Thread locked. A related question is posted in the Precalc Homework section.Callumnc1 said:
Trig identities are mathematical equations that involve trigonometric functions and are true for all values of the variables involved. They are used to simplify expressions and solve trigonometric equations.
Alternative methods can provide different perspectives, simplify complex problems, and offer more efficient solutions. They can also help in gaining a deeper understanding of the relationships between trigonometric functions.
Euler's formula states that \( e^{ix} = \cos(x) + i\sin(x) \). By manipulating this formula, we can derive various trig identities. For example, setting \( x = \theta \) and \( x = -\theta \) and adding or subtracting the resulting equations can yield identities like the Pythagorean identity and angle sum identities.
The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. By considering the coordinates of points on the unit circle, we can derive identities such as \( \sin^2(\theta) + \cos^2(\theta) = 1 \) and the angle addition formulas. The geometric representation helps in visualizing and understanding these identities.
Common tricks include factoring and expanding expressions, using conjugates, and substituting known identities to simplify complex expressions. For instance, expressing functions in terms of sine and cosine and then using known identities can often simplify the process of finding new identities.