The Main Idea

Sum and difference formulas allow us to find the sine, cosine, or tangent of non-special angles by rewriting them in terms of sums or differences of angles we know. These formulas also let us simplify trig expressions and prove identities. They are especially helpful for evaluating angles like [latex]75^\circ[/latex] or [latex]15^\circ[/latex], since those can be written as sums or differences of [latex]30^\circ[/latex], [latex]45^\circ[/latex], and [latex]60^\circ[/latex].

The formulas are:

• [latex]\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B[/latex]

• [latex]\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B[/latex]

• [latex]\tan(A \pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A \tan B}[/latex]

Quick Tips: Using Sum and Difference Formulas

1. Remember the Patterns

   • Sine keeps the same sign: plus for sum, minus for difference.

   • Cosine flips the sign: plus for difference, minus for sum.

   • Tangent has the same sign in the numerator, opposite in the denominator.

2. Use Special Angles

   • Break down unfamiliar angles into sums or differences of [latex]30^\circ[/latex], [latex]45^\circ[/latex], [latex]60^\circ[/latex], or [latex]90^\circ[/latex].

   • Example: [latex]\cos(75^\circ)=\cos(45^\circ+30^\circ)[/latex].

3. Check Signs by Quadrant

   • Always be mindful of whether the angle is in a quadrant where sine, cosine, or tangent is positive or negative.

The Main Idea

Sum and difference formulas are powerful tools for proving that two trigonometric expressions are equal. When verifying identities, you often expand one side using these formulas, then simplify until it matches the other side. This process shows that the two sides are equivalent for all values where both are defined.

The formulas are:

• [latex]\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B[/latex]

• [latex]\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B[/latex]

• [latex]\tan(A \pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A \tan B}[/latex]

Quick Tips: Using Sum and Difference Formulas in Verifications

1. Expand Using a Formula

   • Start with the more complicated side of the identity.

   • Apply the appropriate sum or difference formula.

2. Simplify Carefully

   • Replace sine, cosine, or tangent of special angles with exact values.

   • Combine like terms, simplify fractions, or cancel factors.

   • Look for Target Expressions

     • If the identity involves a single trig function of a sum or difference, try collapsing products into one cosine, sine, or tangent.