Mathematics Dictionary

27 05 Equaliteral triangle ABC inscribed unit circle. P on circle. PA^2 +PB^2 +PC^2
21 07 Equations e.g. solve sin(x) = 0.5
09 08 Equation : Solve 1/cos(x) + 1/sin(x) = 2*Sqr(2)
09 10 Equation : sin(2*x)*sin(5*x) +sin(4*x)*sin(11*x) +sin(9*x)*sin(24*x) = 0
19 00 Ex-central triangle
10 10 Es-circle
- Tangent from A or B or C is s.
- Radius of escile 1 is r1 = s*tan(A/2).
- Radius of escile 2 is r2 = s*tan(B/2).
- Radius of escile 3 is r3 = s*tan(C/2).
- Area of triangle ABC = r1*(s-a) = r2*(s-b) = r3*(s-c).
- 1/r = 1/r1 + 1/r2 + 1/r3 where r is in-radius.
10 13 Ex-circle
- Sine law : a = 2*R*sin(A).
- Area of triangle ABC = a*b*c/(4*R)

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Q06. F

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Q07. G

07 11 GP question
- Angles of triangle in 3 consecutive GPterms and common ratio = 3
- Prove that cos(A)*cos(B) + cos(B)*cos(C) + cos(C)*cos(A) = -1/4
07 16 GP question
- Angles of triangle in 3 consecutive GPterms and common ratio = 2
- Prove that cos(A)*cos(B) + cos(B)*cos(C) + cos(C)*cos(A) = ?
27 13 GP : Sum[((1/2)^n)*cos(n*pi/2))] for n = 1 to infinite
27 13 GP : Sum[((1/2)^n)*cos(n*pi/2))] for n = 1 to infinite
17 02 Graph of x = sin(t) and y = cos(t)
17 03 Graph of x = tan(t) and y = sec(t)
06 06 Graph of y = cot(x)
06 03 Graph of y = cos(x)
06 04 Graph of y = csc(x)
06 05 Graph of y = sec(x)
06 01 Graph of y = sin(x)
06 03 Graph of y = tan(x)

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Q08. H

07 07 Half angle : cos(A/2) = F(cos(A))
07 07 Half angle : sin(A/2) = F(cos(A))
07 07 Half angle : tan(A/2) = F(cos(A))
10 08 Half angle : cos(A/2) = F(s)
10 09 Half angle : sin(A/2) = F(s)
10 07 Half angle : tan(A/2) = F(s)
10 06 Heron Formula
27 06 Hexagon ABCDEF inscribed unit circle, P on circle. PA^2+ ... +PF^2 = 2*6

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Q09. I

10 08 Identity : cos(A/2) = Sqr((s-b)*(s-c)/(b*c))

10 09 Identity : sin(A/2) = Sqr(s*(s-a)/(b*c))

10 05 Identity : sin(A) = 2*Sqr(s*(s-a)*(s-b)*(s-c))/(b*c)

10 09 Identity : tan(A/2) = Sqr(s*(s-a)/((s-b)*(s-c))

10 12 Identity : tan(A/2)*tan(B/2) + tan(B/2)*tan(C/2) + tan(C/2)*tan(A/2) = 1

11 01 Identity : sin(2*A)+sin(2*B)+sin(2*C) = ?

11 02 Identity : cos(2*A)+cos(2*B)+cos(2*C) = ?

11 03 Identity : cos(A)^2+cos(B)^2+cos(B)^2 = ?

11 04 Identity : sin(A)^2+sin(B)^2+sin(C)^2 = ?

11 05 Identity : sin(A)+sin(B)+sin(C) = 4*cos(A/2)*cos(B/2)*cos(C/2)

11 06 Identity : cos(A)+cos(B)+cos(C) = 1 + 4*sin(A/2)*sin(B/2)*sin(C/2)

11 07 Identity : cos(A/2)^2+cos(B/2)^2+cos(B/2)^2 = 2*(1 + sin(A/2)*sin(B/2)*sin(C/2))

11 08 Identity : sin(A/2)^2+sin(B/2)^2+sin(C/2)^2 = 1 - 2*sin(A/2)*sin(B/2)*sin(C/2)

21 04 Inverse trigonometric functions

08 08 Inverse trig : arcsin(sin(A)+sin(B))+arc(sin(A)-sin(B))=pi/2, find ....

10 10 In-circle

Tangent at A = (s-a).
Tahgent at B = (s-b).
Tahgent at C = (s-c).
in-radius r = (s-a)*tan(A/2) = (s-b)*tan(B/2) = (s-c)*tan(C/2).
Area of ABC = r*s.

