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

스케이트 공원에서 한 라이더가 곡선 경사로에 접근합니다. 변화하는 가파른 경사는 라이더의 가속도를 변화시켜 각운동에 의해 형성된 경로를 추적합니다.라이더가 경사로를 위아래로 이동함에 따라 경로의 가파른 각도가 변경되며, 이는 수직 상승과 수평 달리기를 비교하는 탄젠트 함수와 유사합니다.탄젠트는 수평에서 측정된 모든 각도에 대해 사인을 코사인으로 나눈 것으로 정의됩니다.이 정의에서 이중 각도 항등식을 도출할 수 있습니다. 각도의 두 배인 사인과 코사인을 기억하십시오. 이것을 접선의 정의에 대입하면 각도의 두 배에 달하는 접선이라는 결과가 발생합니다. 라이더가 더 높이 발사하고 각도가 두 배가 되면 이 항등식은 새로운 경사를 예측합니다.라이더가 완만한 곡선으로 진입하는 램프 바닥 근처의 작은 각도의 경우 하프 앵글 ID가 유용합니다.사인 및 코사인 반각 항등식을 회상하고 접선 정의에 적용하면 표현이 제공됩니다. 공액을 취하고 1 더하기 또는 1 빼기 코사인을 곱하여 단순화하면 접선 반각 항등식의 두 가지 동등한 형태가 생성됩니다.이러한 정체성은 방향 변화를 분석하고 스케이트 공원 역학에서 곡선 운동을 모델링하는 데 도움이 됩니다.

삼각함수 항등식 ii

Trigonometric Identities II

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삼각함수 항등식 II

&#48176;&#44033; &#48143; &#48152;&#44033; &#49340;&#44033;&#54632;&#49688; &#54637;&#46321;&#49885;&#51008; &#44592;&#48376; &#54633;&#44284; ...

Education

교육

JoVE Core

Math Fundamentals

수학 기초

Unit

단위

Trigonometry

삼각법

KEY TERMS AND CONCEPTS

주요 용어 및 개념

삼각 항등 II

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

radian and degree measure

Degrees to Radians in Circle Measure

Degrees and radians are the two main units used to measure angles. They both describe rotation around a fixed point. A full turn equals 360 degrees or 2π radians, depending on the unit used. Degrees divide a circle into 360 equal parts. This makes them easy to use in everyday math. They are common in navigation, design, and basic geometry. Radians come from the geometry of a circle. One radian is the angle made when the arc length matches the radius of the circle. Since a circle’s...

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Unit Circle Trig: Signs and Symmetry

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Trigonometric Wave Graphs and Asymptotes

Trigonometric functions create repeating waves and curved graphs with clear symmetry. Their shapes come from the unit circle, where a point moves counterclockwise and its vertical and horizontal positions form the sine and cosine values. Sine and cosine graphs are smooth waves with the same period, or repeat length. Their ranges stay bounded between -1 and 1. Sine, cosine, and the other trig functions all show regular patterns, but each one has its own graph shape. Tangent is the ratio of...

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Trig Ratios for Right Triangle Sides

Right triangle trigonometry uses sine, cosine, and tangent to connect an acute angle with the sides of the triangle. These trig ratios describe how side lengths relate to one another in right-angled geometry. They give a clear way to measure the links between angles and sides. Sine compares the side opposite an angle to the hypotenuse. Cosine compares the side adjacent to the angle to the hypotenuse. Tangent compares the opposite side to the adjacent side. Each ratio depends on the angle, not...

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Measuring rocket height with trigonometry uses a fixed ground point and a changing angle of elevation. An observer stands a known horizontal distance from the launch site and watches the object rise. The angle between the ground and the line of sight gives the key information for finding height. The tangent function is the main tool for this calculation. Tangent is the ratio of the vertical side to the adjacent side in a right triangle. When the ground distance is fixed, the object’s height...

the law of sines

Solving Oblique Triangles with Sines

Solving oblique triangles uses the Law of Sines. An oblique triangle is a triangle with no right angle. These triangles need a different trigonometric method because the Pythagorean theorem works only for right triangles. The Law of Sines links each side of a triangle to the sine of its opposite angle. In triangle ABC, side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. The ratios a/sin A, b/sin B, and c/sin C are constant. This relationship helps find...

the law of cosines

Using Cosines to Find Triangle Sides

The Law of Cosines helps find unknown sides and angles in any triangle. It is a key trig formula for non-right triangles, where the Pythagorean Theorem does not give a direct answer. The law links side lengths to the cosine of an angle, so students can solve triangles with missing measurements. For a triangle with sides a, b, and c and opposite angles A, B, and C, the formula is written as a squared equals b squared plus c squared minus 2bc cosine A. Similar equations can be used for the other...

trigonometric identities i

Unit Circle Trig Identities and Sum Formulas

Trigonometric identities link trigonometric functions and stay true for every angle in their domains. A key example is the Pythagorean identity, which comes directly from the geometry of the unit circle. For an angle θ, a point on the unit circle has coordinates (cos θ, sin θ). Because the radius is 1, the Pythagorean Theorem gives the basic identity. That same identity can be rearranged to create another useful form. If it is divided by cos²θ, assuming cos θ is not 0, it leads to a second...

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...

trigonometric identities iii

Cofunction Identities and Complementary Angles

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Inverse Trig Functions and Angle Finding

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Trigonometric equations use one or more trigonometric functions, and they often appear in mathematical modeling. Some are identities, which are true for every value of the variable. Others are conditional equations, which are true only for specific values. Solving these equations usually combines algebra with the basic properties of trigonometric functions. Some equations look like standard algebra problems, so factoring can work well. For example, a quadratic form in sin x can be factored...

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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