Sum and difference formulas

Textbook reading

The problem that opens the lesson

Find cos $15$ degrees exactly without a calculator.

Solution

Begin by identifying the mathematical object and the information that fixes it. Choose angles with known exact values, apply the correct formula with parentheses, simplify radicals, and use quadrant reasoning to check the sign. The relevant conditions are not optional bookkeeping: Tangent sum formulas require denominators to be nonzero and should be used only where all expressions are defined. Following that structure gives $cos(45-30)=\frac{sqrt(6)+sqrt(2)}{4}$ .

Why this works

The formulas preserve both magnitude and sign interactions; they are not obtained by distributing sine or cosine across addition. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Textbook reading

What this lesson is really about

Sum and difference formulas express trig values of combined angles in terms of the separate angles.

The cosine difference formula can be derived from distances or dot products between unit-circle points. The other formulas follow by substitutions, cofunction relationships, and quotient definitions.

The point is not merely to reproduce a formula. A learner should be able to identify the quantities or geometric objects involved, explain why the relationship has its stated form, and recognize when the same idea appears in a graph, table, diagram, or model.

Textbook reading

Why the relationship works

The formulas preserve both magnitude and sign interactions; they are not obtained by distributing sine or cosine across addition.

Textbook reading

A reliable way to work

Choose angles with known exact values, apply the correct formula with parentheses, simplify radicals, and use quadrant reasoning to check the sign.

Tangent sum formulas require denominators to be nonzero and should be used only where all expressions are defined.

After the symbolic work is complete, check the result. Depending on the lesson, this may mean substituting into an original equation, comparing coordinates, examining a graph, checking units, testing an interval, or confirming that every branch of a periodic solution has been included.

Textbook reading

What commonly goes wrong

A common error is writing $sin(a+b)=sin a+sin b$ or losing the negative sign in cosine’s formula.

The repair is to return to the definition and identify the first step where the invalid solution stops describing the original mathematical object. Later algebra cannot rescue a first step that changed the domain, orientation, branch, or meaning of the problem.

Textbook reading

Worked examples

Worked example 1

Find cos $15$ degrees exactly without a calculator.

Solution

Begin by identifying the mathematical object and the information that fixes it. Choose angles with known exact values, apply the correct formula with parentheses, simplify radicals, and use quadrant reasoning to check the sign. The relevant conditions are not optional bookkeeping: Tangent sum formulas require denominators to be nonzero and should be used only where all expressions are defined. Following that structure gives $cos(45-30)=\frac{sqrt(6)+sqrt(2)}{4}$ .

Why this works

The formulas preserve both magnitude and sign interactions; they are not obtained by distributing sine or cosine across addition. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Transfer example

Problem

Derive cos(alpha-beta) from a rotation or distance argument.

Worked development

Choose angles with known exact values, apply the correct formula with parentheses, simplify radicals, and use quadrant reasoning to check the sign. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. The cosine difference formula can be derived from distances or dot products between unit-circle points. The other formulas follow by substitutions, cofunction relationships, and quotient definitions. Then apply the conditions explicitly: Tangent sum formulas require denominators to be nonzero and should be used only where all expressions are defined. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

Angle addition supports exact values, rotations, signal combination, and derivation of double-angle formulas.

Reasoning example

Problem

Find sin $75$ degrees exactly.

Worked development

Choose angles with known exact values, apply the correct formula with parentheses, simplify radicals, and use quadrant reasoning to check the sign. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. The cosine difference formula can be derived from distances or dot products between unit-circle points. The other formulas follow by substitutions, cofunction relationships, and quotient definitions. Then apply the conditions explicitly: Tangent sum formulas require denominators to be nonzero and should be used only where all expressions are defined. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

Angle addition supports exact values, rotations, signal combination, and derivation of double-angle formulas.

Worked example 4: quick check

Find sin $15$ degrees exactly.

Solution

Begin by identifying the mathematical object and the information that fixes it. Choose angles with known exact values, apply the correct formula with parentheses, simplify radicals, and use quadrant reasoning to check the sign. The relevant conditions are not optional bookkeeping: Tangent sum formulas require denominators to be nonzero and should be used only where all expressions are defined. Following that structure gives $\frac{sqrt(6)-sqrt(2)}{4}$ .

Why this works

The formulas preserve both magnitude and sign interactions; they are not obtained by distributing sine or cosine across addition. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Textbook reading

Application and interpretation

Angle addition supports exact values, rotations, signal combination, and derivation of double-angle formulas.

A contextual answer must include units, a meaningful domain, and the assumptions that make the model plausible. An exact mathematical relationship should not be diluted into a decimal unless a measurement or comparison requires it.

Practice