MATH 00/58: Trigonometry Minh-Tam Trinh For the trigonometry unit, we will cover the equivalent of 0.7,.4,.4 in Purcell Rigon Varberg.. Right Triangles Trigonometry is the stuy of triangles in the plane (i.e., -imensional space). The wor comes from Greek trigōnon + metron, meaning triangle + measure. Our broaer goal is to unerstan polygons, i.e., any kin of shape in the plane whose sies are straight line segments. But there are no polygons with only or sies, so the simplest case is sies, i.e., triangles. Also, any polygon with more than sies can be subivie into a bunch of triangles. Usually, we ll be given some of the angles an sie lengths of a triangle, an we want to euce the other angles an sie lengths. As far as this question, it turns out the easiest triangles to stuy are right triangles, those with a right (90 ı ) angle. Of these, the easiest are what we will call 0-60-90 triangles an 45-45-90 triangles... Why right triangles? Below is an example of a right triangle: The sie opposite the right angle (i.e., the re sie) is calle the hypotenuse. The Pythagorean theorem states that if ; ; are the lengths of the sies with those colors, then C D : In short, if you know two sie lengths in a right triangle, then it s easy to fin the remaining one.
MATH 00/58: Trigonometry.. Why 0-60-90 an 45-45-90? A 0-60-90 triangle looks like while a 45-45-90 triangle looks like (Source: Khan Acaemy.) These are the only right triangles where both () the angles an () the proportions between ifferent sies are all nice numbers. The reason for this is actually somewhat subtle, but I will try to explain it below. A regular polygon is one where all the sies have the same length. A -sie regular polygon is an equilateral triangle, a 4-sie regular polygon is a square, etc.: Only three regular polygons have the special property that you can use them to tile the plane in a completely perioic way, as happens in a crystal lattice. The crystallographic regular polygons are the equilateral triangle, the square, an the equilateral hexagon (resp.,, 4, an 6 sies): https://en.wikipeia.org/wiki/regular_polygon https://en.wikipeia.org/wiki/eucliean_tilings_by_convex_regular_polygons
MATH 00/58: Trigonometry This special feature of the numbers, 4 an 6 plays a significant role in a variety of human conventions: E.g., why an analog clock has numbers, why a ay has 4 hours, an why we ivie a full revolution into 60 egrees. To return to trigonometry, the 0-60-90 an 45-45-90 triangles are special because they appear naturally in the geometry of the crystallographic regular polygons: In fact, this observation gives a way to erive the proportions of the sie lengths in these special right triangles. () A 0-60-90 triangle is half of an equilateral triangle: Since all sies in an equilateral triangle have the same length, we euce that D. By the Pythagorean theorem, this forces () A 45-45-90 triangle is half of a square: W W D W p W : Since all sies in a square have the same length, we euce that D. By the Pythagorean theorem, this forces W W D W W p : Above, W W means the proportions of these numbers relative to each other. For example, p p equals in a 0-60-90 triangle an in a 45-45-90 triangle.
MATH 00/58: Trigonometry 4.. Degrees vs Raians Up until now, we ve been measuring all angles in egrees. But from a purist s viewpoint, the ivision of one revolution into 60 egrees is fairly arbitrary. We can instea measure angles in raians: 60 egrees D revolution D raians: Sometimes, raians is abbreviate ra, but more often the wor is just omitte. Below is a conversion chart for some special angles, incluing the ones in 0-60-90 an 45-45-90 triangles. egrees 0 ı 0 ı 45 ı 60 ı 90 ı 0 ı 5 ı 50 ı 80 ı raians 0 6 4 4 5 6 Observe, once again, the appearance of the special numbers, 4, an 6. In the rest of this course, all angles will be in raians unless explicitly marke with the egree symbol ı.. Trigonometric Functions To stuy more general right triangles, we introuce trigonometric functions. The iea behin them is as follows: First, if we know a single angle besies the right angle, then we also know the thir angle, as it must be 80 ı 90 ı D 90 ı. Secon, if we know all of the angles in a triangle, then we know all of the ratios between ifferent sie lengths. We euce that the sie-length ratios are functions that epen only on. If we know these functions together with a single sie length, then we can compute everything about the triangle... Definition via HAO Let be a non-right angle of a right triangle. Label the sies as follows: () H is the hypotenuse. () A is the sie ajacent to that isn t the hypotenuse. () O is the sie opposite. The sine, cosine, an tangent functions are efine by sin D O H ; cos D A H ; tan D O A ; respectively. You can tell them apart using the mnemonic SOH CAH TOA, or S C T O A O H H A
MATH 00/58: Trigonometry 5 Their reciprocals are the cosecant, secant, an cotangent functions efine by csc D H O ; sec D H A ; cot D A O ; sin respectively. Personally, I prefer to write out iscretion. Below are the values of the first three trig functions for special angles: instea of csc, etc., but I leave it to your 0 6 4 sin 0 p p cos p p 0 Note the symmetry above! Also, note that 0 an particular tan an cot 0 are unefine. correspon to egenerate triangles; in.. Definition via the Unit Circle In the xy-plane, consier the circle of raius centere at the origin: Any point.x; y/ on this circle is etermine by the angle forme as we rotate counterclockwise from.; 0/ to.x; y/. That is, the coorinates x an y are functions of. What are they? If.x; y/ is in the first quarant, then x an y are positive numbers that satisfy x C y D : So we can raw a right triangle of sie lengths x, y, an, in which is the angle opposite y. Plugging.H; A; O/ D.; x; y/ into the preceing iscussion, we fin sin D y; cos D x; tan D y x : In other wors, if 0 < <, then.x; y/ D.cos ; sin /. For other values of, the corresponing values of x an y entail a natural extension of cosine an sine. For example, sin D, because if we rotate raians, or 70 egrees, counterclockwise aroun the unit circle from.; 0/, then we are at.0; /.
