Double-angle formulas provide a way to express a trigonometric function with a double frequency, such as , in terms of functions with a frequency of one, such as .
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.
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[edit]
Note: use point P and a diagram. Create diagram.
If is a trigonometric function, then is the double angle function. Suppose there is a point on the unit circle: that is, is consistently one unit away from the origin. The sine function, , can be defined as the coordinate in as the point traces the unit circle starting counterclockwise from ; the tracing is completed at radians, or . When the function is not but , the circumnavigation of the unit circle occurs twice as quickly.
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 with initial speed with the acceleration of gravityg will have a range (denoted ) of
The standard derivation of the range formula always leads to the equation above. However, because
The range equation can be simplified to become
The maximum of both equations occurs at or radians; however, this is more evident in the second version of the equation, since there is only one occurence of theta.