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Part 4: Trigonometry

Part 4: Trigonometry. In part 1 we talked about locating things in two dimensions. We restricted ourselves in that part to the Cartesian system, (x,y), but mentioned there were other systems.

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Part 4: Trigonometry

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  1. Part 4: Trigonometry In part 1 we talked about locating things in two dimensions. We restricted ourselves in that part to the Cartesian system, (x,y), but mentioned there were other systems. Many of the other systems employ at least one angle. Therefore, we must talk about what an angle is and what are the measures we can use to quantify those angles.

  2. Trig:Angles Because space is three dimensional (we’ll talk about this soon), we need angles. What is an angle? How do you measure an angle? θ

  3. Trig: Angles How do you measure an angle? 1) in (fractions of) circles (cycles, rotations, revolutions) 2) in degrees - but what is a degree? Why do they break the circle into 360 equal degrees? 3) in radians - but why use a weird number like 2p for a circle? Why does p have the weird value of 3.1415926535… ?

  4. Trig: Angles Full circle = 360o (Comes from year - full cycle of seasons is broken into 365 days; but 365 is awkward number; use “nicer” number of 360.) Full circle = 2p radians (Comes from definition of angle measured in radians: q = arclength / radius = s / r) s q r

  5. Trig: basic functions Besides angles, we can define functions associated with angles. The trig functions are based on a right triangle: 

  6. Trig: basic functions • sin() = opposite/hypotenuse = y/r • cos() = adjacent/hypotenuse = x/r • tan() = opposite/adjacent = y/x • (The hypotenuse is the side opposite the right angle.) r y  x

  7. Trig: sine function The easiest way to see what the function, sin(θ), looks like is to look at the unit circle where r=1. Then sin(θ) = y/r = y. sin(θ) y 1.0 0.5 1800 90o θ x πradians π/2 radians

  8. Trig: sine function Notice that sin(θ) is symmetric about the y axis of the circle, so sin(θ) is symmetric about 90o or ½π radians (i.e., sin[90o+θ] = sin[90o-θ]). y sin(θ) 1.0 0.5 1800 90o θ x πradians π/2 radians

  9. Trig: sine function Notice that sin(θ) is negative between 180o and 360o. y sin(θ) πradians 3π/2 radians 2πradians 1800 3600 270o θ x -0.5 -1.0

  10. Trig: sine function Notice that sin(θ) repeats after 360oso that sin(θ) = sin(θ+360o). Notice that this makes sin(θ) antisymmetric about the origin which makes it an odd function so that sin(θ) = -sin(-θ). Notice sine is antisymmetric about 180o so sin[180o+θ] = -sin[180o-θ]. sin(θ) 180o 360o θ

  11. Trig: cosine function The easiest way to see what the function, cos(θ), looks like is to look at the unit circle where r=1. Then cos(θ) = x/r = x. sin(θ) y 1.0 0.5 1800 90o θ x πradians π/2 radians -0.5 -1.0

  12. Trig: cosine function We can also use some symmetries to see what cosine looks like. Note that from the definition of sine and cosine and using the unit circle are: sin(θ) = y/r = y andcos(θ) = x/r = x. Since we are dealing with a circle, the cosine function will have exactly the same shape as the sine function, except that it starts at a different place. Sine is maximum at 90o, while cosine is maximum at 0o. (See next slide for graphs).

  13. Trig: cosine function Notice also that cos(θ) is symmetric about the x axis on the circle and y axis of the graph, so cosine is an even function:cos[+θ] = cos[-θ]. sin(θ) cos(θ) 180o 360o θ

  14. Trig: sine and cosine functions Notice that since x and y are different by 90o (horizontal versus vertical on the circle), so sin(θ+90o) = cos(θ) , and sin(θ) = cos(θ-90o) which you can see from the diagram. Also since cosine is an even function, cos(θ) = cos(- θ), cos(θ-90o) = cos(90o- θ)= sin(θ). sin(θ) cos(θ) 180o 360o θ

  15. Trig: sine and cosine functions cos(90o- θ)= sin(θ) This last relation can also be seen from the right triangle diagram and the definitions.The hypotenuse is r for both angles θ and 90o-θ. Therefore, we havesin(θ) = y/r = cos(90o-θ) and cos(θ) = x/r = sin(90o-θ). . r 90o-θ y θ x

  16. Trig: sine and cosine functions Our calculators provide us with a nice way of evaluating both sine and cosine functions. However, we need to always be aware of whether we are using degrees or radians for the angle. Know how to check your calculator to see what it is expecting, and know how to change your calculator to make it work with either degrees or radians based on the type of problems you are encountering.