MATH 00/58: Trigonometry 6 Finally, observe that when we write x D cos an y D sin, the equation for the unit circle becomes the trigonometric ientity cos C sin D : (Above, cos means.cos /, etc.) For right triangles, the ientity above merely says that O C A D, a restatement of the Pythagorean ientity O C A D H. H H.. Graphs of Trig Functions Plotting on the horizontal axis, here is the graph of sin : The graph of cos : The graph of tan :
MATH 00/58: Trigonometry 7 Note how the graphs illustrate the ientities sin. C / D sin ; cos. C / D cos ; tan. C / D tan ; as well as the ientities sin. / D sin ; cos. / D cos ; tan. / D tan ; via a consieration of their symmetries.. Some Sum Formulae Here is a very useful pair of formulae: sin. C ˇ/ D.sin /.cos ˇ/ C.cos /.sin ˇ/; cos. C ˇ/ D.cos /.cos ˇ/.sin /.sin ˇ/: I won t prove them in this course, but once you learn about Taylor series an the natural exponential function, there is a very beautiful proof. To get the corresponing formulae for sin. ˇ/ an cos. ˇ/, just substitute ˇ for ˇ... Double- an Half-Angle Formulae If D D ˇ, then the formulae reuce to sin./ D sin cos ; cos./ D cos sin : Remembering that cos C sin D, we can rewrite the latter result in two ways: cos./ D. sin / sin cos./ D cos C.cos / D sin : D cos : Equivalently, cos D. that sin.// D. C cos.//. Substituting for, we euce sin D r cos ; cos r C cos D : q q For example, sin = 6 D sin D cos = = D D p =4 D =, as expecte. But we can also use the half-angle formulae to compute the values of trig functions at new angles like or 5 8, an hence at values like D C or 8 D 8 C 4.
MATH 00/58: Trigonometry 8 Example. Here is a neat trick that I emonstrate in class. We can rewrite the half-angle formula for cosine as: cos D p C cos : Plugging in D, 4, 8, 6, etc., etc. cos 4 D p ; cos 8 D q C p ; cos q r 6 D C C p ;.. Triple-, Quaruple-, Quintuple-,... If D an ˇ D, then we can use the original formulae in combination with the ouble-angle formulae to show: sin./ D sin./ cos C cos./ sin D sin cos sin cos./ D cos./ cos C sin./ sin D cos C sin cos : Can you fin a formula for sin.4/? For cos 5? How about tan./? 4. Calculus of Trigonometric Functions As this is a calculus class, we re more intereste in the limit properties of trig functions than in trigonometry per se. 4.. Limits The two trigonometric limits to memorize for now are: sin lim!0 cos D an lim!0 D 0: The first follows from the fact that if is close to 0 (i.e., very small), then sin. The secon follows from the fact if is very small, then cos. To illustrate, here is a simultaneous plot of sin an :
MATH 00/58: Trigonometry 9 Here is a simultaneous plot of cos an : One more thing: Look at the graph of tan. Not only oes tan fail to exist at D : : : ; ; ; ; : : :, but the limit of tan at those values oes not exist either. 4.. Derivatives of Sine an Cosine It turns out sin D cos an cos D sin : This shoul be believable from the graphs of sine an cosine. (E.g., what s the slope of sin at D 0?) Below, I explain how to prove sin D cos rigorously using the two limits from the previous subsection. First, sin D lim!0 sin. C / sin.sin /.cos / C.cos /.sin / sin D lim!0 cos sin D.sin / lim C.cos / lim!0!0 : cos Now, plugging in lim!0 cos. D 0 an lim!0 sin D, the last expression simplifies to
MATH 00/58: Trigonometry 0 4.. Derivatives of the Other Trig Functions If you memorize the erivatives of sine an cosine, then you can work out the erivatives of all the other trig functions using the various rules. For instance, Moreover, an tan D.cos /.sin /0.sin /.cos / 0 cos D cos C sin cos D cos : csc D. sin /.cos / D csc cot sec D. cos /. sin / D sec tan cot D.sin /.cos /0.cos /.sin / 0 sin D sin cos sin D sin : A. The Pythagorean Theorem Here is a proof-without-wors: Source: Imgur.