S. D. V. Stephens

notes / Integrated Mathematics / chapter 15: Trigonometry

15.5 Product-to-Sum, Sum-to-Product and Problem Solving

Adding and subtracting the angle addition identities turns products of sines and cosines into sums, and sums back into products. The second half of the page uses all the identities together on harder problems, and the page ends with a summary of the identities.

Product-to-sum identities

Problem 15.5.1. (a) Express in terms of and .

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and That is,

(b) Find a similar identity for .

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And

(c) Find a identity, using too.

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From the computation of in (b):

Problem 15.5.2. Prove using our new identities.

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NO. Let’s try something else:

so

Sum-to-product identities

Problem 15.5.3. (a) Use one of the product-to-sum identities above to show that .

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(b) Find a similar identity for .

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, so

(c) The same for sines:

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Problem 15.5.4. Compute .

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Exercises: products and sums

Exercise 15.5.5. Show that for all such that .

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Exercise 15.5.6. Find the acute angle such that .

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Exercise 15.5.7. , which is like . Why? Just different variables?

Exercise 15.5.8. Express the following without using trig functions:

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First rearrange the top:

Wait, it is more simple if I do more basic grouping/pairing!

Exercise 15.5.9. Prove that .

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Exercise 15.5.10. Find the smallest positive integer solution to

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and with , so this is . Likewise So

Problem solving with identities

Problem 15.5.11. Determine if and are acute angles such that and .

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Square both and add: , so , i.e. .

Problem 15.5.12. Find the sum of the roots of that are between and radians.

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With : , so and are the solutions: and and are solutions, and , which gives and : is largest since and we want .

Problem 15.5.13. Let . Find the smallest positive integer such that is an integer.

Problem 15.5.14. Let such that , , and . Find .

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so take , , , : so and .

Problem 15.5.15. Evaluate

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Idea 1. Pair the outside terms: , which brings in .

Idea 2 (seems better). Pair neighbours (angles in degrees): so the cancels from top and bottom and we are left with Pairing again and again (pairs of pairs, and so on) and tracking how far apart the paired angles are gets stuck: , so after two rounds there are 11 terms and one has to be isolated.

Back to Idea 1: and the next pairs give , , …, always with the same in front. So

Problem 15.5.16. Given any 7 real numbers, prove that there are two of them, and , such that

Problem 15.5.17. Evaluate .

Summary of identities

Basic identities

Angle sum and difference identities

Double angle and half angle identities

Product-to-sum and sum-to-product identities