  17. Trig: sine and cosine functions Why do we continue to have two different major units for angles: degrees and radians? As strange as it may seem, the answer is basically one of convenience. The degree goes way back in history but it is an arbitrary unit that is related to us being on the earth with a year of 365 days. The main advantage is that there are an integer number of degrees in a circle which is an advantage when measuring. The radian measure does NOT have an integer number in one circle, but it is geometrically defined (and so is used a lot in math courses).

  18. Trig: sine and cosine functions Note that once the angle has made a complete circle (gone past 360o = 2π radians), both the sine and cosine functions start repeating themselves:sin(θ+360o) =sin(θ)=sin(θ+2π radians)cos(θ+360o) =cos(θ)= cos(θ+2π radians) Note also that both sine and cosine are limited to a range between –1 and 1.

  19. Trig: sine and cosine functions We can easily calculate several values of sine and cosine from basic geometry: for a 45o angle, x = y, so r2 = 1 = x2 + y2 = x2 + x2 = 2x2, or x = SQRT(1/2) = y, so sin(45o) = √½ = cos(45o).For the 60o angle, cos(60o) = [½r / r] = ½ = sin(30o); sin(60o) = y/r where (with r=1) y = SQRT(r2 – [r½]2) = (SQRT[3]/2) so sin(60o) = √(3)/2 = cos(30o). r r r y=x 60o 45o r/2 x

  20. Trig: sine and cosine functions Here are a few values for sine and cosine that you should know. It helps you keep track of where you are on the oscillating sine and cosine graphs: sin(0o) = cos(90o) = 0 0o = 0 rads sin(30o) = cos(60o) = 0.500 30o = π/6 rads sin(45o) = cos(45o) = 0.707 = √(2)/2 45o = π/4 rads sin(60o) = cos(30o) = 0.866 = √(3)/2 60o = π/3 rads sin(90o) = cos(0o) = 1.00090o = π/2 rads

  21. Trig: sine and cosine functions Because of symmetries, we can extend the table from the previous slide: sine is symmetric about 90o, cosine is antisymmetric about 90o. sin(120o) = sin(60o) = 0.866 = √(3)/2 120o = 2π/3 rads sin(135o) = sin(45o) = 0.707 = √(2)/2 135o = 3π/4 rads sin(150o) = sin(30o) = 0.500 150o = 5π/6 rads sin(180o) = sin(0o) = 0.000180o = π rads cos(120o) = -cos(60o) = -0.500 cos(135o) = -sin(45o) = -0.707 = -√(2)/2 cos(150o) = -cos(30o) = -0.866 = -√(3)/2 cos(180o) = -cos(0o) = -1.000

  22. Trig: sine and cosine functions For angles between 180o = π radians and 270o = 3π/2 radians, both sine and cosine simply change sign when 180o = π radians is added: sin(θ+180o) =-sin(θ)=sin(θ+π radians) cos(θ+180o) =-cos(θ) = cos(θ+π radians)

  23. Trig: sine and cosine functions For angles between 270o = 3π/2 radians and 360o = 2π radians, we can convert to negative angles between –90o = -π/2 radians and 0o by using the fact that both functions repeat over one cycle (±360o = 2π radians) Sine is an antisymmetric (odd) function and cosine is a symmetric (even) function: sin(θ) =-sin(-θ) ; sin(θ-360o) =-sin(360o-θ) cos(θ) = cos(-θ) ; cos(θ-360o) = cos(360o -θ)

  24. Trig functions –domain and range Earlier we defined the terms: domain and range. For linear functions, the domain and range were both usually (-∞, +∞). For quadratic functions, the domain was usually (-∞, +∞), while the range usually went to infinity on one side but was limited to the y value of the vertex on the other side. For exponential functions, the domain was usually (-∞, +∞), but the range was usually (0, +∞). For log functions, the domain was usually (0, +∞) while the range was (-∞, +∞). Note that the domain and range flip between the function and its inverse.

