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Chapter 9: Problem 22

Simplify $$ \sin A \cos A \tan A+\frac{2 \sin A \cos ^{3} A}{\sin 2 A} $$

Short Answer

Expert verified
Question: Simplify the trigonometric expression \( \sin A \cos A (\cot A)+\frac{2 \sin A \cos ^{3} A}{2\sin A \cos A} \). Answer: \( \cos A \)

Step by step solution

01

Rewrite the expression in terms of sine and cosine

We can begin by rewriting the expression in terms of sine and cosine functions: $$ \sin A \cos A (\frac{\sin A}{\cos A})+\frac{2 \sin A \cos ^{3} A}{2\sin A \cos A} $$
02

Simplify the expression

Now we can simplify the expression further by cancelling out common terms and performing simple calculations: $$ \sin A \cos A \cdot \sin A +\frac{2 \sin A \cos^{3} A}{2 \sin A \cos A} $$
03

Cancel common terms in the fraction

In the second term, notice that we have common factors of 2, \(\sin A\), and \(\cos A\) in both the numerator and denominator, so we can cancel them: $$ \sin A \cos A \cdot \sin A +\frac{\cancel{2} \cancel{\sin A} \cos^{\cancel{2}} A}{\cancel{2} \cancel{\sin A} \cancel{\cos A}} $$ This leaves us with: $$ \sin A \cos A \cdot \sin A +\cos^{2} A $$
04

Apply the Pythagorean identity for sine and cosine

Recall that the Pythagorean identity for sine and cosine is given by: $$ \sin^{2}A + \cos^{2}A = 1 $$ We can use this identity to simplify the expression further. Since the first term is \(\sin A \cos A \cdot \sin A = \sin^{2}A \cos A\), we can rewrite the expression as: $$ \sin^{2}A \cos A +\cos^{2}A $$ Now, we can factor out \(\cos A\) from both terms: $$ \cos A (\sin^{2} A + \cos^{2}A) $$ Using the Pythagorean identity, we can replace the expression inside the parentheses with 1: $$ \cos A (1) = \cos A $$ So the original expression simplifies to: $$ \cos A $$

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Convert the following angles in radians to degrees: (a) \(\frac{\pi}{2}\) (b) \(\frac{\pi}{3}\) (c) \(\frac{4 \pi}{3}\) (d) \(1.25 \pi\) (e) \(1.25\) (f) \(9.6314(\mathrm{~g}) 3\)

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