Sum and Difference of Angles Explained — Trigonometry III (JAMB/WAEC Mathematics)

The sum and difference formulas allow you to find the sine, cosine, or tangent of an angle formed by adding or subtracting two other angles, without needing a calculator for angles that can be broken down into well-known values.

This is Part 3 of our Trigonometry series, building on the identities and complementary angles from Part 2. This topic is a regular feature in JAMB and WAEC Mathematics, usually tested by asking students to find the exact value of a non-standard angle like 15° or 75°. This lesson covers each formula with worked examples.

Quick takeaways

• The sum and difference formulas let you calculate the sine, cosine, or tangent of an angle formed by adding or subtracting two known angles.
• sin(A ± B) = sinA cosB ± cosA sinB, and cos(A ± B) = cosA cosB ∓ sinA sinB (note the sign flips between sine and cosine formulas).
• tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB)
• These formulas are especially useful for finding exact values of angles like 15°, 75°, or 105°, by expressing them as a sum or difference of well-known angles (30°, 45°, 60°, 90°).
• Exam questions often ask you to evaluate an expression like sin 75° exactly, without a calculator, by breaking it into 45° + 30°.

What are the sum and difference formulas for sine?

The formulas for finding the sine of a sum or difference of two angles are:

sin(A + B) = sinA cosB + cosA sinB sin(A − B) = sinA cosB − cosA sinB

Notice both formulas share the same structure, differing only in the sign between the two terms, which directly matches whether you’re adding or subtracting the angles.

What are the sum and difference formulas for cosine?

The formulas for cosine work similarly but with an important twist — the sign flips compared to sine:

cos(A + B) = cosA cosB − sinA sinB cos(A − B) = cosA cosB + sinA sinB

This is a frequent source of confusion: for cosine, the addition formula uses a minus sign, while the subtraction formula uses a plus sign — exactly the opposite pattern from sine.

What are the sum and difference formulas for tangent?

The formulas for tangent are:

tan(A + B) = (tanA + tanB) / (1 − tanA tanB) tan(A − B) = (tanA − tanB) / (1 + tanA tanB)

These follow from dividing the sine sum/difference formula by the corresponding cosine sum/difference formula, though at JAMB/WAEC level, it’s generally more practical to memorize the tangent formulas directly rather than deriving them each time.

How do you use these formulas to find exact values of non-standard angles?

Many “unusual” angles can be expressed as the sum or difference of standard angles (30°, 45°, 60°, 90°) whose sine, cosine, and tangent values are already well known. For example, to find sin 75° exactly:

1. Express 75° as a sum of known angles: 75° = 45° + 30°
2. Apply the sine sum formula: sin(45° + 30°) = sin45° cos30° + cos45° sin30°
3. Substitute known exact values for sin45°, cos30°, cos45°, and sin30°
4. Simplify to get the exact value of sin 75°

This same approach works for other non-standard angles like 15° (= 45° − 30°) or 105° (= 60° + 45°).

Common mistakes students make with sum and difference formulas

• Mixing up the sign pattern between the sine formulas (which match addition/subtraction directly) and the cosine formulas (which flip the sign)
• Forgetting the exact values of sin, cos, and tan for the standard angles (30°, 45°, 60°), which are needed to actually apply these formulas
• Choosing an inefficient combination of angles to express a non-standard angle, when a simpler pair (like 45° + 30°) would work just as well
• Attempting to treat sin(A + B) as equal to sinA + sinB, which is a common but incorrect simplification — the formulas must be used in full

Frequently asked questions

Q: What is the formula for sin(A + B)? The formula is sin(A + B) = sinA cosB + cosA sinB.

Q: How does the cosine sum formula differ from the sine sum formula? The cosine sum formula uses a minus sign — cos(A + B) = cosA cosB − sinA sinB — while the sine sum formula uses a plus sign, and this pattern reverses for the corresponding difference formulas.

Q: How can you find the exact value of sin 75° without a calculator? Express 75° as the sum of two standard angles, such as 45° + 30°, then apply the sine sum formula using the known exact values of sin, cos for 45° and 30°.

Q: Is sin(A + B) the same as sinA + sinB? No, this is a common misconception — sin(A + B) must be expanded using the full sum formula (sinA cosB + cosA sinB) and is not simply the sum of the individual sine values.

New to this series? Start with Trigonometry Basics → or review identities and complementary angles →

This is one topic from our full JAMB & WAEC Maths course — see everything covered →