Double-angle identities, quadrant signs and squared forms
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| double angle/ˈdʌbl ˈæŋɡl/ | góc kép |
Which signs survive the calculation?
- Doubling a wheel angle can move its point into another quadrant.
- This lesson studies double angle 倍角: An angle equal to twice the original angle.
The method
- Set B=A=θ in the compound formulas: sin2θ=2sinθ cosθ; cos2θ=cos²θ−sin²θ=1−2sin²θ=2cos²θ−1.
- Where tanθ and tan2θ exist, tan2θ=2tanθ/(1−tan²θ).
- Rearranging gives sin²θ=(1−cos2θ)/2 and cos²θ=(1+cos2θ)/2.
- These identities depend on the full angle 2θ, not on squaring the function.
With cos θ=-3/5 in quadrant II, why is sin θ=+4/5 rather than -4/5?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
A squared identity yields only the magnitude. The stated quadrant fixes the sign: sine is positive in quadrant II, so sin θ=+4/5 and sin2θ=2(4/5)(-3/5)=-24/25.
A worked case
If cosθ=−3/5 and θ is in quadrant II, sinθ=+4/5. Then sin2θ=2(4/5)(−3/5)=−24/25 and cos2θ=9/25−16/25=−7/25. Both are negative, so the doubled angle lies in quadrant III modulo a turn; tan2θ=24/7. The tangent formula gives the same answer from tanθ=−4/3. Recover sin²θ from cos2θ: [1−(−7/25)]/2=16/25. At θ=π/4, the sine/cosine double-angle formulas still work, but tan2θ is undefined. At θ=π/2, tanθ is undefined while tan2θ=0; use sine/cosine rather than the tangent quotient.
Double-angle identities, quadrant signs and squared forms
Set B=A=θ in the compound formulas: sin2θ=2sinθ cosθ; cos2θ=cos²θ−sin²θ=1−2sin²θ=2cos²θ−1
Explain how the angle formula follows from the geometric or coefficient conditions.
For the worked quadrant-II angle, find sin2θ.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
2sinθcosθ=2×(4/5)×(−3/5)=−24/25=−0.96.
Where it goes wrong
- sin2θ means sin(2θ); sin²θ means (sinθ)².
- A squared value gives a magnitude but not a sign; use the stated quadrant before taking a root.
- Never infer the doubled angle’s quadrant from the original one without checking signs.
- Simplified identities do not remove the original tangent exclusions.
Not true: The functions sin2θ and sin²θ have the same value for every angle. sin 2θ=2 sin θ cos θ, while sin²θ means (sin θ)²: at θ=90° they are 0 and 1.
Find cos2θ.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
cos²θ−sin²θ=9/25−16/25=−7/25=−0.28.
The functions sin2θ and sin²θ have the same value for every angle.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
sin 2θ=2 sin θ cos θ, while sin²θ means (sin θ)²: at θ=90° they are 0 and 1.
Applying it
- Choose the cosine form that matches the known quantity: 2cos²θ−1 when cosine is given, or 1−2sin²θ when sine is given.
- Use the squared forms to replace a trig square with a constant and a double-angle term.
- Verify a derived sign against the unit circle and retain exact fractions until a decimal is requested.
Find tan2θ. Give it to 3 significant figures.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Divide sine by cosine: (−24/25)/(−7/25)=24/7.
Match each double-angle form to its expression.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
sin2θ means sin(2θ), never (sin θ)²; the rearranged squared forms trade a square for a double angle in integrations and proofs.
In your course
- AQA A-level Mathematics (7357), specification section E.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Double angle: An angle equal to twice the original angle.
- Set B=A=θ in the compound formulas: sin2θ=2sinθ cosθ; cos2θ=cos²θ−sin²θ=1−2sin²θ=2cos²θ−1.
- Watch for: sin2θ means sin(2θ); sin²θ means (sinθ)².