Lesson 5.11

Sum & Difference Formulas (Cosine)

The cosine sum and difference formulas mirror sine's — but with a crucial sign twist. Master the pattern and you can evaluate cosine at any combination of standard angles.

Introduction

Just as sine has sum and difference formulas, cosine has its own pair. The key difference: the signs are opposite to what you might expect, which makes them tricky — but the pattern is easy to remember once you see it.

Past Knowledge

Sine sum/difference formulas (5.10) and unit circle values.

Today's Goal

Apply formulas to find exact values.

Future Success

All three cosine double-angle formulas (5.13) are derived from these.

Key Concepts

The Cosine Sum & Difference Formulas

The Sign Twist

For cosine, the sign in the formula is opposite to the sign in the argument: uses , and uses . Also note: cosine formulas have matching pairs (cos·cos, sin·sin), not alternating like sine.

Sine vs. Cosine — Side by Side

Formula	Pattern	Sign Rule
	sin·cos ± cos·sin (alternating)	Same sign
	cos·cos ∓ sin·sin (matching)	Opposite sign

Worked Examples

Basic

Evaluating cos 75°

Question: Find the exact value of .

Step 1:

Step 2: Apply the sum formula (minus sign):

Final Answer:

Intermediate

Evaluating cos 15°

Question: Find the exact value of .

Step 1:

Step 2: Apply the difference formula (plus sign):

Final Answer:

Advanced

Deriving the Cofunction Identity

Question: Use the cosine difference formula to prove .

Step 1: Let , :

Step 2: Substitute , :

The cofunction identity is proven using the cosine difference formula. ✓

Common Pitfalls

Getting the Sign Backwards

The most common error: using a + in the cosine sum formula. Remember the mnemonic: “Cosine is contrary” — the sign in the formula is always opposite to the sign in the argument.

Real-Life Applications

Robotics — Joint Angle Computation

Robotic arms calculate the position of their end effector (the tool tip) using rotation matrices that are built from cosine sum formulas. Each joint rotates by some angle, and the net orientation is computed by combining these rotations — exactly like evaluating .