Can Trig Identities be Derived from Easier Formulas?

[hatelove]

I know you can derive the double angle formulas for sin(2a) and cos(2a) from Euler's identity, but is there any way to derive the tan(2a) in a similar manner from an easier formula? What about the addition/subtraction formulas (i.e. sin(a+b), etc.)

 

[SuperSonic4]

I know you can derive the double angle formulas for sin(2a) and cos(2a) from Euler's identity, but is there any way to derive the tan(2a) in a similar manner from an easier formula? What about the addition/subtraction formulas (i.e. sin(a+b), etc.)

Using the addition formula for tan would be the easiest: $\tan(A+B) = \frac{\tan(A)+\tan(B)}{1-\tan(A)\tan(B)}$

Alternatively you can use the fact that $\tan(ax) = \frac{\sin(ax)}{\cos(ax)}$ (where a is a constant) together with your values for sin(2a) and cos(2a).

 

[CaptainBlack]

I know you can derive the double angle formulas for sin(2a) and cos(2a) from Euler's identity, but is there any way to derive the tan(2a) in a similar manner from an easier formula? What about the addition/subtraction formulas (i.e. sin(a+b), etc.)

\[\tan(2a)=\frac{\sin(2a)}{\cos(2a)}=\frac{2\sin(a) \cos(a)}{\cos^2(a)-\sin^2(a)}\]

Now divide top and bottom by \(\cos^2(a)\)

CB

 

[chisigma]

I know you can derive the double angle formulas for sin(2a) and cos(2a) from Euler's identity, but is there any way to derive the tan(2a) in a similar manner from an easier formula? What about the addition/subtraction formulas (i.e. sin(a+b), etc.)

In...

http://mathworld.wolfram.com/TrigonometricAdditionFormulas.html

... a purely geometric way to obtain the sine of the sum of two angles is given...

Kind regards

chi sigma

 

[CellCoree]

Trig identities, such as the double angle formulas and addition/subtraction formulas, can indeed be derived from other simpler formulas. In fact, many of these identities can be derived from the basic trigonometric ratios (sine, cosine, and tangent) and the Pythagorean theorem.

For example, the double angle formulas for sine and cosine can be derived from the Pythagorean identity: sin^2(a) + cos^2(a) = 1. By substituting sin(a) = opposite/hypotenuse and cos(a) = adjacent/hypotenuse, we can obtain the double angle formulas sin(2a) = 2sin(a)cos(a) and cos(2a) = cos^2(a) - sin^2(a).

Similarly, the addition/subtraction formulas can be derived from the trigonometric ratios and the Pythagorean theorem. For instance, the formula for sin(a+b) can be derived by considering a right triangle with angle a and a+b, and using the trigonometric ratios and Pythagorean theorem to find the relationship between the opposite and adjacent sides.

While it may not always be possible to directly derive tan(2a) from a simpler formula, it can be derived from the double angle formula for tangent, which itself can be derived from the double angle formulas for sine and cosine.

In summary, while some trig identities may seem complex, they can often be derived from simpler formulas and concepts. It is important for scientists to understand the fundamental principles and relationships in trigonometry in order to derive and apply these identities in their work.