6.4.2: Homework

Math 373: Trigonometry for Calculus · LibreTexts

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    From LibreTexts, under not declared, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    Reading Questions

    1. What is the primary purpose of the Power Reduction Identities?
    2. From which identities are the Power Reduction Identities derived?
    3. What is the Power Reduction Identity for \(\sin^2(\theta)\)?
    4. How are the Half-Angle Identities derived from the Power Reduction Identities?
    5. What is the Half-Angle Identity for \(\cos(\frac{\theta}{2})\)?
    6. What does the \(\pm\) symbol in the Half-Angle Identities signify?
    7. How do you determine whether to use the positive or negative root when applying a Half-Angle Identity?
    8. If \(\theta\) is in Quadrant IV, in which quadrant must \(\frac{\theta}{2}\) lie?
    9. How can you use the Half-Angle Identities to help graph a function like \(y = 3\sin^2(\frac{x}{2})\)?
    10. To find the exact value of \(\sin(15^\circ)\) using a Half-Angle Identity, what angle would you use for \(\theta\)?

    Homework

    Basic Skills

    For the following exercises, rewrite the expression without exponents.

    1. \(\cos ^{2} (5x)\)

    2. \(\cos ^{2} (6x)\)

    3. \(\sin ^{4} (8x)\)

    4. \(\sin ^{4} \left(3x\right)\)

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

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

    For the following exercises, use the Half-Angle Identities to find the exact value.

    1. \(\cos(75^{\circ})\)

    2. \(\sin(105^{\circ})\)

    3. \(\cos(67.5^{\circ})\)

    4. \(\sin(157.5^{\circ})\)

    5. \(\tan(112.5^{\circ})\)

    6. \(\cos\left( \dfrac{7\pi}{12} \right)\)

    7. \(\sin\left( \dfrac{\pi}{12} \right)\)

    8. \( \sin\left( \dfrac{\pi}{24} \right) \)

    9. \( \cos\left( \dfrac{\pi}{24} \right) \)

    10. \( \sin\left( \dfrac{9\pi}{8} \right) \)

    11. \( \tan\left( \dfrac{3\pi}{8} \right) \)

    12. \(\cos \left( \dfrac{\pi}{8} \right)\)

    13. \(\sin \left( \dfrac{5\pi}{8} \right)\)

    14. \(\tan \left( \dfrac{7\pi}{8} \right)\)

    15. \( \sin\left( -\dfrac{\pi}{8} \right) \)

    16. \( \sin\left( \dfrac{13\pi}{24} \right) \)

    For the following exercises, use the given information about \(\theta\) to find the exact values of

    • \(\sin\left(\dfrac{\theta}{2}\right)\)

    • \(\cos\left(\dfrac{\theta}{2}\right)\)

    • \(\tan\left(\dfrac{\theta}{2}\right)\)

    1. \(\sin(\theta) = -\dfrac{7}{25}\) where \(\dfrac{3\pi}{2} < \theta < 2\pi\)

    2. \(\cos(\theta) = \dfrac{28}{53}\) where \(0 < \theta < \dfrac{\pi}{2}\)

    3. \(\tan(\theta) = \dfrac{12}{5}\) where \(\pi < \theta < \dfrac{3\pi}{2}\)

    4. \(\csc(\theta) = 4\) where \(\dfrac{\pi}{2} < \theta < \pi\)

    5. \(\cos(\theta) = \dfrac{3}{5}\) where \(0 < \theta < \dfrac{\pi}{2}\)

    6. \(\sin(\theta) = -\dfrac{4}{5}\) where \(\pi < \theta < \dfrac{3\pi}{2}\)

    7. \(\cos(\theta) = \dfrac{12}{13}\) where \(\dfrac{3\pi}{2} < \theta < 2\pi\)

    8. \(\sin(\theta) = \dfrac{5}{13}\) where \(\dfrac{\pi}{2} < \theta < \pi\)

    9. \(\sec(\theta) = \sqrt{5}\) where \(\dfrac{3\pi}{2} < \theta < 2\pi\)

    10. \(\tan(\theta) = -2\) where \(\dfrac{\pi}{2} < \theta < \pi\)

    Synthesis Problems

    1. Let \(\theta\) be a Quadrant III angle with \(\cos(\theta) = -\dfrac{1}{5}\). Show that this is not enough information to determine the sign of \(\sin\left(\dfrac{\theta}{2}\right)\) by first assuming \(3\pi < \theta < \dfrac{7\pi}{2}\) and then assuming \(\pi < \theta < \dfrac{3\pi}{2}\) and computing \(\sin\left(\dfrac{\theta}{2}\right)\) in both cases.

    For the following exercises, graph two cycles of the function.

    1. \( y = 8 \sin^2\left( \dfrac{x}{2} \right) \)

    2. \( y = 2 \cos^2\left( \dfrac{x}{2} \right) \)

    3. \( y = -2 \cos^2\left( \dfrac{x}{2} \right) + 4 \)

    4. \( y = \dfrac{1}{3} \cos^2\left( \dfrac{x}{2} \right) - \dfrac{1}{6}\)

    Proofs

    For the following exercises, prove the identity.

    1. \( \tan\left( \dfrac{1}{2} \theta \right) = \csc\left( \theta \right) - \cot\left( \theta \right)\)

    2. \( \tan^2\left( \dfrac{1}{2} \theta \right) = \dfrac{\tan\left( \theta \right) - \sin\left( \theta \right)}{\tan\left( \theta \right) + \sin\left( \theta \right)} \)

    3. \( \sin^2\left( \dfrac{\theta}{2} \right) = \dfrac{\csc\left( \theta \right) - \cot\left( \theta \right)}{2\csc\left( \theta \right)} \)

    4. \( 2 \sin^2\left( \dfrac{\theta}{2} \right) = \dfrac{\sin^2\left( \theta \right)}{1 + \cos\left( \theta \right)} \)

    5. \( 2\cos^2\left( \dfrac{\theta}{2} \right) = \dfrac{\sin^2\left( \theta \right)}{1 - \cos\left( \theta \right)} \)

    6. \( \cos^2\left( \dfrac{\theta}{2} \right) = \dfrac{\tan\left( \theta \right) + \sin\left( \theta \right)}{2 \tan\left( \theta \right)} \)

    7. \( \sec^2\left( \dfrac{\theta}{2} \right) = \dfrac{2 \sec\left( \theta \right)}{\sec\left( \theta \right) + 1}\)

    8. \( \csc^2\left( \dfrac{\theta}{2} \right) = \dfrac{2\sec\left( \theta \right)}{\sec\left( \theta \right) - 1} \)

    9. \( 4 \sin^4\left( \theta \right) = 1 - 2 \cos\left( 2\theta \right) + \cos^2\left( 2\theta \right) \)

    10. \( \cos^4\left( \theta \right) = \dfrac{1}{4} + \dfrac{\cos\left( 2\theta \right)}{2} + \dfrac{\cos^2\left( 2\theta \right)}{4} \)

    Challenge Questions

    1. Without using your calculator, show that \(\dfrac{\sqrt{2 + \sqrt{3}}}{2} = \dfrac{\sqrt{6} + \sqrt{2}}{4}\)

    This page titled 6.4.2: Homework was last modified on Fri, 04 Jul 2025 22:31:39 GMT and is shared under a not declared license and was authored, remixed, and/or curated by Roy Simpson.