Radians, identities and trigonometric equations
| English | 中文 | Pinyin |
|---|---|---|
| radian/ˈreɪdɪən/ | 弧度 | hú dù |
How far does the rim travel?
- A rotating wheel sweeps an arc. Radians let its angle connect directly to the distance travelled along the rim.
- This lesson studies radian 弧度: The angle subtended by an arc whose length equals the radius.
Choose the mathematical structure
- For θ in radians, arc length s=rθ and sector area A=r²θ/2. Use sin²θ+cos²θ=1 and angle identities to simplify or solve. List every solution in the stated interval.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines radian?
The angle subtended by an arc whose length equals the radius.
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.
For r=4 and θ=π/3, s=4π/3 and A=8π/3. Solving sin x=1/2 on 0≤x<2π gives x=π/6 and 5π/6. For sin2x=1/2, solve first on 0≤2x<4π before dividing by 2.
Radians, identities and trigonometric equations
For θ in radians, arc length s=rθ and sector area A=r²θ/2
Compare the model with the worked case and explain one change.
For r=4 and θ=0.5 radians, find arc length.
Arc length=rθ=4×0.5=2.
Test a tempting shortcut
- A calculator's inverse sine gives one principal value. It does not give every solution. Dividing by sin x can discard solutions where sin x=0; check that case first.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The principal inverse sine value is always the only solution in a full turn. This claim is false. Explain which definition or assumption it violates.
For r=4 and θ=0.5 radians, find sector area.
Sector area=r²θ/2=16×0.5/2=4.
The principal inverse sine value is always the only solution in a full turn.
A calculator's inverse sine gives one principal value. It does not give every solution. Dividing by sin x can discard solutions where sin x=0; check that case first.
Interpret a new situation
- Choose an identity that reduces the equation to one trigonometric function. Keep exact familiar angles where possible, and check each solution against the original equation.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
How many solutions does sin x=0.5 have on 0≤x<2π?
The two solutions are π/6 and 5π/6 in the stated interval.
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
- edexcel IAL mathematics; official unit P1. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
- 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.
The angle subtended by an arc whose length equals the radius. Choose the relationship, show the method, check its assumptions and interpret the result.