sin(A + B) is not sin A + sin B. Testing A = B = 30° shows it immediately: sin 60° is not 1.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The sum formulas
sin(A + B) = sin A cos B + cos A sin B. cos(A + B) = cos A cos B − sin A sin B.
The sign flip
Sine keeps the sign of the operation; cosine reverses it. That reversal is the detail people forget.
Double angle
Setting B = A gives sin 2A = 2 sin A cos A and cos 2A = cos²A − sin²A.
Three forms of cos 2A
Using the Pythagorean identity turns it into 1 − 2sin²A or 2cos²A − 1. All three are the same statement.
Exact values
They reach angles the special triangles miss. 75° is 45° + 30°, so its exact value follows from two known ones.
Sine does not distribute
sin(A + B) is not sin A + sin B. Check with 30° and 60°: the left side is 1 and the right about 1.37. Assuming linearity is the most damaging misconception 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. The sign flip in the cosine version is the detail most often lost.
Double angles are the special case
Setting B = A gives sin 2A = 2 sin A cos A and cos 2A = cos²A − sin²A. They are not separate results but the addition formulas with equal angles.
They generate new exact values
Since 75° is 30° + 45°, the addition formula gives sin 75° exactly. Combined with half-angle formulas, the set of angles with known exact values becomes surprisingly large.
Step 2: Try It Yourself
Tap and try it out.
- Angle75° = 5π/12 rad
- x-coordinate0.259
- y-coordinate0.966
- cos 75°0.259
- sin 75°0.966
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 Sana Find an Exact Value
Sana needs sin 75° exactly.
- Step 1
75° is not on a special triangle, but 45° + 30° is 75°.
Step 4: Your Turn
Practice makes it stick.
The Test
Problem 1 of 2
Is sin(A + B) equal to sin A + sin B? 1 yes, 0 no.
The Double
Problem 2 of 2
sin A = 0.6 and cos A = 0.8. What is sin 2A?
Combine the Angles
1 of 8
sin A = 0.6, cos A = 0.8. What is sin 2A?
2 of 8
Same values. What is cos 2A?
3 of 8
sin A = 0.5, cos A = 0.866. What is sin 2A, to three decimal places?
4 of 8
75° is the sum of 45° and which angle, in degrees?
5 of 8
In cos(A + B), what sign sits between the two products? 1 plus, 2 minus.
6 of 8
How many equivalent forms does cos 2A have?
7 of 8
Sort each formula by which sign it uses between the products.
Tap something to move it.
- Empty
- Empty
8 of 8
sin A = 0.8, cos A = 0.6. What is sin 2A?
Step 5: Quick Check
Show what you know.
Question 1 of 2
sin A = 0.6 and cos A = 0.8. What is cos 2A?
Question 2 of 2
Which formula reverses the sign between its products?
What You Learned
- sin(A + B) is not sin A + sin B.
- Sine keeps the operation sign; cosine reverses it.
- Double angle formulas come from setting B equal to A.