We will learn to express trigonometric function of tan 2A in terms of A or tan 2A in terms of tan A. We know if A is a given angle then 2A is known as multiple angles.
How to proof the formula of tan 2A is equals \(\frac{2 tan A}{1 - tan^{2} A}\)?
We know that for two real numbers or angles A and B,
tan (A + B) = \(\frac{tan A + tan B}{1 - tan A tan B }\)
Now, putting B = A on both sides of the above formula we get,
tan (A + A) = \(\frac{tan A + tan A}{1 - tan A tan A }\)
⇒ tan 2A = \(\frac{2 tan A}{1 - tan^{2} A}\)
Note: (i) In the above formula we should note that the angle on the R.H.S. is half of the angle on L.H.S. Therefore, tan 60° = \(\frac{2 tan 30°}{1 - tan^{2} 30°}\).
(ii) The above formula is also known as double angle formulae for tan 2A.
Now, we will apply the formula of multiple angle of tan 2A in terms of A or tan 2A in terms of tan A to solve the below problem.
1. Express tan 4A in terms of tan A
Solution:
tan 4a
= tan (2 ∙ 2A)
= \(\frac{2 tan 2A}{1 - tan^{2} (2A)}\), [Since we know \(\frac{2 tan A}{1 - tan^{2} A}\)]
= \(\frac{2 \cdot \frac{2 tan A}{1 - tan^{2} A}}{1 - (\frac{2 tan A}{1 - tan^{2} A})^{2}}\)
= \(\frac{4 tan A (1 - tan^{2} A)}{(1 - tan^{2} A)^{2} - 4 tan^{2} A}\)
= \(\frac{4 tan A (1 - tan^{2} A)}{1 - 6 tan^{2} A + 4 tan^{4}}\)
11 and 12 Grade Math
From tan 2A in Terms of A to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Dec 06, 23 01:23 AM
Dec 04, 23 02:14 PM
Dec 04, 23 01:50 PM
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.