.

trigonometric identities ii

Double-angle and half-angle trigonometric identities are derived from the fundamental sum and difference formulas and serve as essential tools for simplifying expressions, solving equations, and evaluating integrals. These identities reduce the complexity of trigonometric functions by relating functions of a multiple or fractional angle to functions of a single angle. Their applications extend across mathematics, physics, and engineering, particularly in Fourier analysis, wave mechanics, and calculus.The double-angle identities are obtained by substituting equal angles into the sum formulas. For sine and cosine, they are:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/41035/41035_Equation_1_editor_41035.svg' alt='41035_Equation_1.svg' /></figure>For tangent, the double-angle identity is:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/41035/41035_Equation_2_editor_41035.svg' alt='41035_Equation_2.svg' /></figure>These identities are especially useful for reducing higher powers of trigonometric functions and simplifying integrals.The half-angle identities are derived from the double-angle formulas by solving for sin2&#55349;&#57091; and cos2&#55349;&#57091;. They are expressed as:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:71.57%;'><img style='aspect-ratio:1920/650;' src='https://cloudfront.jove.com/files/ftp_upload/41035/41035_Equation_3_editor_41035.svg' alt='41035_Equation_3.svg' /></figure>The choice of sign depends on the quadrant in which the half-angle lies. These identities are crucial in evaluating trigonometric functions for nonstandard angles and in integration techniques, such as trigonometric substitution.

Trigonometric Identities II

Double-angle and half-angle trigonometric identities are derived from the fundamental sum and difference formulas and serve as essential tools for ...

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Math Fundamentals

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Trigonometry

KEY TERMS AND CONCEPTS

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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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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