Resolved: "Write the trigonometric expression as an algebraic expression...."

DofotheroU

Answered question

2021-09-16

Write the trigonometric expression as an algebraic expression.

\cos (\arcsin x - \arctan 2 x)

odgovoreh

Skilled2021-09-17Added 107 answers

1. Let

u = \arctan 2 x

, then

\tan u = \frac{2 x}{1} = \frac{a d j}{o p p} .

h y p = \sqrt{1 + 4 x^{2}}

3. Then

\cos u = \frac{1}{\sqrt{1 - 4 x^{2}}} = \cos (\arctan 2 x)

4. And

\sin u = \frac{2 x}{\sqrt{1 - 4 x^{2}}} = \sin (\arctan 2 x)

5. Let

v = \arccos x

, then

\cos v = x .

o p p = \sqrt{1 - x^{2}}

7. Then

\sin v = \frac{\sqrt{1 - x^{2}}}{1} = \sin (\arccos x)

\cos (\arcsin x - \arctan 2 x) - \cos (\arcsin x) \cos (\arctan 2 x) + \sin (\arcsin x) \sin (\arctan 2 x) =

= \cos (\arcsin x) \cos (\arctan 2 x) + x \sin (\arctan 2 x) = \frac{2 x^{2} + \sqrt{1 - x^{2}}}{\sqrt{1 - 4 x^{2}}}

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