9.5.2: Homework

Math 375: Pre-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. How can the Product-to-Sum identities be verified?
    2. What is the Product-to-Sum identity used to rewrite a product of two cosines, \(\cos(\alpha)\cos(\beta)\), as a sum?
    3. What is the Sum-to-Product identity used to rewrite a difference of two sines, \(\sin(\alpha) - \sin(\beta)\), as a product?
    4. According to the text, is it beneficial to memorize the Sum-to-Product and Product-to-Sum identities?
    5. What two different methods are mentioned in Example 2 for finding the exact value of \(\cos(15^\circ) - \cos(75^\circ)\)?
    6. In Example 1, what previously learned identity is used to rewrite \(\cos(-4\theta)\) as \(\cos(4\theta)\)?
    7. When might you need to use a Sum-to-Product or Product-to-Sum identity?

    Homework

    Basic Skills

    For the following exercises, write the given product as a sum.

    1. \(\cos(3\theta)\cos(5\theta)\)

    2. \(\sin(2\theta)\sin(7\theta)\)

    3. \(\sin(9\theta)\cos(\theta)\)

    4. \(\cos(2\theta) \cos(6\theta)\)

    5. \(\sin(3\theta) \sin(2\theta)\)

    6. \(\cos(\theta) \sin(3\theta)\)

    For the following exercises, write the given sum as a product.

    1. \(\cos(3\theta) + \cos(5\theta)\)

    2. \(\sin(2\theta) - \sin(7\theta)\)

    3. \(\cos(5\theta) - \cos(6\theta)\)

    4. \(\sin(9\theta) - \sin(-\theta)\)

    5. \(\sin(\theta) + \cos(\theta)\)

    6. \(\cos(\theta) - \sin(\theta)\)

    For the following exercises, find the exact values of the trigonometric expressions.

    1. \( 2 \cos\left( \dfrac{\pi}{5} \right) \cos\left( \dfrac{\pi}{3} \right)\)

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

    3. \( \cos\left( \dfrac{5\pi}{12} \right) + \cos\left( \dfrac{7\pi}{12} \right)\)

    Proofs

    1. Verify the Product to Sum Identities.

    2. Verify the Sum to Product Identities.

    For the following exercises, prove the identity.

    1. \(\dfrac{\sin(x) + \sin(y)}{\cos(x) + \cos(y)} = \tan\left(\dfrac{x + y}{2}\right)\)

    2. \(\dfrac{\sin(x) - \sin(y)}{\cos(x) - \cos(y)} = -\cot\left(\dfrac{x + y}{2}\right)\)

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

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

    5. \(\dfrac{\cos(x) - \cos(3x)}{\sin(3x) - \sin(x)} = \tan(2x)\)

    6. \(-\tan(4x) = \dfrac{\cos(3x) - \cos(5x)}{\sin(3x) - \sin(5x)} \)

    7. \(32\sin^{2}(\theta) \cos^{4}(\theta) = 2 + \cos(2\theta) - 2\cos(4\theta) - \cos(6\theta)\)

    8. \(32\sin^{4}(\theta) \cos^{2}(\theta) = 2 - \cos(2\theta) - 2\cos(4\theta) + \cos(6\theta)\)

    This page titled 9.5.2: Homework was last modified on Tue, 08 Jul 2025 17:45:52 GMT and is shared under a not declared license and was authored, remixed, and/or curated by Roy Simpson.