The cos3x formula (triple angle formula for cosine) is: cos3x = 4cos³x − 3cosx. This formula expresses the cosine of three times an angle in terms of the cosine of the angle itself. It is derived from the compound angle formula cos(A+B) and the double angle formula cos2x = 2cos²x − 1.
cos3x = 4cos³x − 3cosx (triple angle formula for cosine).
sin3x = 3sinx − 4sin³x (triple angle formula for sine).
tan3x = (3tanx − tan³x)/(1 − 3tan²x).
Derived using compound angle formula: cos(2x + x).
cos³x = (3cosx + cos3x)/4 — useful for integration of cos³x.
sin³x = (3sinx − sin3x)/4 — useful for integration of sin³x.
Example: cos3(60°) = 4(1/2)³ − 3(1/2) = −1 = cos180° ✓
Formula: cos3x = 4cos³x − 3cosx
Derivation: Step 1: Write cos3x = cos(2x + x) cos3x = cos2x·cosx − sin2x·sinx (compound angle formula)
Step 2: Substitute double angle formulas: • cos2x = 2cos²x − 1 • sin2x = 2sinx·cosx
Step 3: cos3x = (2cos²x − 1)·cosx − (2sinx·cosx)·sinx = 2cos³x − cosx − 2sin²x·cosx
Step 4: Replace sin²x = 1 − cos²x: = 2cos³x − cosx − 2(1 − cos²x)·cosx = 2cos³x − cosx − 2cosx + 2cos³x = 4cos³x − 3cosx
Answer: cos3x = 4cos³x − 3cosx ✓
Alternately written: cosx can be expressed using cos3x: 4cos³x = 3cosx + cos3x cos³x = (3cosx + cos3x)/4
sin3x formula: sin3x = 3sinx − 4sin³x
Derivation of sin3x: sin3x = sin(2x + x) = sin2x·cosx + cos2x·sinx = 2sinx·cos²x + (1 − 2sin²x)·sinx = 2sinx(1 − sin²x) + sinx − 2sin³x = 2sinx − 2sin³x + sinx − 2sin³x = 3sinx − 4sin³x ✓
tan3x formula: tan3x = (3tanx − tan³x) / (1 − 3tan²x)
Summary of triple angle formulas: • sin3x = 3sinx − 4sin³x • cos3x = 4cos³x − 3cosx • tan3x = (3tanx − tan³x)/(1 − 3tan²x)
Alternate forms (for integration): • sin³x = (3sinx − sin3x)/4 • cos³x = (3cosx + cos3x)/4 These are very useful for integrating sin³x and cos³x.
Example 1: Find cos3x if cosx = 1/2 (x = 60°) cos3x = 4cos³x − 3cosx = 4(1/2)³ − 3(1/2) = 4(1/8) − 3/2 = 1/2 − 3/2 = −1
Check: cos3(60°) = cos180° = −1 ✓
Example 2: Find cos3x if cosx = √3/2 (x = 30°) cos3x = 4(√3/2)³ − 3(√3/2) = 4(3√3/8) − 3√3/2 = 3√3/2 − 3√3/2 = 0
Check: cos3(30°) = cos90° = 0 ✓
Example 3: Prove that cos3x = 4cos³x − 3cosx using x = 45°: cos3(45°) = cos135° = −1/√2 4cos³(45°) − 3cos(45°) = 4(1/√2)³ − 3(1/√2) = 4/(2√2) − 3/√2 = 2/√2 − 3/√2 = −1/√2 ✓
Application in integration: ∫cos³x dx = ∫(3cosx + cos3x)/4 dx = (3sinx)/4 + sin3x/12 + C
cos3x = 4cos³x − 3cosx. This is the triple angle formula for cosine. It is derived from cos(2x+x) using compound and double angle formulas.
cos3x = cos(2x+x) = cos2x·cosx − sin2x·sinx = (2cos²x−1)cosx − 2sinx·cosx·sinx = 2cos³x − cosx − 2sin²x·cosx = 2cos³x − cosx − 2(1−cos²x)cosx = 2cos³x − cosx − 2cosx + 2cos³x = 4cos³x − 3cosx.
sin3x = 3sinx − 4sin³x. It is derived from sin(2x+x) = sin2x·cosx + cos2x·sinx = 2sinx·cos²x + (1−2sin²x)sinx = 3sinx − 4sin³x.
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