ANSWERS: 7
  • They relate the ratio of sides of a triangle.. E.g. Imagine a right angle triangle, i'll try draw one .................../| ................H./.| S1 ................./..| ................/S2_| We'll call the angle between H and Side 2, theta1. 2 and 1, is a right angle, and h and S1, theta 2. Sin Theta1 = S1/H Sin Theta2 = S2/H Cos Theta 1 = S2/H Cos Theta2 = S1/H Tan theta 1 = S1/S2, Tan Theta 2 = S2/S1 that's the basics of trigs, the way I learnt to remember it is... Some Old Hags (S = O/H) Cannot Always Hide (C = A / H) Their Old Age (T = O/A) E.g. Opposite = O, Adjacent = A, Hypotenuse = H
  • For a right angled triangle the sine is the ratio between the opposite side from the angle and the hypotenuse: sin A = opposite/hypotenuse Cosine is the ratio between the adjacent side from the angle and the hypotenuse: cos A = adjacent/hypotenuse Tangent is the ratio between the sine and the cosine: tan A = sin/cos Now there are many equations using these, i.e. sin^2 theta = 1 - cos^2 theta, sin (A +/- B) = sin A * cos B +/- cos A * sin B etc. but the only thing you need to realize is that these are merely equations using the functions of sine, cosine and tangent. There is a rule of remembering which is which but I don't know if it'll help: SOHCAHTOA (think of it as an native american word); sin A = O/H, cos A = A/H & tan A = O/A (the tan = O/A means tan A = the operation for the Opposite, 'sine', over the operation for the Adjacent, 'cosine'). There are the inversions: cosec A = 1/sin A, secant A = 1/ cos A & cotangent = 1/tan A. These are abreviated to csc A, sec A & cot A. A good way of remembering which is which is the third letter of the name is the function inverted: coSec A = 1/Sin A, seCant A = 1/Cos A & coTangent A = 1/Tan A.
  • sin – the sine function cos – the cosine function tan – the tangent function csc – the cosecant function sec – the secans function cot – the cotangent function sin(x) represents the sine function. cos(x) represents the cosine function. tan(x) represents the tangent function sin(x)/cos(x). csc(x) represents the cosecant function 1/sin(x). sec(x) represents the secant function 1/cos(x). cot(x) represents the cotangent function cos(x)/sin(x). The arguments have to be specified in radians, not in degrees. E.g., use to specify an angle of . All trigonometric functions are defined for complex arguments. Floating point values are returned for floating point arguments. Floating point intervals are returned for floating point interval arguments. Unevaluated function calls are returned for most exact arguments. Translations by integer multiples of are eliminated from the argument. Further, arguments that are rational multiples of lead to simplified results; symmetry relations are used to rewrite the result using an argument from the standard interval . Explicit expressions are returned for the following arguments: . Cf. example 2. The result is rewritten in terms of hyperbolic functions, if the argument is a rational multiple of I. Cf. example 3. The functions expand and combine implement the addition theorems for the trigonometric functions. Cf. example 4. The trigonometric functions do not respond to properties set via assume. Use simplify to take such properties into account. Cf. example 4. sec(x) and csc(x) are immediately rewritten as 1/cos(x) and 1/sin(x), respectively. Use rewrite to rewrite expressions involving tan and cot in terms of sin and cos. Cf. example 5. The inverse functions are implemented by arcsin, arccos, arctan, arccsc, arcsec, and arccot, respectively. Cf. example 6. The float attributes are kernel functions, i.e., floating point evaluation is fast. Example 1: We demonstrate some calls with exact and symbolic input data: sin(PI), cos(1), tan(5 + I), csc(PI/2), sec(PI/11), cot(PI/8) sin(-x), cos(x + PI), tan(x^2 - 4) Floating point values are computed for floating point arguments: sin(123.4), cos(5.6 + 7.8*I), cot(1.0/10^20) Floating point intervals are computed for interval arguments: sin(0 ... 1), cos(20 ... 30), tan(0 ... 5) For the functions with discontinuities, the result may be a union of intervals: csc(-1 ... 