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Mathematics Dictionary
Dr. K. G. Shih

Values of functions

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Symbol Defintion

Example : GE = Greater and Equal

TR 03 00 | - Outlines

TR 03 01 | - Values of angle 0 degrees

TR 03 02 | - Special value for angle is 30 degrees

TR 03 03 | - Special value for angle is 45 degrees

TR 03 04 | - Special value for angle is 60 degrees

TR 03 05 | - Special value for angle 90 degrees

TR 03 06 | - Summary of values for 0, 30, 45, 60, 90 degrees

TR 03 07 | - Use special angle to prove sin(A+B) = sin(A)*cos(A) + cos(A)*sin(B)

TR 03 08 | - Use special angle to prove sin(A-B) = sin(A)*cos(A) - cos(A)*sin(B)

TR 03 09 | - Use special angle to prove sin(2*A) = 2*sin(A)*cos(A)

TR 03 10 | - Exercises

Answers

TR 03 01. Special angles 0 degrees

Values of 0 degrees of functions

1. sin(0) = 0
- Since sin(A) = Opp/Hyp.
- If angle A tends to zero, Opp also tends to zero.
- Hence sin(0) = 0.
2. cos(0) = 1
- Since cos(A) = Adj/Hyp.
- If angle A tends to zero, Adj also tends to Hyp.
- Hence cos(0) = 1.
3. tan(0) = infinite
- Since tan(A) = Opp/Adj.
- If angle A tends to zero, Opp also tends to zero.
- Hence tan(0) = 0.
4. csc(0) = infinite
- Since csc(A) = 1/sin(A).
- Hence csc(0) = 1/0 = infinite.
5. sec(0) = 1
- Since sec(A) = 1/cos(A).
- Hence sec(0) = 1/1 = 1.
6. cot(0) = infinite
- Since cot(A) = 1/tan(A).
- Hence tan(0) = 1/0 = infinite.

TR 03 02. Special value for angle is 30 degrees

Let angle A = 30 and angle C = 90

Geometric theory
- If Opposite side of angle A is 1, then Hypothese is 2 (Geometric theory).
- Hence Adjacent of angle A is Sqr(2^2 - 1) = Sqr(3)
The values of the ratios are
- sin(30) = Opp/Hyp = 1/2.
- cos(30) = Adj/Hyp = Sqr(3)/2.
- Tan(30) = Opp/Adj = 1/Sqr(3) = Sqr(3)/3.
- csc(30) = Hyp/Opp = 2.
- sec(30) = Hyp/Adj = 2/Sqr(3) = 2*Sqr(3)/3.
- cot(30) = Adj/Opp = Sqr(3).

TR 03 03. Special value for angle is 45 degrees

Let angle A = 45 and angle C = 90

Geometric theory
- If Opposite side of angle A is 1 and adjacent side is also 1.
- Hence hypothese Sqr(1^2 + 1^2) = Sqr(2)
The values of the ratios are
- sin(45) = Opp/Hyp = Sqr(2)/2.
- cos(45) = Adj/Hyp = Sqr(2)/2.
- Tan(45) = Opp/Adj = 1.
- csc(45) = Hyp/Opp = 2/Sqr(2) = Sqr(2).
- sec(45) = Hyp/Adj = 2/Sqr(2) = Sqr(2).
- cot(45) = Adj/Opp = 1.

TR 03 04. Special value for angle is 60 degrees

Let angle A = 60 and angle C = 90

Geometric theory
- If hypothese is 2, then adjacent side of A is 1.
- Hence Oppsite side of angle A is Sqr(2^2 - 1) = Sqr(3)
The values of the ratios are
- sin(60) = Opp/Hyp = Sqr(3)/2.
- cos(60) = Adj/Hyp = 1/2.
- Tan(60) = Opp/Adj = 1/Sqr(3) = Sqr(3)/3.
- csc(60) = Hyp/Opp = 2/Sqr(3) = 2*Sqr(3)/3..
- sec(60) = Hyp/Adj = 2.
- cot(60) = Adj/Opp = 1/Sqr(3) = Sqr(3)/3.

