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Saturday, June 30, 2012

trignometry


sin(theta) = a / ccsc(theta) = 1 / sin(theta) = c / a
cos(theta) = b / csec(theta) = 1 / cos(theta) = c / b
tan(theta) = sin(theta) / cos(theta) = a / bcot(theta) = 1/ tan(theta) = b / a


sin(-x) = -sin(x)
csc(-x) = -csc(x)
cos(-x) = cos(x)
sec(-x) = sec(x)
tan(-x) = -tan(x)
cot(-x) = -cot(x)
sin^2(x) + cos^2(x) = 1tan^2(x) + 1 = sec^2(x)cot^2(x) + 1 = csc^2(x)
sin(x y) = sin x cos y cos x sin y
cos(x y) = cos x cosy sin x sin y

tan(x y) = (tan x tan y) / (1  tan x tan y)
sin(2x) = 2 sin x cos x
cos(2x) = cos^2(x) - sin^2(x) = 2 cos^2(x) - 1 = 1 - 2 sin^2(x)
tan(2x) = 2 tan(x) / (1 - tan^2(x))
sin^2(x) = 1/2 - 1/2 cos(2x)
cos^2(x) = 1/2 + 1/2 cos(2x)
sin x - sin y = 2 sin( (x - y)/2 ) cos( (x + y)/2 )
cos x - cos y = -2 sin( (x - y)/2 ) sin( (x + y)/2 )
Trig Table of Common Angles
angle030456090
sin^2(a)0/41/42/43/44/4
cos^2(a)4/43/42/41/40/4
tan^2(a)0/41/32/23/14/0


Given Triangle abc, with angles A,B,C; a is opposite to A, b opposite B, c opposite C:
a/sin(A) = b/sin(B) = c/sin(C) (Law of Sines)
c^2 = a^2 + b^2 - 2ab cos(C)b^2 = a^2 + c^2 - 2ac cos(B)
a^2 = b^2 + c^2 - 2bc cos(A)
(Law of Cosines)

(a - b)/(a + b) = tan [(A-B)/2] / tan [(A+B)/2] (Law of Tangents)


Trigonometric Tables
(Math | Trig | Tables)
PI = 3.141592... (approximately 22/7 = 3.1428)
radians = degrees x PI / 180 (deg to rad conversion)
degrees = radians x 180 / PI (rad to deg conversion)

