Lesson 5.12

Sum & Difference Formulas (Tangent)

The tangent sum and difference formulas complete the trio. Derived from the sine and cosine formulas, they let you evaluate tangent at non-standard angles with a single, elegant fraction.

Introduction

Since , we can derive tangent sum/difference formulas by dividing the sine formula by the cosine formula. The result is a compact fraction involving only tangent values.

Past Knowledge

Sine and cosine sum/difference formulas (5.10–5.11), quotient identity.

Today's Goal

Apply formulas to evaluate non-standard angles.

Future Success

The double-angle tangent formula (5.13) follows directly from letting .

Key Concepts

The Tangent Sum & Difference Formulas

Sign Pattern

The numerator sign matches the argument: for sum, for difference. The denominator sign is opposite: for sum, for difference. Think: “numerator agrees, denominator disagrees.”

The Complete Sum/Difference Family

Function	Sum Formula Sign	Lesson
sin	Same as argument	5.10
cos	Opposite of argument	5.11
tan	Numerator same, denominator opposite	5.12

Worked Examples

Basic

Evaluating tan 75°

Question: Find the exact value of .

Step 1:

Step 2: Apply the formula:

Step 3: Multiply numerator and denominator by 3:

Rationalize by multiplying by :

Final Answer:

Intermediate

Evaluating tan 15°

Question: Find the exact value of .

Step 1:

Step 2: Apply the difference formula:

Final Answer:

Advanced

Undefined Case

Question: What happens when you try ?

Step 1: Apply the formula:

Step 2: Division by zero — is undefined, which matches the vertical asymptote on the tangent graph!

The formula correctly produces “undefined” when the tangent does not exist.

Common Pitfalls

Mixing Up Numerator and Denominator Signs

The numerator sign matches the argument, but the denominator sign is opposite. For : numerator is , denominator is . Swapping these gives a wrong answer.

Real-Life Applications

Computer Graphics — Rotation Matrices

When a 3D game engine rotates an object, it combines rotation angles using sum formulas. The tangent form is especially useful for computing view frustum angles in camera systems, where the field of view is often expressed as a tangent ratio.