Angle-Sum and Double-Angle Identities

Develop sum and difference formulas, double-angle forms.

Lesson 2 of 42 parts Practice Flashcards

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Sum and Difference Formulas

Objective: Explain sum and difference formulas, apply the rule, and justify relevant restrictions.

Sine of a sum satisfies sin(α+β)=sin α cos β+cos α sin β; cosine satisfies cos(α+β)=cos α cos β−sin α sin β. Replacing β by −β gives difference formulas. These formulas can evaluate angles such as 75° using familiar angles. Trigonometric functions do not generally distribute over addition: sin(α+β) is not sin α+sin β.

Formulas

$sin(α+β)=sin α cos β+cos α sin β$

Worked examples

Try itGiven sin α=7/25, cos α=24/25, sin β=3/5, and cos β=4/5, find sin(α+β).Show answer

Step 1: Use sin(α+β)=sin α cos β+cos α sin β. Step 2: Substitute (7/25)(4/5)+(24/25)(3/5). Step 3: Combine the fractions to obtain 4/5.

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Sum and Difference Formulas

Objective: Explain sum and difference formulas, apply the rule, and justify relevant restrictions.

Formulas

$sin(α+β)=sin α cos β+cos α sin β$

Worked examples

Try itGiven sin α=7/25, cos α=24/25, sin β=3/5, and cos β=4/5, find sin(α+β).Show answer

Step 1: Use sin(α+β)=sin α cos β+cos α sin β. Step 2: Substitute (7/25)(4/5)+(24/25)(3/5). Step 3: Combine the fractions to obtain 4/5.

Double-Angle Forms

Objective: Explain double-angle forms, apply the rule, and justify relevant restrictions.

Setting α=β in the sum formulas gives sin(2θ)=2 sin θ cos θ and cos(2θ)=cos²θ−sin²θ. The Pythagorean identity yields equivalent cosine forms 1−2 sin²θ and 2 cos²θ−1. Choose the form that matches known information. These identities describe a doubled angle; they do not multiply the original function value by two.

Formulas

$cos(2θ)=1−2 sin²θ=2 cos²θ−1$

Worked examples

Try itIf sin θ=7/10, find cos(2θ).Show answer

Step 1: Use cos(2θ)=1−2 sin² θ. Step 2: Substitute 1−2(7/10)². Step 3: Simplify to 1/50.

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