sin(A + B) is not sin A + sin B. Sine is not a multiplier, so it does not distribute.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
A quick check
sin(90° + 90°) = sin 180° = 0, but sin 90° + sin 90° = 2. They disagree completely.
The real formulas
sin(A + B) = sin A cos B + cos A sin B, and cos(A + B) = cos A cos B − sin A sin B.
Double angles follow
Setting B = A gives sin 2A = 2 sin A cos A and cos 2A = cos²A − sin²A.
Sine does not distribute
sin(A + B) is not sin A + sin B. Test it: sin(30° + 60°) = sin 90° = 1, but sin 30° + sin 60° ≈ 1.37. The functions are not linear, and assuming they are is the most damaging error in trigonometry.
The addition formulas
sin(A + B) = sin A cos B + cos A sin B, and cos(A + B) = cos A cos B − sin A sin B. Note the sign flip in the cosine formula — it is the detail most often lost, and it changes the answer entirely.
Double angles are the special case
Setting B = A gives sin 2A = 2 sin A cos A and cos 2A = cos²A − sin²A. The double angle formulas are not separate results; they are the addition formulas with both angles equal.
They produce new exact values
Since 75° = 30° + 45°, the addition formula gives sin 75° exactly. Compound angle formulas extend the handful of known exact values to many more, which is one of their main practical uses.
Step 2: Try It Yourself
Tap and try it out.
- Angle30° = π/6 rad
- x-coordinate0.866
- y-coordinate0.500
- cos 30°0.866
- sin 30°0.500
The cosine is the x-coordinate and the sine is the y-coordinate. They are not merely equal — they are the same two numbers, which is why angles past 90° cause no trouble.
Step 3: Watch an Example
One step at a time.
Watch Nadia Disprove the Shortcut
Nadia tests whether sin(A + B) = sin A + sin B.
- Step 1
She picks A = 90° and B = 90°.
Step 4: Your Turn
Practice makes it stick.
The Test
Problem 1 of 2
What is sin 180°?
The Double
Problem 2 of 2
sin A = 0.6 and cos A = 0.8. What is sin 2A?
Compound Angles
1 of 4
sin A = 0.8, cos A = 0.6. What is sin 2A?
2 of 4
cos A = 0.6, sin A = 0.8. What is cos 2A?
3 of 4
Is sin(A + B) equal to sin A + sin B?
4 of 4
What is sin 90° + sin 90°?
Step 5: Quick Check
Show what you know.
Question 1 of 2
sin A = 0.5 and cos A = 0.866. What is sin 2A, to two places?
Question 2 of 2
Why is sin(A + B) not sin A + sin B?
What You Learned
- sin(A + B) is not sin A + sin B.
- sin(A + B) = sin A cos B + cos A sin B.
- Setting B = A gives the double angle formulas.