All-angle definitions, exact values and periodic graphs
| English | 中文 | Pinyin |
|---|---|---|
| unit circle/ˈjuːnɪt ˈsɜːkl/ | 单位圆 | dān wèi yuán |
A wheel turns beyond a right angle. Which coordinate gives its vertical position after more than half a turn?
- A wheel turns beyond a right angle. Which coordinate gives its vertical position after more than half a turn?
- This lesson studies unit circle 单位圆: The circle of radius one centred at the coordinate origin.
Choose the mathematical structure
- 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.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines unit circle?
The circle of radius one centred at the coordinate origin.
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.
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.
7π/6 is in quadrant III; sine is −sin(π/6)=−1/2.
Test a tempting shortcut
- 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.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Sine and cosine repeat after π radians for every input angle. This claim is false. Explain which definition or assumption it violates.
Find tan(3π/4).
3π/4 is in quadrant II; tangent is −tan(π/4)=−1.
Sine and cosine repeat after π radians for every input angle.
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.
Interpret a new situation
- 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.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the maximum of y=2 sin(3x−π)+1.
Sine can reach 1, so 2×1+1=3.
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.
The circle of radius one centred at the coordinate origin. Choose the relationship, show the method, check its assumptions and interpret the result.