Question 7.5.6: Finding Function Values Using Identities Evaluate each expre...
Finding Function Values Using Identities
Evaluate each expression without using a calculator.
(a) \cos (\arctan \sqrt{3} + \arcsin \frac{1}{3} ) (b) \tan (2 \arcsin \frac{2}{5})(a) Let A = \arctan \sqrt{3} and B = \arcsin\frac{1}{3} . Therefore, \tan A = \sqrt{3} and \sin B = \frac{1}{3} . Sketch both A and B in quadrant I, as shown in Figure 25, and use the Pythagorean theorem to find the unknown side in each triangle.
\cos (\arctan\sqrt{3} +\arcsin\frac{1}{3} ) Given expression
= \cos(A + B) Let A = \arctan\sqrt{3} and B = \arcsin \frac{1}{3}.
= \cos A \cos B – \sin A \sin B Cosine sum identity
= \frac{1}{2} •\frac{2 \sqrt{2}}{3} – \frac{\sqrt{3}}{2} • \frac{1}{3} Substitute values using Figure 25.
= \frac{2\sqrt{2} – \sqrt{3}}{ 6} Multiply and write as a single fraction.
(b) Let B = \arcsin\frac{2}{5} , so that \sin B = \frac{2}{5}. Sketch angle B in quadrant I, and use the Pythagorean theorem to find the length of the third side of the triangle.
See Figure 26.
\tan (2 \arcsin \frac{2}{5}) Given expression
=\frac{ 2 (\frac{ 2}{\sqrt{21}} )}{ 1 – (\frac{ 2}{\sqrt{21}})²} Use \tan 2B = \frac{2 \tan B}{1 – \tan² B} with
\tan B = \frac{2}{\sqrt{21}} from Figure 26.
= \frac{\frac{4}{\sqrt{21} } }{1-\frac{4}{21} } Multiply and apply the exponent.
=\frac{\frac{4}{\sqrt{21} } • \frac{\sqrt{21} }{\sqrt{21} } }{\frac{17}{21} } Rationalize in the numerator.
Subtract in the denominator.
= \frac{\frac{4\sqrt{21}}{21 } }{\frac{17}{21} } Multiply in the numerator.
= \frac{4\sqrt{21}}{17} Divide; \frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} ÷ \frac{c}{d} = \frac{a}{b} • \frac{d}{c} .
