Porisma

Worked solution

Use first principles and the sine small-angle limit

Let f:x↦sin⁡xf:x\mapsto\sin x. Using the first principles of differentiation, show that f′(x)=cos⁡xf^{\prime}(x)=\cos x.

CJC · Lecture appendix · Example B

Apply the printed sum-to-product identity

[sin⁡(x+h)−sin⁡x]/h=2cos⁡(x+h/2)sin⁡(h/2)/h=[sin⁡(h/2)/(h/2)]cos⁡(x+h/2)[\sin(x+h)-\sin x]/h=2\cos(x+h/2)\sin(h/2)/h=[\sin(h/2)/(h/2)]\cos(x+h/2). The source gives sin⁡P−sin⁡Q=2cos⁡[(P+Q)/2]sin⁡[(P−Q)/2]\sin P-\sin Q=2\cos[(P+Q)/2]\sin[(P-Q)/2]. Both copies say this formula is not tested in the H2 Math syllabus; the Teacher yellow note additionally tells the lecturer that the factor formula is no longer in the syllabus. It is used here to explain the appendix derivation, not presented as a new required memorisation rule.

How did this part go?

Enter and graph the auxiliary function

In radian mode, the source calculator steps are Y=, enter Y1=sin⁡(X/2)/(X/2)Y_1=\sin(X/2)/(X/2), then GRAPH. The plotted curve has a missing value at zero even though the pixel display does not clearly show the hole. 2nd → GRAPH opens its table: at X=0X=0 the source screen reports ERROR, because the expression is 0/00/0.

How did this part go?

Inspect the source table near zero

Press 2nd → WINDOW for table setup: TblStart=0, ΔTbl=1E−7, independent and dependent modes both Auto. Press 2nd → GRAPH again. The source screen still shows ERROR at zero and displayed values 1 at 10−7,2×10−7,…,10−610^{-7},2\times10^{-7},\ldots,10^{-6}. These finite-precision values suggest the limiting value 1; they are not exact equalities at nonzero inputs or a proof from finitely many samples.

How did this part go?

Complete the source argument with its stated limit

Using the standard radian limit lim⁡t→0sin⁡t/t=1\lim_{t\to0}\sin t/t=1 and continuity of cosine, f′(x)=lim⁡h→0[sin⁡(h/2)/(h/2)]lim⁡h→0cos⁡(x+h/2)=1⋅cos⁡x=cos⁡xf^{\prime}(x)=\lim_{h\to0}[\sin(h/2)/(h/2)]\lim_{h\to0}\cos(x+h/2)=1\cdot\cos x=\cos x. The source motivates the first limit using its calculator observations; accepting that limit makes this a first-principles derivation. A calculator table alone does not establish the two-sided limit.

How did this part go?

Compare your sketch

Inspect the sine quotient near zero

  1. 1. Enter the quotient

    Open Y= and enter the whole quotient. Use radians for trigonometric input.

    Y1
    sin(X/2)/(X/2)

  2. 2. Compare coarse inputs

    Open TABLE with 2nd → GRAPH. The retained source displays these coarse values.

    Compare coarse inputs
    XY1 display
    0ERROR
    1.95885
    2.84147
    3.665
    4.45465
    5.23939
    6.04704
    7−.1002
    8−.1892
    9−.2172
    10−.1918

  3. 3. Make the increment smaller

    Press 2nd → WINDOW for TABLE SETUP. Set TblStart=0 and ΔTbl=1E−7, with independent/dependent entries on Auto. Reopen TABLE.

    Make the increment smaller
    XY1 display
    0ERROR
    1E−71
    2E−71
    1E−61

At zero the quotient remains undefined: ERROR. The near-zero 1s are rounded displays, not exact equalities or a defined value at zero. Finite samples suggest a limit; they do not prove it. The original screens retain every intermediate table row.

View original calculator screens
Expand diagram

Use the scrollbars or arrow keys to inspect details.

Mathematical reconstruction of sin(X/2)/(X/2) with an open circle at (0,1), all printed coarse and near-zero calculator table values and setup start 0/step 1E−7. Zero is ERROR; near-zero values display 1 due to rounding. Original screenshots retained in source evidence.
Mathematical reconstruction of sin(X/2)/(X/2) with an open circle at (0,1), all printed coarse and near-zero calculator table values and setup start 0/step 1E−7. Zero is ERROR; near-zero values display 1 due to rounding. Original screenshots retained in source evidence.
Mathematical reconstruction of sin(X/2)/(X/2) with an open circle at (0,1), all printed coarse and near-zero calculator table values and setup start 0/step 1E−7. Zero is ERROR; near-zero values display 1 due to rounding. Original screenshots retained in source evidence.
Open original screens (new tab)

Common mistakes

Use radians. A display rounded to 1 near zero is numerical evidence, not proof that a quotient equals 1.

Solution notes

Both originals print the complete appendix working. Source increments are written δx; this lesson uses h. No printed marks or past-paper attribution. Domain qualifications are independently authored. Appendix letters remain in the source label and identity; internal number 0 does not renumber them as printed numbered examples.

Read the hint