Exploring the realm of trigonometry, this content delves into double-angle and half-angle formulas, their derivations, and applications. It covers the sine, cosine, tangent, secant, cosecant, and cotangent functions, providing insights into their use in solving equations, integrating, and differentiating in calculus, as well as their importance in physics, engineering, and geometry for analyzing wave motion and designing structures.
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Double-angle formulas are derived from the sum formulas of trigonometric functions and provide a method to express functions of double angles in terms of single angles
Formula Derivation
The double-angle formula for sine, sin(2θ) = 2sin(θ)cos(θ), is derived from the sum formula for sine
Application Example
The sine double-angle formula can be used to calculate the sine of an angle that is twice another known angle
Formula Derivation
The double-angle formula for cosine, cos(2θ), can be derived using the sum formula for cosine and has three equivalent expressions
Application Example
The cosine double-angle formula can be used to compute the cosine of a double angle from the known sine or cosine of the original angle
Formula Derivation
The double-angle formula for tangent, tan(2θ) = 2tan(θ) / (1 - tan^2(θ)), is derived from the reciprocal of the tangent sum formula
Application Example
The tangent double-angle formula simplifies finding the tangent of a double angle based on the tangent of the original angle
Secant Double-Angle Formula
The double-angle formula for secant, sec(2θ) = 1 / (cos^2(θ) - sin^2(θ)), is derived from the reciprocal of the cosine double-angle formula
Cosecant Double-Angle Formula
The double-angle formula for cosecant, csc(2θ) = 1 / (2sin(θ)cos(θ)), comes from the sine double-angle formula
Cotangent Double-Angle Formula
The double-angle formula for cotangent, cot(2θ) = (cot^2(θ) + 1) / (2cot(θ)), is derived from the reciprocal of the tangent double-angle formula
Half-angle formulas are derived from the corresponding double-angle formulas and are useful for calculating trigonometric functions of half angles
Formula Derivation
The half-angle formula for sine, sin(θ/2) = ±√((1 - cos(θ))/2), is obtained by setting θ as 2φ in the cosine double-angle formula and solving for sin(φ)
Application Example
The sine half-angle formula facilitates the calculation of the sine for half an angle when the cosine of the full angle is known
Formula Derivation
The half-angle formula for cosine, cos(θ/2) = ±√((1 + cos(θ))/2), is similarly derived
Application Example
The cosine half-angle formula is useful for determining the cosine of half an angle when the cosine of the full angle is known
Tangent Half-Angle Formula
The half-angle formula for tangent, tan(θ/2) = ±√((1 - cos(θ)) / (1 + cos(θ))), is derived from the half-angle formulas for sine and cosine
Secant Half-Angle Formula
The half-angle formula for secant, sec(θ/2) = ±2 / (1 + cos(θ)), is derived by taking the reciprocal of the cosine half-angle formula
Cosecant Half-Angle Formula
The half-angle formula for cosecant, csc(θ/2) = ±2 / (1 - cos(θ)), is derived by taking the reciprocal of the sine half-angle formula
Cotangent Half-Angle Formula
The half-angle formula for cotangent, cot(θ/2) = (1 + cos(θ)) / sin(θ), is derived from the reciprocal of the tangent half-angle formula
Double-angle and half-angle formulas are useful for simplifying trigonometric expressions and solving equations involving trigonometric functions
These formulas are instrumental in calculus for integrating and differentiating trigonometric functions
In physics, engineering, and geometry, double-angle and half-angle formulas are crucial for analyzing wave motion, oscillations, and designing structures and mechanical systems
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Double-angle formula for sine
sin(2θ) = 2sin(θ)cos(θ)
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Derivation source of double-angle formulas
Derived from sum formulas of trigonometric functions
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Example application of sine double-angle formula
sin(60°) using θ=30°: sin(60°) = 2sin(30°)cos(30°) = √3/2