Rad Deg Sin Cos Tan Csc Sec Cot 
.0000 00 .0000 1.0000 .0000 ----- 1.0000 ----- 90 1.5707 
.0175 01 .0175 .9998 .0175 57.2987 1.0002 57.2900 89 1.5533 
.0349 02 .0349 .9994 .0349 28.6537 1.0006 28.6363 88 1.5359 
.0524 03 .0523 .9986 .0524 19.1073 1.0014 19.0811 87 1.5184 
.0698 04 .0698 .9976 .0699 14.3356 1.0024 14.3007 86 1.5010 
.0873 05 .0872 .9962 .0875 11.4737 1.0038 11.4301 85 1.4835 
.1047 06 .1045 .9945 .1051 9.5668 1.0055 9.5144 84 1.4661 
.1222 07 .1219 .9925 .1228 8.2055 1.0075 8.1443 83 1.4486 
.1396 08 .1392 .9903 .1405 7.1853 1.0098 7.1154 82 1.4312 
.1571 09 .1564 .9877 .1584 6.3925 1.0125 6.3138 81 1.4137 
.1745 10 .1736 .9848 .1763 5.7588 1.0154 5.6713 80 1.3953 
.1920 11 .1908 .9816 .1944 5.2408 1.0187 5.1446 79 1.3788 
.2094 12 .2079 .9781 .2126 4.8097 1.0223 4.7046 78 1.3614 
.2269 13 .2250 .9744 .2309 4.4454 1.0263 4.3315 77 1.3439 
.2443 14 .2419 .9703 .2493 4.1336 1.0306 4.0108 76 1.3265 
.2618 15 .2588 .9659 .2679 3.8637 1.0353 3.7321 75 1.3090 
.2793 16 .2756 .9613 .2867 3.6280 1.0403 3.4874 74 1.2915 
.2967 17 .2924 .9563 .3057 3.4203 1.0457 3.2709 73 1.2741 
.3142 18 .3090 .9511 .3249 3.2361 1.0515 3.0777 72 1.2566 
.3316 19 .3256 .9455 .3443 3.0716 1.0576 2.9042 71 1.2392 
.3491 20 .3420 .9397 .3640 2.9238 1.0642 2.7475 70 1.2217 
.3665 21 .3584 .9336 .3839 2.7904 1.0711 2.6051 69 1.2043 
.3840 22 .3746 .9272 .4040 2.6695 1.0785 2.4751 68 1.1868 
.4014 23 .3907 .9205 .4245 2.5593 1.0864 2.3559 67 1.1694 
.4189 24 .4067 .9135 .4452 2.4586 1.0946 2.2460 66 1.1519 
.4363 25 .4226 .9063 .4663 2.3662 1.1034 2.1445 65 1.1345 
.4538 26 .4384 .8988 .4877 2.2812 1.1126 2.0503 64 1.1170 
.4712 27 .4540 .8910 .5095 2.2027 1.1223 1.9626 63 1.0996 
.4887 28 .4695 .8829 .5317 2.1301 1.1326 1.8807 62 1.0821 
.5061 29 .4848 .8746 .5543 2.0627 1.1434 1.8040 61 1.0647 
.5236 30 .5000 .8660 .5774 2.0000 1.1547 1.7321 60 1.0472 
.5411 31 .5150 .8572 .6009 1.9416 1.1666 1.6643 59 1.0297 
.5585 32 .5299 .8480 .6249 1.8871 1.1792 1.6003 58 1.0123 
.5760 33 .5446 .8387 .6494 1.8361 1.1924 1.5399 57 .9948 
.5934 34 .5592 .8290 .6745 1.7883 1.2062 1.4826 56 .9774 
.6109 35 .5736 .8192 .7002 1.7434 1.2208 1.4281 55 .9599 
.6283 36 .5878 .8090 .7265 1.7013 1.2361 1.3764 54 .9425 
.6458 37 .6018 .7986 .7536 1.6616 1.2521 1.3270 53 .9250 
.6632 38 .6157 .7880 .7813 1.6243 1.2690 1.2799 52 .9076 
.6807 39 .6293 .7771 .8098 1.5890 1.2868 1.2349 51 .8901 
.6981 40 .6428 .7660 .8391 1.5557 1.3054 1.1918 50 .8727 
.7156 41 .6561 .7547 .8693 1.5243 1.3250 1.1504 49 .8552 
.7330 42 .6691 .7431 .9004 1.4945 1.3456 1.1106 48 .8378 
.7505 43 .6820 .7314 .9325 1.4663 1.3673 1.0724 47 .8203 
.7679 44 .6947 .7193 .9657 1.4396 1.3902 1.0355 46 .8029 
.7854 45 .7071 .7071 1.0000 1.4142 1.4142 1.0000 45 .7854 
COs Sin Cot Sec CSC Tan Deg Rad 
 
Trig Table of Common Angles
angle (degrees) 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 360 = 0
angle (radians) PI/6PI/4PI/3PI/22/3PI3/4PI5/6PIPI 7/6PI 5/4PI 4/3PI 3/2PI 5/3PI 7/4PI 11/6PI 2PI = 0
sin(a)(0/4)(1/4)(2/4)(3/4)(4/4)(3/4)(2/4)(1/4)(0/4)-(1/4)-(2/4)-(3/4)-(4/4)-(3/4)-(2/4)-(1/4)(0/4)
COs(a)(4/4)(3/4)(2/4)(1/4)(0/4)-(1/4)-(2/4)-(3/4)-(4/4)-(3/4)-(2/4)-(1/4)(0/4)(1/4)(2/4)(3/4)(4/4)
tan(a)(0/4)(1/3)(2/2)(3/1)(4/0)-(3/1)-(2/2)-(1/3)-(0/4)(1/3)(2/2)(3/1)(4/0)-(3/1)-(2/2)-(1/3)(0/4)
Those with a zero in the denominator are undefined. They are included solely to demonstrate the pattern.
 unit circle picture 
  