TR 03 05. Values for angle 90 degrees

Values of 90 degrees of function

1. sin(90) = 0
- Since sin(A) = Opp/Hyp.
- If angle A tends to 90 degrees, Opp also tends to Hyp.
- Hence sin(90) = 1.
2. cos(90) = 1
- Since cos(A) = Adj/Hyp.
- If angle A tends to 90, Adj also tends to 0.
- Hence cos(90) = 0.
3. tan(90) = infinite
- Since tan(A) = Opp/Adj.
- If angle A tends to 90 degrees, Adj also tends to zero.
- Hence tan(90) = infinite.
4. csc(90) = infinite
- Since csc(A) = 1/sin(A).
- Hence csc(90) = 1.
5. sec(90) = 1
- Since sec(A) = 1/cos(A).
- Hence sec(90) = 1/0 = infinite.
6. cot(90) = 0
- Since cot(A) = 1/tan(A).
- Hence tan(90) = 1/infinite = 0.

TR 03 06. Summary of values for 0, 30, 45, 60, 90 degrees

Angle A =	0	30	45	60	90
sin(A) =	0	1/2	Sqr(2)/2	Sqr(3)/2	1
cos(A) =	1	Sqr(3)/2	Sqr(2)/2	1/2	0
tan(A) =	0	Sqr(3)/3	1	Sqr(3)	+∞
csc(A) =	+∞	2	Sqr(2)	2*Sqr(3)/3	0
sec(A) =	1	2*Sqr(3)/3	Sqr(2)	2	+∞
cot(A) =	+∞	Sqr(3)	1	Sqr(3)/3	0

TR 03 07. Use special angle to prove that sin(A + B) = sin(A)*cos(A) + cos(A)*sin(B)

Use 30 and 60 degrees

sin(30)*cos(60) + cos(30)*sin(60) = +sin(90)
- LHS = (1/2)*(1/2) + (Sqr(3)/2)*(Sqr(3)/2) = 1/4 + 3/4 = 1.
- RSH = sin(90) = 1.

Use 45 and 45 degrees

sin(45)*cos(45) + cos(45)*sin(45) = +sin(90)
- LHS = (Sqr(2)/2)*(Sqr(2)/2) + (Sqr(2)/2)*(Sqr(2)/2) = 1/2 + 1/2 = 1
- RSH = sin(90) = 1.

TR 03 08. Use special angle to prove that sin(A - B) = sin(A)*cos(A) -cos(A)*sin(B)

Use 30 and 60 degrees

sin(30)*cos(60) - cos(30)*sin(60) = -sin(30)
- LHS = (1/2)*(1/2) - (Sqr(3)/2)*(Sqr(3)/2) = 1/4 - 3/4 = -1/2.
- RSH = -sin(30) = -1/2.

Ese 45 and 45 degrees

sin(60)*cos(30) - cos(60)*sin(30) = +sin(30)
- LHS = (Sqr(3)/2)*(Sqr(3)/2) + (1/2)*(1/2) = 3/2 - 1/2 = 1/2
- RSH = sin(30) = 1/2.

TR 03 09. Use special angles prove that sin(2*A) = 2*sin(A)*cos(A)

Let A = 30 degrees

LHS = sin(2*A) = sin(60) = Sqr(3)/2.
RHS = 2*sin(A)*cos(A) = 2*(1/2)*(Sqr(3)/2) = Sqr(3)/2.

Let A = 45 degrees

LHS = sin(2*A) = sin(90) = 1.
RHS = 2*sin(A)*cos(A) = 2*(Sqr(2)/2)*(Sqr(2)/2) = 1.

TR 03 10. Exercises : Use special angles 30 and 60 to prove that

Questions

1. cos(A+B) = cos(A)*cos(B) - sin(A)*sin(B).
2. cos(A-B) = cos(A)*cos(B) + sin(A)*sin(B).
3. tan(A+B) = (tan(A) + tan(B))/(1 - tan(A)*tan(B).
4. tan(A-B) = (tan(A) - tan(B))/(1 + tan(A)*tan(B).

TR 03 00. Outlines

Values of 0 degrees

sin(0) = 0, cos(0) = 1 and tan(0) = 0

Values of 30 degrees

sin(30) = 0.5, cos(30) = Sqr(3)/2 and tan(0) = Sqr(3)/3

Values of 45 degrees

sin(45) = Sqr(2)/2, cos(45) = Sqr(2)/2 and tan(45) = 1

Values of 90 degrees

sin(90) = 1, cos(90) = 0 and tan(90) = infinite
tang(89.999999) = +infinite
tang(90.000001) = +infinite

From diagrams compare the answer

Diagrams Program 02 01, 02 02, ....

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