  25. Trig functions – domain and range But with the functions of sine and cosine, we can see a real use for the concepts of domain and range. The normal ranges for both sine and cosine are [-1,1]. While the domain for both sine and cosine can be (-∞, +∞), if we limit the domain to be [0, 90o], both sine and cosine are one-to-one functions and inverse functions should exist.

  26. Trig functions – domain and range Limiting the trig functions to a domain of [0, 90o], however, does not let them go through their full range of [-1,+1]. For the sine function, we need a domain of [-90o, +90o], and for the cosine we need a domain of [0, 180o].

  27. Trig functions: slope We have seen before that the slope of a straight line is constant, the slope of a parabola depends on the x value, and the slope of an exponential depends on the y value. What about the slope of an oscillating function (sine and cosine)?

  28. Trig functions: slope Notice that near θ = 0o the slope is positive, but that the slope gets less positive until it reaches zero slope at θ = 90o and then the slope become negative and reaches its steepest negative at about θ = 180o and then it becomes less steep until it again reaches zero slope at θ = 270o and then becomes positive until it reaches its steepest slope at θ = 360o . So we see that the slope of the sine wave alsooscillates similar to what the sine function does. However, the slope oscillates 90o out of phase with the function. sin(θ) 180o 360o θ

  29. Trig functions: slope What about the slope of an oscillating function (sine and cosine)? We had a nice picture on the previous slide, but we can calculate the difference quotient (slope near a point) using the sine function to get a more quantitative answer. You might want to calculate the difference quotients near the following points (with h = Δθ = .01 radians): 0 rads, π/6 rads, π/4 rads, π/3, rads, π/2 rads, 2π /3 rads, 3π/4 rads, π rads, 3π/2 rads, and 2π rads. What does this remind you of? Would we get similar results for the cosine function?

  30. Inverse trig functions Recall that an inverse “undoes” what the original function did. In general, an inverse function, g(x), operating on the original function, f(x), gives the original value of x back again: g(f(x)) = x. We call the inverse of the sine function the Arcsin(x) or sin-1(x). Note that sin-1[sin(θ)] = θ. Since the value of sin(θ) is a number between –1 and 1, (the range of the sine function), the domain of the sin-1(x) function is a number between –1 and 1, and the range is an angle between –90o and +90o .

  31. Inverse trig functions sin(θ) has domain [-90o, 90o] & range [-1, 1] sin-1(x) has domain [-1, 1] & range [-90o, 90o]. In a similar way, cos(θ) has domain [0o, 180o] & range [-1, 1] cos-1(x) has domain [-1, 1] & range [0o, 180o]. To see the different ranges of arcsine and arccosine, try using your calculator to find sin-1(-0.5) and then find cos-1(-0.5).

  32. Inverse trig functions Can you draw the graph of the arcsine (sin-1) and the arccosine (cos-1) functions? You need to be careful since the x axis will have to be in numbers and the y axis will have to be in either degrees or radians! Will there be any asymptotic behavior? Are there any symmetries? +1 sin(θ) -90o θ +90o -1

  33. Inverse trig functions Note that there are no asymptotic behaviors. Note that the sine function is an odd function so it is symmetric through the origin, so the sin-1 function should also be antisymmetric. +90o +1 sin-1(x) sin(θ) -90o θ +90o -1 x +1 -1 -90o

  34. Inverse trig functions The graphs look the same but the dotted symmetry line is different if you use radians instead of degrees. From this, we see that if you use radians and have small angles, sin(θ) ≈θ .For example, sin(0.1 rad) = .0988 ≈ 0.1 . π/2 +1 sin-1(x) sin(θ) π/2 θ +π/2 -1 x +1 -1 -π/2

  35. Inverse trig functions Can you draw the graph of the arcsine and the arccosine functions? You need to be careful since the x axis will have to be in numbers and the y axis will have to be in either degrees or radians! Any asymptotes or symmetries? +1 cos(θ) 90o θ 180o -1

  36. Inverse trig functions Again no asymptotes, but there is an antisymmetry through 90o . +180o +1 cos-1(x) cos(θ) 90o 90o θ 180o -1 -1 +1 x

  37. Trig: tangent function While the sine and cosine functions relate the rectangular (Cartesian) coordinates (x,y) to the polar coordinates (r, θ), there is another function that directly relates θ to x and y without going through the radius, r. It is the tangent function: tan(θ) = y/x. This function is directly related to the sine and cosine functions: tan(θ) = y/x = (y/r) / (x/r) = sin(θ) / cos(θ).

