18.3: Exercises
- Page ID
- 49068
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Find the trigonometric function values.
- \(\sin\left(\dfrac{5\pi}{12}\right)\)
- \(\cos\left(\dfrac{5\pi}{12}\right)\)
- \(\tan\left(\dfrac{\pi}{12}\right)\)
- \(\sin\left(\dfrac{7\pi}{12}\right)\)
- \(\cos\left(\dfrac{11\pi}{12}\right)\)
- \(\sin\left(\dfrac{2\pi}{3}\right)\)
- \(\sin\left(\dfrac{5\pi}{6}\right)\)
- \(\cos\left(\dfrac{3\pi}{4}\right)\)
- \(\tan\left(\dfrac{13\pi}{12}\right)\)
- \(\cos\left(-\dfrac{\pi}{12}\right)\)
- \(\sin\left(\dfrac{11\pi}{12}\right)\)
- \(\sin\left(\dfrac{29\pi}{12}\right)\)
- Answer
-
- \(\dfrac{\sqrt{2}+\sqrt{6}}{4}\)
- \(\dfrac{\sqrt{6}-\sqrt{2}}{4}\)
- \(2-\sqrt{3}\)
- \(\dfrac{\sqrt{2}+\sqrt{6}}{4}\)
- \(\dfrac{-(\sqrt{2}+\sqrt{6})}{4}\)
- \(\dfrac{\sqrt{3}}{2}\)
- \(\dfrac{1}{2}\)
- \(\dfrac{-\sqrt{2}}{2}\)
- \(2-\sqrt{3}\)
- \(\dfrac{\sqrt{2}+\sqrt{6}}{4}\)
- \(\dfrac{\sqrt{6}-\sqrt{2}}{4}\)
- \(\dfrac{\sqrt{2}+\sqrt{6}}{4}\)
Simplify the function \(f\) using the addition and subtraction formulas.
- \(f(x)=\sin\left(x+\dfrac{\pi}{2}\right)\)
- \(f(x)=\cos\left(x-\dfrac{\pi}{4}\right)\)
- \(f(x)=\tan\left(\pi-x\right)\)
- \(f(x)=\sin\left(\dfrac{\pi}{6}-x\right)\)
- \(f(x)=\cos\left(x+\dfrac{11\pi}{12}\right)\)
- \(f(x)=\cos\left(\dfrac{2\pi}{3}-x\right)\)
- Answer
-
- \(\sin \left(x+\dfrac{\pi}{2}\right)=\cos (x)\)
- \(\cos \left(x-\dfrac{\pi}{4}\right)=\dfrac{\sqrt{2}}{2}(\sin (x)+\cos (x))\)
- \(\tan (\pi-x)=-\tan (x)\)
- \(\sin \left(\dfrac{\pi}{6}-x\right)=\dfrac{1}{2} \cos (x)-\dfrac{\sqrt{3}}{2} \sin (x)\)
- \(\cos \left(x+\dfrac{11 \pi}{12}\right)=\left(\dfrac{-(\sqrt{2}+\sqrt{6})}{4}\right) \cdot \cos (x)-\left(\dfrac{\sqrt{6}-\sqrt{2}}{4}\right) \cdot \sin (x)\)
- \(\cos \left(\dfrac{2 \pi}{3}-x\right)=\dfrac{-1}{2} \cdot \cos (x)+\dfrac{\sqrt{3}}{2} \cdot \sin (x)\)
Find the exact values of the trigonometric functions applied to the given angles by using the half-angle formulas.
- \(\cos\left(\dfrac{\pi}{12}\right)\)
- \(\tan\left(\dfrac{\pi}{8}\right)\)
- \(\sin\left(\dfrac{\pi}{24}\right)\)
- \(\cos\left(\dfrac{\pi}{24}\right)\)
- \(\sin\left(\dfrac{9\pi}{8}\right)\)
- \(\tan\left(\dfrac{3\pi}{8}\right)\)
- \(\sin\left(\dfrac{-\pi}{8}\right)\)
- \(\sin\left(\dfrac{13\pi}{24}\right)\)
- Answer
-
- \(\dfrac{\sqrt{2+\sqrt{3}}}{2}=\dfrac{\sqrt{2}+\sqrt{6}}{4}\)
- \(\sqrt{2}-1\)
- \(\dfrac{\sqrt{8-2 \sqrt{2}-2 \sqrt{6}}}{4}\)
- \(\dfrac{\sqrt{8+2 \sqrt{2}+2 \sqrt{6}}}{4}\)
- \(-\dfrac{\sqrt{2-\sqrt{2}}}{2}\)
- \(1+\sqrt{2}\)
- \(-\dfrac{\sqrt{2-\sqrt{2}}}{2}\)
- \(\dfrac{\sqrt{2+\sqrt{2+\sqrt{3}}}}{2}\)
Find the exact values of the trigonometric functions of \(\dfrac \alpha 2\) and of \(2\alpha\) by using the half-angle and double angle formulas.