 
  


Hyperbolic Definitions

sinh(x) = ( e x - e -x )/2csch(x) = 1/sinh(x) = 2/( e x - e -x )
cosh(x) = ( e x + e -x )/2
sech(x) = 1/cosh(x) = 2/( e x + e -x )
tanh(x) = sinh(x)/cosh(x) = ( e x - e -x )/( e x + e -x )
coth(x) = 1/tanh(x) = ( e x + e -x)/( e x - e -x )

cosh 2(x) - sinh 2(x) = 1
tanh 2(x) + sech 2(x) = 1
coth 2(x) - csch 2(x) = 1

Inverse Hyperbolic Definitions

arcsinh(z) = ln( z + sqrt(z 2 + 1) )arccosh(z) = ln( z  sqrt(z 2 - 1) )
arctanh(z) = 1/2 ln( (1+z)/(1-z) )
arccsch(z) = ln( (1+(1+z 2) )/z )
arcsech(z) = ln( (1(1-z 2) )/z )
arccoth(z) = 1/2 ln( (z+1)/(z-1) )

Relations to Trigonometric Functions

sinh(z) = -i sin(iz)csch(z) = i csc(iz)
cosh(z) = cos(iz)
sech(z) = sec(iz)
tanh(z) = -i tan(iz)
coth(z) = i cot(iz)


Overview
Math | Algebra Function |Overview)


Trig Functions: OverviewUnder its simplest definition, a trigonometric (literally, a "triangle-measuring") function, is one of the many functions that relate one non-right angle of a right triangle to the ratio of the lengths of any two sides of the triangle (or vice versa).
Any trigonometric function (f), therefore, always satisfies either of the following equations:
f(q) = a / b   OR   f(a / b) = q,
where q is the measure of a certain angle in the triangle, and a and b are the lengths of two specific sides.This means that
  • If the former equation holds, we can choose any right triangle, then take the measurement of one of the non-right angles, and when we evaluate the trigonometric function at that angle, the result will be the ratio of the lengths of two of the triangle's sides.
  • However, if the latter equation holds, we can chose any right triangle, then compute the ratio of the lengths of two specific sides, and when we evaluate the trigonometric function at any that ratio, the result will be measure of one of the triangles non-right angles. (These are called inverse trig functions since they do the inverse, or vice-versa, of the previous trig functions.)
This relationship between an angle and side ratios in a right triangle is one of the most important ideas in trigonometry. Furthermore, trigonometric functions work for any right triangle. Hence -- for a right triangle -- if we take the measurement of one of the triangles non-right angles, we can mathematically deduce the ratio of the lengths of any two of the triangle's sides by trig functions. And if we measure any side ratio, we can mathematically deduce the measure of one of the triangle's non-right angles by inverse trig functions. More importantly, if we know the measurement of one of the triangle's angles, and we then use a trigonometric function to determine the ratio of the lengths of two of the triangle's sides, and we happen to know the lengths of one of these sides in the ratio, we can then algebraically determine the length of the other one of these two sides. (i.e. if we determine that a / b = 2, and we know a = 6, then we deduce that b = 3.)Since there are three sides and two non-right angles in a right triangle, the trigonometric functions will need a way of specifying which sides are related to which angle. (It is not-so-useful to know that the ratio of the lengths of two sides equals 2 if we do not know which of the three sides we are talking about. Likewise, if we determine that one of the angles is 40°, it would be nice to know of which angle this statement is true.
Under a certain convention, we label the sides as oppositeadjacent, and hypotenuse relative to our angle of interest qfull explanation
triangle 1 triangle 2
As mentioned previously, the first type of trigonometric function, which relates an angle to a side ratio, always satisfies the following equation:
f(q) = a / b.
Since given any angle q, there are three ways of choosing the numerator (a), and three ways of choosing the denominator (b), we can create the following nine trigonometric functions:
f(q) = opp/oppf(q) = opp/adjf(q) = opp/hyp
f(q) = adj/oppf(q) = adj/adjf(q) = adj/hyp
f(q) = hyp/oppf(q) = hyp/adjf(q) = hyp/hyp