  38. Trig: tangent function tan(θ) = y/x = sin(θ) / cos(θ) This function is a little more “tricky” than either sine or cosine. Can you see where the “tricky” parts come in?

  39. Trig: tangent function tan(θ) = y/x = sin(θ) / cos(θ) Since the value of cos(θ) equals zero when θ = 90o = π/2 radians, the tangent function will approach infinity as θ approaches 90o from below 90o, but since cos(θ) is less than zero (negative) for θ greater than 90o, the tangent function will approach negative infinity as θ approaches 90o from above 90o. This gives asymptotic behavior!

  40. Trig: tangent function tan(θ) = y/x = sin(θ) / cos(θ) The second “tricky” thing about tangent is trying to determine a domain for it in which an inverse function can be defined. The normal domain for sine to define the inverse sine function was [-90o, 90o], while the normal domain for the cosine to define the inverse cosine was [0, 180o]. What should the normal domain be for the tangent function to define the inverse tangent function, called arctan(x) or tan-1(x) ?

  41. Trig: arctangent function One way is simply to plot the tan(θ) function and then plot the tan-1(x) relation:

  42. Trig: arctangent function As you can see, the tan(θ) function is one-to-one on the interval from –90o to +90o, and alternately on the interval from 0o to 180o. The usual choice for use with the inverse tangent is from –90o to +90o . Using your calculator, see whether tan-1(-0.5) is less than 0o or greater than +90o. +3 tan(θ) -90o θ +90o +180o +270o -3

  43. Trig: arctangent function Note that tan(θ)≈ θ (red curve below almost falls on dotted line) for angles close to zero as long as the angle is measured in radians. Try checking this on your calculator: tan(.1 rads) ≈ .100335 .1 rads = 5.73o tan(.3 rads) ≈ .309336 .3 rads = 17.2o What will the tan-1(x) function (in this range) look like? Note that tangent is anti- symmetric (odd function), and note the asymptotic behavior near ±90o . +3 tan(θ) -90o θ +90o -3

  44. Trig: arctangent function Since tan(θ)goes through the origin and has anti-symmetry, tan-1(x) will also go through the origin and have antisymmetry. Since tan(θ)approaches infinity as the angle approaches 90o (vertical asymptote), tan-1(x) will approach 90o as x approaches infinity (horizontal asymptote). +90o +3 tan-1(θ) tan(θ) -90o θ +90o x -3 +3 -90o -3

  45. Trig: example of vectors One of the main uses of the trig functions is in the expression of vectors. A vector is simply a quantity that not only has a magnitude (amount), but it also has a direction.

  46. Trig: Vectors How do we work with a quantity that needs two (or more) numbers to specify it (like position does)? We can work either with a group of numbers sometimes put in parenthesis, or we can work with unit vectors that we add together: (x,y) or x*x + y*y(where x indicates the x direction and x indicates how far in that direction).

  47. Trig: Vectors • The individual numbers in the vector are called components of the vector. • There are two common ways of expressing a vector in two dimensions: rectangular and polar.

  48. Trig: Vectors • Rectangular (x,y) is often used on city maps. Streets generally run East-West or North-South. The distance East or West along a street give one distance, and the distance along the North-South street gives the second distance - all measured from some generally accepted origin.

  49. Trig: Vectors • If you are at home (the origin), and travel four blocks East and then three blocks North, you will end up at position A. • Position A (relative to your house) is then (4 blocks, 3 blocks) where the first number indicates East (+) or West (-), and the second number indicates North (+) or South (-).

  50. Trig: Vectors • If you had a helicopter or could walk directly there, it would be shorter to actually head straight there. How do you specify 3the location of pointA this way? 4

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