sin(y + z) sin y + sin z (y + z)sin23 = y sin 23 + z sin 23 [sin x][sinx] = [sinx] 2 = sin 2 x sin x 2 = sin ( x 2) = sin(x x) ...

Transcription

1 Sec. 01 notes Some Observations Not Linear Doubling an angles does NOT usually double the ratio. Compare sin10 with sin 20, for example. generally,... sin(2x) 2 sinx sin IS NOT a number If x, y and z are real numbers, then certainly, x(y + z) = xy + xz is true by the Distributive Law, but sin is not a real number, and the Distributive Law does not apply to it. generally,... sin(y + z) sin y + sin z sin 23 IS a number By the very definition sin of some angle, θ, IS a ratio. Thus, its a real number, thus if follows all Real Number Axioms, such as DL, Associations Laws, Commutativity Laws, etc... generally,... (y + z)sin23 = y sin 23 + z sin 23 Special Exponent Notation By convention, the following notation has been adopted for exponents. Special Exponent Notation [sin x][sinx] = [sinx] 2 = sin 2 x By convention, the following notation has been adopted for exponents. Special Exponent Notation sin x 2 = sin ( x 2) = sin(x x) By convention, the 1 exponent has been reserved for composition inverse, not multiplicative inverse of the functions. That is.. 1 sin x sin 1 x sec, csc, or cot NOT on most Calculators... however, 1 sin x = [sinx] 1 To calculate sec23 for example, you will most likely calculate cos23, then find the reciprocal. sec NOT on CALCULATOR, Do not confuse with cos 1 button which IS on most scientific calculators. hands pg. 1

2 Some Observations cos10 cos30 = 1 3 tan 100 tan 50 = 2 tan( ) = tan (100 ) + tan (50 ) 4. Generally, tan (xy) = tan(x) tan (y). 5. Generally, tan(2y) = 2 tan(y). 6. Generally, sin(2y) = 2 sin(y). 7. Generally, cos( 5y) = 5 cos(y). 8. Generally, cos(10x) x = cos10 9. NOT LINEAR: (a) Calculate cos10 cos (b) Calculate cos20 cos (c) Does doubling an angle generally double the corresponding cosine ratio? most definitely not.. (d) Does doubling an angle double the adjacent to hypothenuse ratio? Draw a for each of the ref. triangles. hands pg. 2

3 done above... cos2x = 2 cosx generally false... may occasionally be true for certain values of x. 10. NOT LINEAR: (a) Calculate cos20 cos (b) Calculate cos60 1/2 (c) Does tripling an angle generally triple the corresponding cosine ratio? most definitely not.. (d) Does tripling an angle generally triple the adjacent to hypothenuse ratio? Draw a for each of the ref. triangles. most definitely not.. cos3x = 3 cosx generally false... may occasionally be true for certain values of x. 11. NOT LINEAR: (a) Calculate sin (b) Calculate sin (c) Does doubling an angle generally double the sine ratio? hands pg. 3

4 most definitely not.. (d) Does doubling an angle generally double the opp to hypothenuse ratio? most definitely not.. sin 2x = 2 sinx generally false... may occasionally be true for certain values of x. 12. cos is not a number : (a) Calculate cos (b) Calculate cos15 cos (c) Calculate cos (d) : cos45 = cos( ) = cos30 + cos cos(x + y) = cosx + cosy the above exercises show this is generally not true. 13. sin is not a number : (a) Calculate sin 100 (b) Calculate sin 80 (c) Calculate sin 180 hands pg. 4

5 (d) (TRUE OR FALSE): sin 180 = sin( ) = sin sin 80 sin(x + y) = sin x + sin y the above exercises show this is generally not true. 14. sin is not a number : (a) Calculate sin (b) Calculate sin (c) Calculate sin 540 (d) (TRUE OR FALSE): sin540 = sin( ) = sin sin 360 this is true = 0... lucky shot... sin(x + y) = sin x + sin y generally not true.. try... sin( ) sin 90 + sin90 since sin is not a number: 5c 5w = c w True, you should try to explain... 5w 3w = 5 3 hands pg. 5

6 True, so long as these are non zero constants... you should try to explain... xyzc xyzw = c w True, so long as these are non zero constants... you should try to explain... (d) (TRUE OR FALSE explain your answer): sin c sinw = c w not true.. sin is not a non zero constant.. in fact it is not a constant.. it is a function (e) (TRUE OR FALSE explain your answer): sin 30 sin 60 = not true.. sin is not a non zero constant.. in fact it is not a constant.. it is a function (f) (TRUE OR FALSE explain your answer): (xy)z = x(yz) True, so long as these are constants... associative law of mult... (g) (TRUE OR FALSE explain your answer): (sin y)z = sin(yz) not true... sin is not a constant.. no associative law for this.. (h) (TRUE OR FALSE explain your answer): xy = yx True, so long as these are constants... commutativity law of mult... (i) (TRUE OR FALSE explain your answer): siny = y sin NOT True, so long as these are NOT constants... NO commutativity law of mult... here sin23 IS a number: 5 sin23 7 sin23 = 5 7 ( 5 + 3)sin 23 = 5sin sin23 hands pg. 6

7 (d) (TRUE OR FALSE explain your answer): (e) (TRUE OR FALSE explain your answer): ( 3t)(sin 23 ) = ( 3)(t sin 23 ) 5(sin 23 ) = (sin 23 )5 if x(sin 23 ) = 7 then x = 7 sin sin is not a number: 5c = c5 xyzw = wxyz sinw = w sin 18. sin is not a number: if 5c = x then c = x 5 if yc = x then c = x y if sinc = x then c = x sin hands pg. 7