Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus


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1 Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus Objectives: This is your review of trigonometry: angles, six trig. functions, identities and formulas, graphs: domain, range and transformations. Angle Measure Angles can be measured in ways, in degrees or in radians. The following picture shows the relationship between the two measurements for the most frequently used angles. Notice, degrees will always have the degree symbol above their measure, as in 5, whereas radians are real number without any dimensions, so the number 5 without any symbol represents an angle of 5 radians. An angle is made up of an initial side (positioned on the positive xaxis) and a terminal side (where the angle lands). It is useful to note the quadrant where the terminal side falls. Rotation direction Positive angles start on the positive xaxis and rotate counterclockwise. Negative angles start on the positive xaxis, also, and rotate clockwise. Conversion between radians and degrees when radians are given in terms of DEGREES RADIANS: The official formula is θ θ radians 80 Ex. Convert 0 into radians SOLUTION: 0 radians RADIANS DEGREES: The conversion formula is θ radians θ Ex. Convert 5 into degrees. SOLUTION:
2 For your own reference, radian 57.0 A radian is defined by the radius of a circle. If you measure off the radius of the circle, then take the straight radius and curved it along the edge of the circle, the angle this arc marks off measures radian. Arc length: when using radians you can determine the arc length of the intercepted arc using this formula: Arc length (radius) (degree measure in radians) OR s rθ There may be times you ll use variations of this formula. Ex. Find the length of the arc pictured here: SOLUTION: s rθ you know the values of r and θ 5 5 s 5 units for the arc length.
3 The Trigonometric Ratios The six trigonometric ratios are defined in the following way based on this right triangle and the angle θ adj. adjacent side to angle θ opp. opposite side to angle θ hyp. hypotenuse of the right triangle SOH CAH TOA sinθ opp. hyp. cosθ adj. hyp. tanθ opp. adj. Reciprocal functions cscθ hyp. opp. secθ hyp. adj. cotθ adj. opp. Ex. Find the exact values of all trigonometric functions of the angle θ shown in the figure. SOLUTION: first you ll need to determine the rd side using a + b c a + 5 So for the angle labeled θ, ADJACENT, OPPOSITE 5 and HYPOTENUSE a opp 5 sinθ hyp hyp cscθ opp 5 adj cosθ hyp hyp secθ adj opp 5 tanθ adj adj cotθ opp 5 Special Angles The following triangles will help you to memorize the trig functions of these special angles 0 0 5
4 If the triangles are not your preferred way of memorizing exact trig. ratios, then use this table. θ θ RAD sinθ cosθ tanθ cscθ secθ cotθ but even better than this is the unit circle. The trig. ratios are defined as sin t y cost x tan t csct y, x 0 x, y 0 y sect, x 0 x x cot t, y 0 y
5 Domain and Period of Sine and Cosine Considering the trigonometric ratios as functions where the INPUT values of t come from values (angles) on the unit circle, then you can say the domain of these functions would be all real numbers. Domain of Sine and Cosine: All real numbers Based on the way the domain values can start to cycle back over the same points to produce the same OUTPUT over and over again, the range is said be periodic. But still, the largest value that sine and cosine OUTPUT on the unit circle is the value of, the lowest value of OUTPUT is. Range of Sine and Cosine: [, ] Since the real line can wrap around the unit circle an infinite number of times, we can extend the domain values of t outside the interval [ 0, ]. As the line wraps around further, certain points will overlap on the same ( x, y ) coordinates on the unit circle. Specifically for the functions sine and cosine, for any value sint and cost if we add to t we end up at the same ( x, y ) point on the unit circle. Thus sin( t+ n ) sint and cos( t+ n ) cost for any integer n multiple of. These cyclic natures of the sine and cosine functions make them periodic functions. Graphs of Sine and Cosine Below is a table of values, similar to the tables we ve used before. We re going to start thinking of how to get the graphs of the functions y sin x and y cos x. x 0 y sin x y cos x Now, if you plot these yvalues over the xvalues we have from the unwrapped unit circle, we get these graphs. One very misleading fact about these pictures is the domain of the function remember that the functions of sine and cosine are periodic and they exist for input outside the interval [0, ]. The domain of these functions is all real numbers and these graphs continue to the left and right in the same sinusoidal pattern. The range is [,].
