First-principles sine and cosine derivatives
| English | 中文 | Pinyin |
|---|---|---|
| radian limit | 弧度极限 | hú dù jí xiàn |
A rotating point has sine and cosine coordinates. How do its tiny coordinate changes lead to exact derivative formulas?
- A rotating point has sine and cosine coordinates. How do its tiny coordinate changes lead to exact derivative formulas?
- This lesson studies radian limit 弧度极限: A small-angle limit used with angles measured in radians.
Choose the mathematical structure
- In radians, sin h/h tends to 1 and (cos h−1)/h tends to 0 as h tends to zero. Using angle-addition formulas in the difference quotient gives d(sin x)/dx=cos x and d(cos x)/dx=−sin x. These limits determine the tangent gradients; they require radian input.
- 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 limit?
A small-angle limit used with angles measured in radians.
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 sin x, [sin(x+h)−sin x]/h=sin x[(cos h−1)/h]+cos x[sin h/h], tending to cos x. For cos x the quotient is cos x[(cos h−1)/h]−sin x[sin h/h], tending to −sin x. Near zero, the unit-circle inequality sin h<h<tan h for 0<h<π/2 bounds sin h/h between cos h and 1; negative h has the same ratio. Also (cos h−1)/h=−2sin²(h/2)/h tends to zero, using sin(h/2)/(h/2)→1. At x=π/6, the sine gradient is √3/2 and the cosine gradient is −1/2. For sine at x=0, h=0.1 gives sin(0.1)/0.1≈0.998334; at h=−0.1 the ratio is the same.
First-principles sine and cosine derivatives
In radians, sin h/h tends to 1 and (cos h−1)/h tends to 0 as h tends to zero
Connect a derivative calculation to its limit or gradient sign interpretation.
Find the derivative of sin x at x=0.
The sine proof gives cos0=1.
Test a tempting shortcut
- A small-angle approximation is not an exact finite-h equality. The cosine derivative has a minus sign. If a variable θ is measured in degrees, d sin(πθ/180)/dθ=(π/180)cos(πθ/180); the radian formula cannot be copied without that factor. Do not use the derivative formula itself to prove the starting limit.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
The derivative of cos x is positive sin x for radian x. This claim is false. Explain which definition or assumption it violates.
Find the derivative of cos x at x=π/2.
The cosine proof gives −sin(π/2)=−1.
The derivative of cos x is positive sin x for radian x.
A small-angle approximation is not an exact finite-h equality. The cosine derivative has a minus sign. If a variable θ is measured in degrees, d sin(πθ/180)/dθ=(π/180)cos(πθ/180); the radian formula cannot be copied without that factor. Do not use the derivative formula itself to prove the starting limit.
Interpret a new situation
- Start with the original difference quotient and use angle addition before applying the two radian limits. Retain the fixed sin x/cos x factors and their signs. Check exact-point gradients against whether the graph is rising or falling; the proof applies at any real radian x, not only near x=0.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the derivative of sin x at x=π.
The sine proof gives cosπ=−1.
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 · G. 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.
A small-angle limit used with angles measured in radians. Choose the relationship, show the method, check its assumptions and interpret the result.