Double-angle identities, quadrant signs and squared forms
| English | Português |
|---|---|
| double angle/ˈdʌbl ˈæŋɡl/ | ângulo duplo |
Doubling a wheel angle can move its point into another quadrant. Which signs survive the calculation?
- Doubling a wheel angle can move its point into another quadrant. Which signs survive the calculation?
- This lesson studies double angle 倍角: An angle equal to twice the original angle.
Choose the mathematical structure
- 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.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines double angle?
An angle equal to twice the original angle.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
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.
2sinθcosθ=2×(4/5)×(−3/5)=−24/25=−0.96.
Test a tempting shortcut
- 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.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The functions sin2θ and sin²θ have the same value for every angle. This claim is false. Explain which definition or assumption it violates.
Find cos2θ.
cos²θ−sin²θ=9/25−16/25=−7/25=−0.28.
The functions sin2θ and sin²θ have the same value for every angle.
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.
Interpret a new situation
- 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.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find tan2θ.
Divide sine by cosine: (−24/25)/(−7/25)=24/7.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · E. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
An angle equal to twice the original angle. Choose the relationship, show the method, check its assumptions and interpret the result.