Summary
Highlights
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.
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.
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.
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.
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)].
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.
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).
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).
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.