The three diagonal functions shown in red always equal one. They are degenerate and, therefore, are of no use to us. We therefore remove these degenerate functions and assign labels to the remaining six, usually written in the following order:

sine(q) = opp/hypcosecant(q) = hyp/opp
cosine(q) = adj/hypsecant(q) = hyp/adj
tangent(q) = opp/adjcotangent(q) = adj/opp

Furthermore, the functions are usually abbreviated: sine (sin), cosine (cos), tangent (tan) cosecant (csc), secant (sec), and cotangent (cot).
Do not be overwhelmed. By far, the two most important trig functions to remember are sine and cosine. All the other trig functions of the first kind can be derived from these two functions. For example, the functions on the right are merely the multiplicative inverse of the corresponding function on the left (that makes them much less useful). Furthermore, the sin(x) / COs(x) = (opp/hyp) / (adj/hyp) = opp / adj = tan(x). Therefore, the tangent function is the same as the quotient of the sine and cosine functions (the tangent function is still fairly handy).

sine(q) = opp/hypCSC(q) = 1/sin(q)
COs(q) = adj/hypsec(q) = 1/COs(q)
tan(q) = sin(q)/COs(q)cot(q) = 1/tan(q)

Let's examine these functions further. You will notice that there are the sine, secant, and tangent functions, and there are corresponding "co"-functions. They get their odd names from various similar ideas in geometry. You may suggest that the cofunctions should be relabeled to be the multiplicative inverses of the corresponding sine, secant, and tangent functions. However, there is a method to this madness. Acofunction of a given trig function (f) is, by definition, the function obtained after the complement its parameter is taken. Since the complement of any angle q is 90° - q, the the fact that the following relations can be shown to hold:
sine(90° - q) = cosine(q)
secant(90° - q) = cosecant(q)
tangent(90° - q) = cotangent(q)
thus justifying the naming convention.The trig functions evaluate differently depending on the units on q, such as degrees, radians, or grads. For example, sin(90°) = 1, while sin(90)=0.89399.... explanation
Just as we can define trigonometric functions of the form
f(q) = a / b
that take a non-right angle as its parameter and return the ratio of the lengths of two triangle sides, we can do the reverse: define trig functions of the form
f(a / b) = q
that take the ratio of the lengths of two sides as a parameter and returns the measurement of one of the non-right angles.
Inverse Functions
arcsine(opp/hyp) = qarccosecant(hyp/opp) = q
arccosine(adj/hyp) = qarcsecant(hyp/adj) = q
arctangent(opp/adj) = qarccotangent(adj/opp) = q

As before, the functions are usually abbreviated: arcsine (arcsin), arccosine (arccos), arctangent (arctan) arccosecant (arccsc), arcsecant (arcsec), and arccotangent (arccot). According to the standard notation for inverse functions (f-1), you will also often see these written as sin-1, cos-1, tan-1 csc-1, sec-1, and cot-1Beware: There is another common notation that writes the square of the trig functions, such as (sin(x))2 as sin2(x). This can be confusing, for you then might then be lead to think that sin-1(x) = (sin(x))-1, which is not true. The negative one superscript here is a special notation that denotes inverse functions (not multiplicative inverses).