- \(\sin(\alpha)=\dfrac{4}{5}\), and \(\alpha\) in quadrant I
- \(\cos(\alpha)=\dfrac{7}{13}\), and \(\alpha\) in quadrant IV
- \(\sin(\alpha)=\dfrac{-3}{5}\), and \(\alpha\) in quadrant III
- \(\tan(\alpha)=\dfrac{4}{3}\), and \(\alpha\) in quadrant III
- \(\tan(\alpha)=\dfrac{-5}{12}\), and \(\alpha\) in quadrant II
- \(\cos(\alpha)=\dfrac{-2}{3}\), and \(\alpha\) in quadrant II
- Answer
-
- \(\sin \left(\dfrac{\alpha}{2}\right)=\dfrac{\sqrt{5}}{5}, \cos \left(\dfrac{\alpha}{2}\right)=\dfrac{2 \sqrt{5}}{5}, \tan \left(\dfrac{\alpha}{2}\right)=\dfrac{1}{2}, \sin (2 \alpha)={\dfrac{24}{25}}, \cos (2 \alpha)=\dfrac{-7}{25}, \tan (2 \alpha)=\dfrac{-24}{7}\)
- \(\sin \left(\dfrac{\alpha}{2}\right)=\dfrac{\sqrt{39}}{13}, \cos \left(\dfrac{\alpha}{2}\right)=\dfrac{-\sqrt{130}}{13}, \tan \left(\dfrac{\alpha}{2}\right)=\dfrac{-\sqrt{30}}{10}, \sin (2 \alpha)=\dfrac{-28 \sqrt{30}}{169}, \cos (2 \alpha)=\dfrac{-71}{169}, \tan (2 \alpha)=\dfrac{28 \sqrt{30}}{71} \)
- \(\sin \left(\dfrac{\alpha}{2}\right)=\dfrac{3 \sqrt{10}}{10}, \cos \left(\dfrac{\alpha}{2}\right)=\dfrac{-\sqrt{10}}{10}, \tan \left(\dfrac{\alpha}{2}\right)=-3, \sin (2 \alpha)=\dfrac{24}{25}, \cos (2 \alpha)=\dfrac{7}{25}, \tan (2 \alpha)=\dfrac{24}{7}\)
- \(\sin \left(\dfrac{\alpha}{2}\right)=\dfrac{2 \sqrt{5}}{5}, \cos \left(\dfrac{\alpha}{2}\right)=\dfrac{-\sqrt{5}}{5}, \tan \left(\dfrac{\alpha}{2}\right)=-2, \sin (2 \alpha)=\dfrac{24}{25}, \cos (2 \alpha)=\dfrac{-7}{25}, \tan (2 \alpha)=\dfrac{-24}{7}\)
- \(\sin \left(\dfrac{\alpha}{2}\right)=\dfrac{5 \sqrt{26}}{26}, \cos \left(\dfrac{\alpha}{2}\right)=\dfrac{\sqrt{26}}{26}, \tan \left(\dfrac{\alpha}{2}\right)=5, \sin (2 \alpha)=\dfrac{-120}{169}, \cos (2 \alpha)=\dfrac{119}{169}, \tan (2 \alpha)=\dfrac{-120}{119}\)
- \(\sin \left(\dfrac{\alpha}{2}\right)=\dfrac{\sqrt{30}}{6}, \cos \left(\dfrac{\alpha}{2}\right)=\dfrac{\sqrt{6}}{6}, \tan \left(\dfrac{\alpha}{2}\right)=\sqrt{5}, \sin (2 \alpha)=\dfrac{-4 \sqrt{5}}{9}, \cos (2 \alpha)=\dfrac{-1}{9}, \tan (2 \alpha)=4 \sqrt{5}\)