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Precalculus 9-12 / Week 06 / Friday
5/6
Week 06 · The Addition Formulas

Friday

Double angles and a review
// Two oars at an angle
⏱ about 20 min

Friday: Double Angles and a Review

Comet sets both oars to exactly 30°, one on top of the other, then swings the top one another 30°. "a plus a."

"So a + b with b equal to a," Wren says. "What do you notice when you put a in for b?"

"sin 2a = sin a cos a + cos a sin a," Comet reads. "Both terms are the same. Two sin a cos a."

"Check it," Wren says. "Twice sin 30° cos 30° is 2 × 1/2 × √3/2 = √3/2. And sin 60° is √3/2."

Nova projects the week's wall: six formulas and a table of twelve angles. "Would you like a hint? Every one came from the picture."

"Double angle, half angle, difference," Comet says. "One proof, many formulas."

"Today we read them all without slowing down," Wren says. "Then the oars go back on the rack."

The double-angle formulas

Put b = a into each addition formula.

sin 2a = 2 sin a cos a. The two mixed terms are equal, so they double.

cos 2a = cos²a - sin²a. With sin²a + cos²a = 1 this is also 2cos²a - 1 or 1 - 2sin²a.

tan 2a = 2 tan a / (1 - tan²a).

Check with a = 30°: cos 60° = (√3/2)² - (1/2)² = 3/4 - 1/4 = 1/2. Correct.

Where the crew used the formulas this week

QuestionFormulaAnswer
sin 75°, the oars at 45° and 30° moresin a cos b + cos a sin b(√2 + √6)/4
cos 75°cos a cos b - sin a sin b(-√2 + √6)/4
sin 15°, the oar swung backsin a cos b - cos a sin b(-√2 + √6)/4
tan 75°(tan a + tan b)/(1 - tan a tan b)2 + √3
sin 60°, both oars at 30°2 sin a cos a√3/2

Formulas around the Boathouse

  • Two hinged oars, a folding ruler and a door on its hinge all add angles. Their sines and cosines mix by the formulas.
  • A rower's two strokes at different angles combine into one direction. Week 10 adds those as vectors.
  • The water level wave in week 7 is written with sines. Shifting it by a quarter turn uses sin(x + 90°) = cos x.

Which answer makes sense?

A classmate writes cos 60° = 2 cos 30° = 1.73. That is bigger than 1, so no cosine can equal it.

The double-angle formula gives cos 60° = 1/2. Doubling the angle never doubles the value.

DOUBLE ANGLES
  • Read the question.
  • Tap your answer.
Both oars at 30°, so a = b = 30°. Using sin(a + a), what is the exact value of sin 60°?
Using cos(a + a), what is the exact value of cos 60°?
Using tan(a + a) with a = 30°, what is the exact value of tan 60°?
Both oars at 45°. Using sin(a + a), what is the exact value of sin 90°?
MIXED SET
  • Read the question.
  • Tap your answer.
Use the addition formula: what is the exact value of sin(45° + 30°) = sin 75°?
Use the subtraction formula: what is the exact value of cos 15°?
Use the addition formula: what is the exact value of tan(60° + 45°) = tan 105°?
Use the addition formula with a = 90°: what is the exact value of cos(90° + 30°) = cos 120°?
sin 2a = 2 sin a times which function of a? Type one word.
sin 60° = ? Type the exact value.
cos 60° = ? Type the exact value.
Week reviewTrue or false?
sin 2a = 2 sin a.?
sin 2a = 2 sin a cos a.?
cos 2a = cos²a - sin²a.?
sin 75° = (√2 + √6)/4 and sin 15° = (-√2 + √6)/4.?
The subtraction formulas need a second picture to prove.?
WHY THIS EXERCISEThese are the facts that make the oars, the ruler and the wave one problem.
Draw the two oars at 30° and 30° more. Beside them write sin 60° two ways: from the table and from 2 sin 30° cos 30°.

A full week from one picture. Tomorrow is Dock Day, and two rulers hinged at home show the family why sines do not add.

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