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Mathematics LibreTexts

9.3E: Double-Angle, Half-Angle, and Reduction Formulas (Exercises)

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    56112
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    For the following exercises, find the exact value.

    20. Find \(\sin (2 \theta), \cos (2 \theta),\) and \(\tan (2 \theta)\) given \(\cos \theta=-\frac{1}{3}\) and \(\theta\) is in the interval \(\left[\frac{\pi}{2}, \pi\right]\).

    21. Find \(\sin (2 \theta), \cos (2 \theta),\) and \(\tan (2 \theta)\) given sec \(\theta=-\frac{5}{3}\) and \(\theta\) is in the interval \(\left[\frac{\pi}{2}, \pi\right]\).

    22. \(\sin \left(\frac{7 \pi}{8}\right)\)

    23. \(\sec \left(\frac{3 \pi}{8}\right)\)

    For the following exercises, use Figure 1 to find the desired quantities.

    Image of a right triangle. The base is 24, the height is unknown, and the hypotenuse is 25. The angle opposite the base is labeled alpha, and the remaining acute angle is labeled beta.

    Figure 1

    24. \(\sin (2 \beta), \cos (2 \beta), \tan (2 \beta), \sin (2 \alpha), \cos (2 \alpha),\) and \(\tan (2 \alpha)\)

    25. \(\sin \left(\frac{\beta}{2}\right), \cos \left(\frac{\beta}{2}\right), \tan \left(\frac{\beta}{2}\right), \sin \left(\frac{\alpha}{2}\right), \cos \left(\frac{\alpha}{2}\right),\) and \(\tan \left(\frac{\alpha}{2}\right)\)

    For the following exercises, prove the identity.

    26. \(\frac{2 \cos (2 x)}{\sin (2 x)}=\cot x-\tan x\)

    27. \(\cot x \cos (2 x)=-\sin (2 x)+\cot x\)

    For the following exercises, rewrite the expression with no powers.

    28. \(\cos ^{2} x \sin ^{4}(2 x)\)

    29. \(\tan ^{2} x \sin ^{3} x\)

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