User:Gracenotes/TrigPages

Wikipedia page + stream of consciousness = !!!!!

Double-angle formulas provide a way to express a trigonometric function with a double frequency, such as $\sin 2x$, in terms of functions with a frequency of one, such as $\sin x$.

Double-angle formula

This text is copied directly from List of trigonometric identities, and all of it will go to the section "Double-angle formulas for the trigonometric functions."

These can be shown by substituting x = y in the addition theorems, and using the Pythagorean formula. Or use de Moivre's formula with n = 2.

$\sin(2x)=2\sin(x)\cos(x)\ ={\frac {2\tan(x)}{1+\tan ^{2}(x)}}$

$\cos(2x)=\cos ^{2}(x)-\sin ^{2}(x)=2\cos ^{2}(x)-1=1-2\sin ^{2}(x)\ ={\frac {1-\tan ^{2}(x)}{1+\tan ^{2}(x)}}$

$\tan(2x)={\frac {2\tan(x)}{1-\tan ^{2}(x)}}$

$\cot(2x)={\frac {\cot(x)-\tan(x)}{2}}$

The double-angle formula can also be used to find Pythagorean triples. If (a, b, c) are the lengths of the sides of a right triangle, then (a² − b², 2ab, c²) also form a right triangle, where angle B is the angle being doubled. If a² − b² is negative, take its opposite and use the supplement of 2B in place of 2B.

Properties of double-angle trigonometric formulas

Note: use point P and a diagram. Create diagram.

The sine function can be defined by the point rotating in a circular pattern.

If $f(\theta )$ is a trigonometric function, then $f(2\theta )$ is the double angle function. Suppose there is a point $(x,y)$ on the unit circle: that is, $(x,y)$ is consistently one unit away from the origin. The sine function, $\sin x$, can be defined as the $y$ coordinate in $(x,y)$ as the point traces the unit circle starting counterclockwise from $(0,1)$; the tracing is completed at $2\pi$ radians, or $360^{\circ }$. When the function is not $\sin \theta$ but $\sin 2\theta$, the circumnavigation of the unit circle occurs twice as quickly.

Procedure for finding double-angle formulas

Diagram to be added, as well as text delineating process.

Suggested content: angle addition, trig relations to each other, and de Moivre formula.

Double-angle formulas for the trigonometric functions

This

Sine function

$\sin(2x)=\sin(x+x)=\sin(x)\cos(x)+\sin(x)\cos(x)=2\sin(x)\cos(x)\,\!$

Cosine function

$\cos(2x)=\cos(x+x)=\cos(x)\cos(x)-\sin(x)\sin(x)=\cos ^{2}(x)-\sin ^{2}(x)\,\!$

Tangent function

$\tan(2x)=\tan(x+x)={\frac {\tan(x)+\tan(x)}{1-\tan(x)\tan(x)}}={\frac {2\tan(x)}{1-\tan ^{2}(x)}}\,\!$

$\tan(2x)={\frac {\sin(2x)}{\cos(2x)}}={\frac {2\sin(x)\cos(x)}{\cos ^{2}(x)-\sin ^{2}(x)}}\,\!$

${\frac {2\sin(x)\cos(x)\cdot {\frac {1}{\cos ^{2}(x)}}}{\cos ^{2}(x)-\sin ^{2}(x)\cdot {\frac {1}{\cos ^{2}(x)}}}}={\frac {2\tan(x)}{1-\tan ^{2}(x)}}\,\!$

Applications

Kinematics

In kinematics, a branch of classical mechanics, it is possible to calculate how far an object such as a ball will travel if launched at an angle at a certain velocity. The horizontal distance that it travels is called range.

Given uniform gravity and no wind or drag, an object launched at angle of elevation $\theta$ with initial speed $v$ with the acceleration of gravity g will have a range (denoted $R$) of

$R={\frac {2v^{2}\cos(\theta )\sin(\theta )}{g}}$

The standard derivation of the range formula always leads to the equation above. However, because

$2\cos(\theta )\sin(\theta )=\sin(2\theta )$

The range equation can be simplified to become

$R={\frac {v^{2}\cdot 2\cos(\theta )\sin(\theta )}{g}}={\frac {v^{2}\sin(2\theta )}{g}}$

The maximum of both equations occurs at $45^{\circ }$ or ${\tfrac {\pi }{4}}$ radians; however, this is more evident in the second version of the equation, since there is only one occurence of theta.

Calculus

There is no way to take the indefinite integral

$\int \sin ^{2}(x)dx$

using integration by substitution (u-substitution), integration by parts, or other common integration methods. However,

$\cos(2x)=1-2\sin ^{2}(x)$

$\sin ^{2}(x)={\frac {1-\cos(2x)}{2}}$

$\int \sin ^{2}(x)\ dx=\int {\frac {1}{2}}-{\frac {\cos(2x)}{2}}\ dx$

Evaluated, the latter integral is

${\frac {x}{2}}-{\frac {\sin(2x)}{4}}\ +C$

And so using double-angle formulas, an otherwise complicated indefinite integral becomes easier to evaluate.