Monday
Monday: Angles Add, Sines Do Not
Two cardboard oars lie hinged at one end on the workbench, a protractor under the hinge.
"First oar at 45° from the bench edge," Comet says, swinging it. "Second oar 30° past the first. So 75° in all."
"We know sin 45° and sin 30°," Wren says. "What is sin 75°? What do you notice if you just add them?"
Comet adds. "√2/2 plus 1/2 is about 1.21. That cannot be right. Sine never passes 1."
Nova projects two right triangles, one stacked on the other. "Would you like a hint? The second oar tilts the second triangle."
"So its height is not sin 30° straight up," Wren says. "Part of it leans sideways. We need a formula."
"Then let us build one," Comet says. "What can we make from two triangles?"
The addition formulas
For any angles a and b: sin(a + b) = sin a cos b + cos a sin b.
And cos(a + b) = cos a cos b - sin a sin b. Notice the minus sign: cosine of a sum subtracts.
Angles add, but their sines and cosines mix. Each term has one factor from a and one from b.
Thursday proves both from a picture. Today you use them and check that they give sensible answers.
A solved problem to study
- The crew's oars: 45° and 30° more. Find sin 75° exactly.
- Let a = 45° and b = 30°. The formula reads sin 75° = sin 45° cos 30° + cos 45° sin 30°.
- Put in the special values: (√2/2)(√3/2) + (√2/2)(1/2).
- Multiply each pair: √6/4 + √2/4.
- Add over the common bottom: sin 75° = (√2 + √6)/4.
- Check with decimals: (√2 + √6)/4 ≈ 0.97. That is less than 1, and bigger than sin 45° ≈ 0.71. It makes sense.
Compare with the tempting mistake: sin 45° + sin 30° ≈ 1.21, which no sine can equal.
Every number here is the crew's own protractor reading from Nova's log, not a fact about real oars.
| Angle | sin (exact) | cos (exact) | tan (exact) | sin (decimal) | cos (decimal) |
|---|---|---|---|---|---|
| 15° | (-√2 + √6)/4 | (√2 + √6)/4 | 2 - √3 | 0.26 | 0.97 |
| 75° | (√2 + √6)/4 | (-√2 + √6)/4 | 2 + √3 | 0.97 | 0.26 |
| 105° | (√2 + √6)/4 | (√2 - √6)/4 | -2 - √3 | 0.97 | -0.26 |
- Read the question.
- Tap your answer.
| Statement | True or false? |
|---|---|
| sin(a + b) = sin a + sin b. | ? |
| sin(a + b) = sin a cos b + cos a sin b. | ? |
| cos(a + b) = cos a cos b - sin a sin b. | ? |
| sin 75° = (√2 + √6)/4, which is about 0.97. | ? |
| cos(a + b) = cos a cos b + sin a sin b. | ? |
Check that your answer is about 0.26 with a calculator. Positive and small, as a 75° angle should be.
Strong start. Tomorrow the formulas run backwards to a difference, and tangent gets its own.