Introduction to Sum and Difference Formulas in Trigonometry (Precalculus - Trigonometry 25)

Overview

Professor Leonard introduces the sum and difference formulas in trigonometry, explaining their utility in finding exact trigonometric values for angles not on the unit circle, such as 105° or 15°. He demonstrates how to break down these angles into sums or differences of known unit circle angles (e.g., 60° + 45° or 60° - 45°) and apply the corresponding sine, cosine, and tangent formulas. The lesson also covers working backward to simplify expressions and handling negative angles and reciprocal trigonometric functions like secant.

Key takeaways

Sum and difference formulas allow exact calculation of trigonometric values for angles not on the unit circle by breaking them into sums/differences of known angles.
For sine, sin(α ± β) = sin(α)cos(β) ± cos(α)sin(β); signs are the same.
For cosine, cos(α ± β) = cos(α)cos(β) ∓ sin(α)sin(β); signs are opposite.
For tangent, tan(α ± β) = (tan(α) ± tan(β)) / (1 ∓ tan(α)tan(β)); numerator sign matches, denominator is opposite.
Even functions (like secant, cosine) allow changing the sign of the angle argument (f(-x) = f(x)), while odd functions (like sine, tangent) flip the sign of the output (f(-x) = -f(x)).
Trigonometric identities are two-way streets; expressions matching the right side can be simplified to the left side (e.g., sin(α)cos(β) - cos(α)sin(β) simplifies to sin(α - β)).

Chapters

0:00 Introduction to Sum and Difference Formulas

Sum and difference formulas allow calculation of exact trigonometric values for angles not on the unit circle.
These formulas are useful for simplifying expressions and in calculus.
They apply to sine, cosine, and tangent functions when two angles are added or subtracted.

2:15 Calculating Sine of 105 Degrees

Represent 105° as a sum of two unit circle angles: 60° + 45°.
Apply the sine sum formula: sin(α + β) = sin(α)cos(β) + cos(α)sin(β).
Substitute known values: sin(60°) = √3/2, cos(45°) = √2/2, cos(60°) = 1/2, sin(45°) = √2/2.
Simplify to get the exact value: (√6 + √2) / 4.

12:34 Calculating Sine of π/12

Represent π/12 as a difference of two unit circle angles: 4π/12 - 3π/12, which simplifies to π/3 - π/4.
Apply the sine difference formula: sin(α - β) = sin(α)cos(β) - cos(α)sin(β).
Substitute known values: sin(π/3) = √3/2, cos(π/4) = √2/2, cos(π/3) = 1/2, sin(π/4) = √2/2.
Simplify to get the exact value: (√6 - √2) / 4.

22:02 Calculating Cosine of 165 Degrees

Represent 165° as a sum of two unit circle angles: 120° + 45°.
Apply the cosine sum formula: cos(α + β) = cos(α)cos(β) - sin(α)sin(β).
Substitute known values: cos(120°) = -1/2, cos(45°) = √2/2, sin(120°) = √3/2, sin(45°) = √2/2.
Simplify to get the exact value: (-√2 - √6) / 4.

33:20 Calculating Tangent of 19π/12

Represent 19π/12 as a sum of two unit circle angles: 16π/12 + 3π/12, which simplifies to 4π/3 + π/4.
Apply the tangent sum formula: tan(α + β) = (tan(α) + tan(β)) / (1 - tan(α)tan(β)).
Substitute known values: tan(4π/3) = √3, tan(π/4) = 1.
Simplify and rationalize the denominator to get -2 - √3.

45:25 Calculating Tangent of 15 Degrees

Represent 15° as a difference of two unit circle angles: 60° - 45°.
Apply the tangent difference formula: tan(α - β) = (tan(α) - tan(β)) / (1 + tan(α)tan(β)).
Substitute known values: tan(60°) = √3, tan(45°) = 1.
Simplify and rationalize the denominator to get 2 - √3.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Professor Leonard.