All-angle definitions, exact values and periodic graphs
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| unit circle/ˈjuːnɪt ˈsɜːkl/ | đường tròn đơn vị |
Which coordinate gives its vertical position after more than half a turn?
- A wheel turns beyond a right angle.
- This lesson studies unit circle 单位圆: The circle of radius one centred at the coordinate origin.
The method
- For a signed angle θ from the positive x-axis, the unit-circle point is (cosθ,sinθ); tanθ=sinθ/cosθ where cosθ≠0.
- Sine and cosine have period 2π and range [−1,1].
- Tangent has period π, every real output and vertical asymptotes θ=π/2+kπ for integer k.
- Sine/tangent are odd functions; cosine is even.
Why is sin(7π/6)=-1/2 rather than +1/2?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The unit-circle point at 7π/6 is (-√3/2,-1/2). The reference angle π/6 supplies the size 1/2 and the third quadrant supplies the minus sign.
A worked case
Special triangles give (sinθ,cosθ): at 0 use (0,1); π/6 gives (1/2,√3/2); π/4 gives (√2/2,√2/2); π/3 gives (√3/2,1/2); π/2 gives (1,0); π gives (0,−1). Divide to obtain tangent, excluding π/2. Reference angles and quadrant signs give sin(7π/6)=−1/2, cos(4π/3)=−1/2 and tan(3π/4)=−1. Subtract complete turns before finding the quadrant: sin(19π/6)=sin(7π/6). For y=2 sin(3x−π)+1, amplitude=2, period=2π/3, centre line y=1 and range [−1,3]. Writing 3x−π=3(x−π/3) gives a right shift π/3.
All-angle definitions, exact values and periodic graphs
For a signed angle θ from the positive x-axis, the unit-circle point is (cosθ,sinθ); tanθ=sinθ/cosθ where cosθ≠0
Explain each condition before using the corresponding trigonometric formula.
Find sin(7π/6) exactly as a decimal.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
7π/6 is in quadrant III; sine is −sin(π/6)=−1/2.
Where it goes wrong
- A negative angle is clockwise; do not automatically make its sine positive.
- Tan(π/2) is undefined rather than a very large finite number.
- Multiply the input frequency to shorten the period; adding a vertical shift does not change the period.
- The sine graph crosses its centre line rather than always crossing y=0.
Not true: Sine and cosine repeat after π radians for every input angle. Sine and cosine have period 2π; after π radians they change sign.
Find tan(3π/4).
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
3π/4 is in quadrant II; tangent is −tan(π/4)=−1.
Sine and cosine have period 2π.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. Sine and cosine have period 2π; after π radians they change sign.
Applying it
- Build a graph from a full cycle of key points, its symmetry and period.
- For cosθ use reflection symmetry about θ=0; for sinθ/tanθ use origin symmetry.
- Keep exact reference values, then use a calculator only when an unfamiliar angle needs a decimal.
- State angle mode and the domain interval.
Find the maximum of y=2 sin(3x−π)+1.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Sine can reach 1, so 2×1+1=3.
Match each function property to the correct function.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
On the unit circle the point is (cos θ, sin θ); tangent is their ratio and blows up wherever cosine vanishes.
In your course
- AQA A-level Mathematics (7357), specification section E.
- Unit circle: The circle of radius one centred at the coordinate origin.
- For a signed angle θ from the positive x-axis, the unit-circle point is (cosθ,sinθ); tanθ=sinθ/cosθ where cosθ≠0.
- Watch for: A negative angle is clockwise; do not automatically make its sine positive.