Radians, identities and trigonometric equations
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| radian/ˈreɪdɪən/ | radian |
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.
The method
- 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.
Solving sin x=1/2 on 0≤x<2π, why are there two solutions?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The inverse sine returns only the principal value π/6. Sine is also positive in the second quadrant, giving π-π/6=5π/6. Both must be listed within the stated interval.
A worked case
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.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Arc length=rθ=4×0.5=2.
Where it goes wrong
- 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.
Not true: The principal inverse sine value is the only solution in a full turn. Sine takes most values twice in a full turn: sin x=0.5 gives 30° and 150°.
For r=4 and θ=0.5 radians, find sector area.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Sector area=r²θ/2=16×0.5/2=4.
The principal inverse sine value is the only solution in a full turn.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Sine takes most values twice in a full turn: sin x=0.5 gives 30° and 150°.
Applying it
- 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.
How many solutions does sin x=0.5 have on 0≤x<2π?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The two solutions are π/6 and 5π/6 in the stated interval.
Match each radian formula to the quantity it gives.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The arc and sector formulae require θ in radians; one radian is the angle whose arc equals the radius.
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.
- Radian: The angle subtended by an arc whose length equals the radius.
- For θ in radians, arc length s=rθ and sector area A=r²θ/2.
- Watch for: A calculator's inverse sine gives one principal value.