Even more of shhets 1-3, but from the same days as sheet 4. Class indolent but, they say, working on other subjects. I do hope so. Meanwhile the generation of questions continues and I’m capturing 25% of them or less to put here. I wrote most of this while they were doing an earlier sheet.
DJS 20130314
We explored a range of questions dancing along the overlap between Cartesian geometry and vector geometry. Students are encouraged to use each system to bolster their grasp of the other. Teachers are encouraged to try drawing i vertically and j horizontally sometimes. Finding the point of the vector equation where the i component is zero will provided a direct and obvious correlation to the explicit y=mx+c Cartesian form. I rote a sequence of problems chasing this down.
20130314 and 15 produced satisfaction among the class that those topics were ‘cracked’, meaning understood, so I moved on to reviewing trig identities and calculus.
FPM Sample 5
1. Differentiate y = e²ˣ + 3e⁻³ˣ wrt x; attempt to find and classify svs.
2. Integrate y = e²ˣ + 3e⁻³ˣ wrt x between ln2 and ln3, giving an exact answer.
3. Solve 2 sin (3θ - π/8) = 1 for 0<θ<2π
4. A circle has area 36 sq units. What angle is subtended by a sector of perimeter of 10 units? Doesn't need a calculator...
5. Find exactly, without a calculator, tan 2φ if sinφ = 28/53
6. Find an expression for cos 4θ entirely in terms of cos θ.
TRIG: I would seem that Edexcel provide the expansion of sin (A±B) and cos (A±B) each time it is wanted, at the head of the relevant question. Students of FPM, C1-3 are encouraged to learn these thoroughly, plus the matching one for tan; also the double and triple angle formulae, especially the three forms of cos2θ.
sin (A+B) = sinA cosB + cosA sinB; cos (A+B) = cosA cosB - sinA sinB;
7. Write down the 3 expansions of cos 2θ;
8. Develop an expansion of cos 3θ only in cos θ and sin 3θ only in sin θ. They form a general form of af³+bf where f is a trig function and a & b are 3 and 4 of differing signs. The triple angle formulae are staple questions at A-level.
9. Try to find expansions cos 4θ only in cos θ and sin 4θ only in sin θ. Attempt a prediction before you begin. The cos version is something like kc⁴-kc²+1.
10. Use the knowledge that (i) sin(-x)=-sin x (it is an ‘odd’ function) and that cos x is ‘even’ to establish and (ii) the double angle formulae to establish tan (A±B), and similarly tan 2θ and tan 3θ (rarely asked in exams).
Students are expected to be able to solve sin (nx +c) =k, for ranges of x, either in degrees ( 0<x<360, -180<x<180 or radians, say 0<x<2π, it being assumed that the hint of units is obvious enough). Many students do not grasp that if 0<x<2π then 0<nx+c<2nπ+c follows, so they rarely give themselves enough answers at their first arc-trig list to make their answers sufficient. It certainly helps to sketch the sine for cosine function enough to find the right number of results.
11. Solve 2 sin 3x = 1 for 0<x<2π (so 0<3x<6π)
12. Solve 5 sin (2x-π/6) = 4 for 0<x<2π
13. Solve 5 sin (3x - 30º) = 3 for 0<x<360º
14. Solve 13 cos (4x - π/4) = 5 for 180º<x<180º
It would be better for ambitious students to practise writing generalised answers, eg sin¹(0.5) = (2n-1) π/2 ± π/6, nεN
15. Write tan 3ψ as a function in tan ψ. Many 3s.
16. By writing π/12 as π/3-π/4 find the exact value of tan π/12
17. A point is symmetrically connected to the four corners of a square (of side 2λ) so that the angle between any two edges of the four planes created is 30°. (a) Find the angle between the medians of two opposing faces. (b) Find the angle between the medians of two adjacent faces.
18. Write a definition of ODD and EVEN functions. Show that cosine is even; decide which the tangent function is; write an example of an even polynomial; try write an example of an exponential function that is odd.
DJS 20130315 &18
559/216
π/12 {7, 23, 55, 741, 103, 119}
5π/9
2520/1241
4c⁴-4c²+1