6 Amplitude When the sine or cosine function has a coefficient in front, such as the value of a in the equation y asin x or y acos x, this causes the graph to stretch or shrink its yvalues. This is referred to as the amplitude. Ex. Compare the graphs of y cos x y cos x y cos x y cos x amplitude amplitude amplitude ½ amplitude with reflection in xaxis Amplitude is an absolute value quantity. When the coefficient is negative, this causes an xaxis reflection. Period If there is a coefficient within the argument in front of the x, this will change the length of the function s period. The usual cycle for sine and cosine is on the interval 0 x, but here s how this can change Let b be a positive real number. The period of y asin bx and y acos bx is found this way: 0 bx divide by b 0 x b NOTE: If b >, this will cause the graph to shrink horizontally because the period will be less than. If 0 < b <, the graph will stretch horizontally making the period greater than. You ll need to adjust the key points of the graph when the period changes! Key points are found by dividing the period length into increments. Ex. Sketch a graph of y sin( x ) by hand. SOLUTION: The coefficient a will effect the range of the graph. The amplitude is. Now the period is determined by taking the argument expression inside the function and solving this inequality 0 x multiply by on all sides 0 x 8 You should divide the interval 0 x 8 into equal increments.
7 Ex. Sketch the graph of y 8cos(0 x). SOLUTION: The amplitude here is 8. The negative sign means the graph of cosine will be reflected in the xaxis. The period for the graph will be 0 0x 0 x 5 The period here is so much smaller than usual, when you graph it on the calculator, it looks too narrow. You ll need to scale the graph down so you can get an accurate picture of the wave. One full period of this graph is shown in red above. Don t forget, just because you re only graphing one full cycle of the function doesn t mean it stops there these graphs continue on in a periodic motion. Translations or Phase Shift For the equation y asin( bx+ c) and y acos( bx+ c) you can determine the phase shift in a way similar to determining the period of the function. c c Set up an inequality 0 bx + c solve for x x b b This new interval represents where the usual cycle for the sine or cosine graph gets shifted to on the xaxis. Ex. Sketch the function y 0.5cos( x ) +. SOLUTION: Amplitude 0.5, Period, now determine the phase shift interval. 7 0 x + subtract x x 7 So, one full cycle of this function s graph will be on the interval x. After you determine the interval for the phase shift, I recommend labeling the xaxis first with all the critical points. Don t position the yaxis until you ve labeled all the points first, then you can decide where the yaxis should fall. The critical points are at x,,, 5 and 7
8 Ex. Graph the function f( x) sin( x ) SOLUTION: This one s got it all! Amplitude with a reflection, period b, phase shift 0 x x + x AND a vertical shift by units. The vertical shift is easy to manage, just prepare the function as you would normally, but shift the xaxis portion of the graph up units in the ydirection. On these vertically shifted problems, it may help to draw in a dotted xaxis to help determine your critical points and sketch the graph. Then when you determine where the y axis falls, you can draw a solid xaxis where it should go. Identities and Formulas Here s a listing of some of the various formulas and identities from trig. which we ll use through calculus. Reciprocal Identities sin u cscu cos u secu tan u cot u cscu sin u sec u cosu cot u tanu Pythagorean Identities sin u+ cos u also cos u sin u and sin u cos u + tan u sec u also tan sec u u and sec u tan u + cot u csc u also cot csc u u and csc u cot u Quotient Identities sin u tanu cosu cosu cot u sinu Even / Odd Identities ODDS sin( u) sinu csc( u) cscu tan( u) tanu cot( u) cot u EVENS cos( u) cosu sec( u) secu
9 These formulas will be used a little less often, but when you need them they are located inside your book cover. You will not have to memorize these. Angle Sum and Difference Formulas sin( α + β) sinα cosβ + cosα sinβ sin( α β) sinαcosβ cosαsinβ cos( α + β) cosα cosβ sinα sinβ cos( α β) cosαcosβ+ sinαsinβ tanα + tanβ tanα tanβ tan( α+ β) tan( α β) tanαtanβ + tanαtanβ The Double Angle Formulas For Sine For Cosine For Tangent cosu cos u sin u tanu sin u sinucosu cosu cos u tan u tan u cosu sin u Power Reduction (a.k.a. HalfAngle) Formulas cosx sin x + cosx cos x The Product to Sum formulas Sum to Product Formulas ( ) ( ) ( ) ( ) ( ) ( ) sinα cos β sin α + β + sin α β cosα cos β cos α + β + cos α β sinα sin β cos α β cos α +β x + y x y sin x+ sin y sin cos x + y x y sin x sin y cos sin x + y x y cos x+ cos y cos cos x + y x y cos x cos y sin sin
10 Trigonometric Equations We ll have to solve some trigonometric equations incidentally throughout calculus. Here s a quick review on how the unit circle can help solve these equations. Ex) Solve the equation cscx + 0 on the interval [0, ). SOLUTION: First: Solve for the trig function involved cscx + 0 cscx csc x csc x If csc x then this also means sin x this will help you identify the angles on the unit circle. Second: Identify the angles which satisfy the equation: Sine is defined by as the ycoordinates on the unit circle. You re looking for the unit circle angles where the ycoordinate is /. This happens in two places 5 x and x. The solutions are 5 x and x.
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