Mr Davis 97 said:I have before me that ##\cos 2x + \sin 2x = \sqrt{2} \cos (2x - \pi / 4)##. Where does this expression on the right come from? I tried to look on the internet but I couldn't really articulate it well enough to find anything on it.
A linear combination of sin and cos involves multiplying each function by a constant and then adding them together. The resulting expression is a linear combination of the original functions.
Linear combination of sin and cos is important in mathematics and physics because it allows for the representation of complex periodic functions as a combination of simpler functions. It also has applications in signal processing, vibration analysis, and other fields.
Sin and cos are both trigonometric functions, but they have different definitions and properties. Sin represents the ratio of the opposite side to the hypotenuse in a right triangle, while cos represents the ratio of the adjacent side to the hypotenuse. In terms of their graphs, sin is an odd function while cos is an even function.
To solve a linear combination of sin and cos, you can use the trigonometric identities and properties to simplify the expression and rewrite it in terms of either sin or cos. You may also need to use algebraic techniques to solve for any unknown constants in the expression.
Some examples of linear combinations of sin and cos include the Fourier series, which represents any periodic function as a combination of sin and cos functions at different frequencies. Another example is the parametric equations of a circle, which can be expressed as a linear combination of sin and cos.