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 &minus; b2, 2ab, c2) also form a right triangle, where angle B is the angle being doubled. If a2 &minus; 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.

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

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

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

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.