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Q10. J

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Q11. K

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Q12. L

10 02 Law of cosine

10 01 Law of sine

14 01 Lim[sin(x)/x] = 1 as x tends to zero
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Q13. M

07 17 Maximum : y = sin(x) + cos(x)

07 18 Maximum : y = sin(x) + Sqr(3)*cos(x)

14 06 Maximum : y = (sec(A)^2 - tan(A))/(sec(A)^2 + tan(A))

14 07 Minimum : y = cos(A)^2 - 2*cos(A) + 3

07 23 Multiple angle : Show that 5 + 8*cos(x) + 4*cos(2*x) + cos(3*x) GE 0
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Q14. N

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Q15. O

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Q16. P

16 00 Parametric equations
17 00 Pedal triangle
07 09 Product of function to sum : cos(A)*cos(B)
07 08 Product of function to sum : cos(A)*sin(B)
07 08 Product of function to sum : sin(A)*cos(B)
07 08 Product of function to sum : sin(A)*sin(B)
07 14 Product of function to sum : S(n) = cos(x) +cos(3*x) +... +cos((2*n-1)*x)
07 15 Product of function to sum : S(n) = sin(x)^2 +sin(2*x)^2 +... +sin(n*x)^2
05 01 Pythagorean Relation : sin(x)^2 + cos(x)^2 = 1
05 02 Pythagorean Relation : tan(x)^2 + 1 = sec(x)^2
05 03 Pythagorean Relation : 1 + cot(x)^2 = csc(x)^2

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Q17. Q

09 07 Quadratic equation : x^2 - 6*x + 16*sin(A) = 0 has integral roots

14 04 Quadratic function : cos(2*x) + sin(x) = q has real roots

14 05 Quadratic function : x^2 - 2*x*sin(pi*x/2) + 1 = 0 has real roots
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Q18. R

Graphs of functions

15 04 R = sec(p*A/q)^M
15 02 R = sin(p*A/q)^M
15 03 R = tan(p*A/q)^M
15 05 R = 1 + 1*sin(p*A/q)^M
15 06 R = 2 + 4*sin(p*A/q)^M
15 07 R = 4 + 2*sin(p*A/q)^M
15 08 R = 1 + 1*sec(p*A/q)^M
15 09 R = 2 + 4*sec(p*A/q)^M
15 10 R = 4 + 2*sec(p*A/q)^M
15 11 R = 1 + 1*tan(p*A/q)^M
15 12 R = 2 + 4*tan(p*A/q)^M
15 13 R = 4 + 2*tan(p*A/q)^M

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Q19. S

01 03 Sector : Area of sectors = (r^2)*A/2 where A in radians
01 02 Sector : Arc length s = r*A where A in radians
13 02 Series : cos(x)
13 01 Series : sin(x)
13 03 Series : tan(x)
13 05 Series : arccos(x)
13 04 Series : arcsin(x)
13 05 Series : arctan(x)
07 14 Series : S(n) = cos(x) + cos(3*x) + .... + cos((2*n-1)*x)
07 15 Series : S(n) = sin(x)^2 + sin(2*x)^2 + .... + sin(n*x)^2
10 01 Sine Law : a/sin(A) = b/sin(B) = c/sin(C) = 2*R
04 09 sin(1) + sin(2) + ... + sin(359) + sin(360) = 0
14 02 sin(18)
14 03 sin(9), sin(36)
07 24 sin(20)*sin(40)*sin(60)*sin(80) = 3/16
07 21 sin(A) and sin(B) have AP mid term sin(x) and GP mid term sin(y), prove ...
11 05 sin(A)+sin(B)+sin(C) = 4*cos(A/2)*cos(B/2)*cos(C/2), if A+B+C = pi
07 01 sin(A+B) = ?
07 03 sin(A-B) = ?
07 07 sin(A/2) = F(cos(A))
10 09 sin(A/2) = F(s)
07 25 sin(pi/16)^4 + sin(3*pi/16)^4 + sin(5*pi/16)^4 + sin(7*pi/16)
04 09 sin(1) + sin(2) + ..... + sin(359) + sin(360) = 0
27 05 Square ABCD inscribed unit circle, P on circle. PA^2 +PB^2 +PC^2 +PD^2 = 2*4
07 09 Sum of functions : cos(A) + cos(B)
07 09 Sum of functions : sin(A) + sin(B)
07 19 Sum of functions : cos(2*A) + cos(2*B) + cos(2*C) = ? if A+B+C=pi
07 19 Sum of functions : sin(A)^2 + sin(B)^2 + sin(C)^2 = ? if A+B+C=pi
07 19 Sum of functions : cos(A) + cos(B) + cos(2) = ? if A+B+C=pi
07 20 Sum of functions : sin(A)+sin(B)=p and cos(A)+cos(B)=q, find sin(A+B)
07 22 Sum of functions : If A and B are acute angles sin(A+B) LT sin(A)+sin(B)