How to Use Product to Sum and Sum to Product Formulas in Trig (Precalculus - Trigonometry 29)

Summary

This video covers the derivation and application of product-to-sum and sum-to-product trigonometric formulas. The instructor demonstrates how to convert products of sine and cosine functions into sums or differences, and vice-versa, using various examples. The video also shows how these formulas can be used to simplify expressions and solve trigonometric equations, emphasizing the importance of these transformations in calculus and other advanced mathematics topics.

Highlights

Applying Product-to-Sum Formulas to Express as a Sum (Sine times Cosine)00:17:39

This example shows how to express sin(4θ)cos(6θ) as a sum using the sin(A)cos(B) formula. The result is 1/2[sin(10θ) - sin(2θ)], emphasizing the use of odd/even function properties for simplification.

Applying Product-to-Sum Formulas to Express as a Sum (Cosine times Cosine)00:21:34

The final product-to-sum example expresses cos(3θ)cos(4θ) as a sum. The result is 1/2[cos(θ) + cos(7θ)], demonstrating the even property of cosine to handle negative angles.

Introduction to Sum-to-Product Formulas00:24:55

Four new sum-to-product formulas are introduced: sin(A) + sin(B), sin(A) - sin(B), cos(A) + cos(B), and cos(A) - cos(B). The instructor details how these formulas convert sums of trigonometric functions into products.

Applying Sum-to-Product Formulas (Sines and Cosines)00:26:30

The first sum-to-product example converts cos(2θ) + cos(4θ) into 2cos(3θ)cos(θ). Subsequent examples cover cos(θ/2) - cos(3θ/2), which simplifies to 2sin(θ)sin(θ/2), and sin(θ/2) - sin(3θ/2), which simplifies to -2sin(θ/2)cos(θ).

Solving Trigonometric Equations Using Sum-to-Product Formulas00:36:48

The video concludes by demonstrating how sum-to-product formulas can simplify solving trigonometric equations. The equation sin(2θ) + sin(4θ) = 0 is transformed into 2sin(3θ)cos(θ) = 0, allowing for the use of the zero product property. Solutions are found for both sin(3θ) = 0 and cos(θ) = 0, with careful consideration of the period and substitutions for the 3θ term.

Applying Product-to-Sum Formulas to Evaluate Expressions (Cosine times Cosine)00:13:47

The second example applies the cos(A)cos(B) formula to evaluate cos(285°)cos(195°). The angles are simplified to 90° and 120° (by subtracting 360° from 480°), leading to an exact value of -1/4.

Introduction to Product-to-Sum Formulas and Proof00:00:29

The video introduces product-to-sum and sum-to-product formulas, explaining their utility in converting between products and sums of sine and cosine functions. The first product-to-sum formula is derived using the sum and difference formulas for cosine: cos(A)cos(B) = 1/2[cos(A-B) + cos(A+B)].

Deriving Additional Product-to-Sum Formulas00:03:36

The derivation continues with sin(A)sin(B) = 1/2[cos(A-B) - cos(A+B)] by subtracting the cosine sum and difference formulas. The last product-to-sum formula, sin(A)cos(B) = 1/2[sin(A+B) + sin(A-B)], is derived by adding the sine sum and difference formulas.

Applying Product-to-Sum Formulas to Evaluate Expressions (Sine times Sine)00:09:16

The first example demonstrates using the sin(A)sin(B) formula to find the exact value of sin(285°)sin(75°). The angles are converted to 210° and 360°, which are on the unit circle, simplifying the calculation to -1/4(-√3 - 1).