1), tan(1 ... 2) Example 2: Some special values are implemented: sin(PI/10), cos(2*PI/5), tan(123/8*PI), cot(-PI/12) Translations by integer multiples of are eliminated from the argument: sin(x + 10*PI), cos(3 - PI), tan(x + PI), cot(2 - 10^100*PI) All arguments that are rational multiples of are transformed to arguments from the interval : sin(4/7*PI), cos(-20*PI/9), tan(123/11*PI), cot(-PI/13) Example 3: Arguments that are rational multiples of I are rewritten in terms of hyperbolicfunctions: sin(5*I), cos(5/4*I), tan(-3*I) For other complex arguments, use expand to rewrite the result: sin(5*I + 2*PI/3), cos(5/4*I - PI/4), tan(-3*I + PI/2) expand(sin(5*I + 2*PI/3)), expand(cos(5/4*I - PI/4)),expand(tan(-3*I + PI/2)) Example 4: The expand function implements the addition theorems: expand(sin(x + PI/2)), expand(cos(x + y)) The combine function uses these theorems in the other direction, trying to rewrite products of trigonometric functions: combine(sin(x)*sin(y), sincos) The trigonometric functions do not immediately respond to properties set via assume: assume(n, Type::Integer): sin(n*PI), cos(n*PI) Use simplify to take such properties into account: simplify(sin(n*PI)), simplify(cos(n*PI)) assume(n, Type::Odd): sin(n*PI + x), simplify(sin(n*PI + x)) y := cos(x - n*PI) + cos(n*PI - x): y, simplify(y) delete n, y: Example 5: Various relations exist between the trigonometric functions: csc(x), sec(x) Use rewrite to obtain a representation in terms of a specific target function: rewrite(tan(x)*exp(2*I*x), sincos), rewrite(sin(x), cot) Example 6: The inverse functions are implemented by arcsin, arccos etc.: sin(arcsin(x)), sin(arccos(x)), cos(arctan(x)) Note that arcsin(sin(x)) does not necessarily yield x, because arcsin produces values with real parts in the interval : arcsin(sin(3)), arcsin(sin(1.6 + I)) Example 7: Various system functions such as diff, float, limit, or series handle expressions involving the trigonometric functions: diff(sin(x^2), x), float(sin(3)*cot(5 + I)) limit(x*sin(x)/tan(x^2), x = 0) series((tan(sin(x)) - sin(tan(x)))/sin(x^7), x = 0) It's been a very long time... I hope this helps.
  • I would have to say that the trig functions very simply, are a ratio of the sides of a triangle when compared to a certain angle in a right triangle. THe sine function is the ratio of a side opposite the angle over the hypotenuse. The cosine function is the ratio of the side closest to the angle over the hypotenuse. The tangent function is the ratio of the side opposite the angle over the side closest to the angle, but not the hypotenuse. Sine and cosine values vary between -1 and 1, while tangent goes from - infinity to infinity. The values are also periodic, and repeat every 2 pi. To help you better understand if you still don't get it, I would suggest finding a graph of sine and cosine together, and also a chart for the values of sine, cosine, tangent, etc, with radians and degrees. I'll make you one if you need me to.
  • I would personally, as well as the other answers, look into the Taylor series expansions of the functions. It all makes alot more sense when you use the series expansions, you can see the relationship of the functions to each other better. It depends at what level you want to know it all I suppose. SOHCAHTOA is fine if you are just using the fncs, but infinite series are the way to get your head wrapped around what they are. My maths lecturers always hammered home that the inifinite series were the definitions of the functions, everything else was just useful shortcuts.
  • It's just ratios. Think of it as ratios and nothing more. When there is a right triangle, the hypotenuse, opposite, and adjacent leg always have the same ratio to each other, and trig uses the ratios to figure out unknowns.
  • check out these articles for a simple tool and tutorial that will make trig simple enough for ANYBODY to do! http://www.ehow.com/how_5428511_pass-part-ii-unknown-angles.html http://www.ehow.com/how_5227490_pass-mind-part-unknown-sides.html

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