Trig Functions: The Functionstriangle 1 triangle 2

sine(q) = opp/hypcosecant(q) = hyp/opp
cosine(q) = adj/hypsecant(q) = hyp/adj
tangent(q) = opp/adjcotangent(q) = adj/opp

The functions are usually abbreviated: sine (sin), cosine (cos), tangent (tan) cosecant (csc), secant (sec), and cotangent (cot).
It is often simpler to memorize the the trig functions in terms of only sine and cosine:

sine(q) = opp/hypcsc(q) = 1/sin(q)
cos(q) = adj/hypsec(q) = 1/COs(q)
tan(q) = sin(q)/cos(q)cot(q) = 1/tan(q)


Inverse Functions
arcsine(opp/hyp) = qarccosecant(hyp/opp) = q
arccosine(adj/hyp) = qarcsecant(hyp/adj) = q
arctangent(opp/adj) = qarccotangent(adj/opp) = q

The functions are usually abbreviated:
arcsine (arcsin)
arccosine (arccos)
arctangent (arctan)
arccosecant (arccsc)
arcsecant (arcsec)
arccotangent (arccot).
According to the standard notation for inverse functions (f-1), you will also often see these written as sin-1, cos-1, tan-1 arccsc-1, arcsec-1, and arccot-1Beware, though, there is another common notation that writes the square of the trig functions, such as (sin(x))2 as sin2(x). This can be confusing, for you then might then be lead to think that sin-1(x) = (sin(x))-1, which is not true. The negative one superscript here is a special notation that denotes inverse functions (not multiplicative inverses).


Trig Functions: Unit Modes
The trig functions evaluate differently depending on the units on q. For example, sin(90°) = 1, while sin(90)=0.89399.... If there is a degree sign after the angle, the trig function evaluates its parameter as a degree measurement. If there is no unit after the angle, the trig function evaluates its parameter as a radian measurement. This is because radian measurements are considered to be the "natural" measurements for angles. (Calculus gives us a justification for this. A partial explanation comes from the formula for the area of a circle sector, which is simplest when the angle is in radians).
Calculator note: Many calculators have degreeradian, and grad modes (360° = 2p rad = 400 grad). It is important to have the calculator in the right mode since that mode setting tells the calculator which units to assume for angles when evaluating any of the trigonometric functions. For example, if the calculator is in degree mode, evaluating sine of 90 results in 1. However, the calculator returns 0.89399... when in radian mode. Having the calculator in the wrong mode is a common mistake for beginners, especially those that are only familiar with degree angle measurements.
For those who wish to reconcile the various trig functions that depend on the units used, we can define the degree symbol (°) to be the value (PI/180). Therefore, sin(90°), for example, is really just an expression for the sine of a radian measurement when the parameter is fully evaluated. As a demonstration, sin(90°) = sin(90(PI/180)) = sin(PI/2). In this way, we only need to tabulate the "natural" radian version of the sine function. (This method is similar to defining percent % = (1/100) in order to relate percents to ratios, such as 50% = 50(1/100) = 1/2.)

Trig: Labeling Sides
Since there are three sides and two non-right angles in a right triangle, the trigonometric functions will need a way of specifying which sides are related to which angle. (It is not-so-useful to know that the ratio of the lengths of two sides equals 2 if we do not know which of the three sides we are talking about. Likewise, if we determine that one of the angles is 40°, it would be nice to know of which angle this statement is true.
We need a way of labeling the sides. Consider a general right triangle:
triangle 1 triangle 2
A right triangle has two non-right angles, and we choose one of these angles to be our angle of interest, which we label "q." ("q" is the Greek letter "theta.")
We can then uniquely label the three sides of the right triangle relative to our choice of q. As the above picture illustrates, our choice of q affects how the three sides get labeled.
We label the three sides in this manner: The side opposite the right angle is called the hypotenuse. This side is labeled the same regardless of our choice of q. The labeling of the remaining two sides depend on our choice of theta; we therefore speak of these other two sides as beingadjacent to the angle q or opposite to the angle q. The remaining side that touches the angle q is considered to be the side adjacent to q, and the remaining side that is far away from the angle q is considered to beopposite to the angle q, as shown in the picture.





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