TRIGONOMETRY AP Calculus AB Summer Review TRIGONOMETRY AND CALCULUS AP Calculus AB, SJHS 2014-2015 Calculus requires a thorough knowledge of the basic functions, which includes the transcendental functions Thus, you are expected to understand polynomial, exponential, logarithmic, and trigonometric functions Trigonometry occurs very often in calculus, and without knowing the algebra of trig functions several calculus concepts will be very difficult 2 ALGEBRA OF TRIGONOMETRY The most basic trigonometry is understanding the algebra trig functions AP Calculus AB, SJHS 2014-2015 This means the methods of rearranging trig functions (as a whole) in addition/subtraction, multiplication/division, and exponents to solve equations This also includes the use of “inverse trig functions” to pull information from a trigonometric equation Several trig equations require a calculator to solve angles in either degrees or radians 3 DEGREES Degree measurement is a standard measurement for angles in some applied sciences, such as mechanics or optics It is based on the idea that a circle turns across 360° for one revolution Degrees are a completely made up unit; they have no physical basis, but are still a useful reference unit AP Calculus AB, SJHS 2014-2015 4 RADIANS The radian is based on the proportion of a circle length to its radius AP Calculus AB, SJHS 2014-2015 Every circle has this relationship, and it helps to define a circle This is not the definition of a circle! By definition, there 2π radians in one revolution of any circle You can also find the length of a sector, area of a sector, and area of a circle (as well as several other quantities) using angles in radians Most math (and theoretical science) is done in radians 5 CONVERSION OF DEGREES TO RADIANS Since we have the following formulae for 1 revolution: We can deduce that the following is true: AP Calculus AB, SJHS 2014-2015 1rev 360 Radians: 1rev 2 Degrees: 2 360 Notice that radians have no units when written down, but if you wish you may write “rad” as a unit (Ex: 2π rad) Thus, if we ever need to convert, we can use either of the following factors 360 or 2 2 360 Note: you may have learned a less significant, but still correct, relationship of 180°=π; this proportion works, but it loses the definition of a revolution It is better to think of angles and radians in terms of revolutions, not just a meaningless simplified proportion 6 EXACT VALUES There are few things that need to be memorized in math, and exact values of some angles are a necessity You should become familiar with the following degree measurements in terms of radians: 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, 330°, 360° Although these degree measurements appear to have no significant connections, they are very closely connected in radians AP Calculus AB, SJHS 2014-2015 7 CONNECTIONS OF THE 30N° ANGLES In radian measurement, the 30n°angles take on Notice that 30° is one-sixth of a semi-circle (pi radians) This makes 60° one-third of a semi-circle AP Calculus AB, SJHS 2014-2015 n reduced values of 6 n By this logic, all 60n° angles take on reduced values of 3 8 CONNECTIONS OF THE 45N° ANGLES In radian measurement, the 45n°angles take on AP Calculus AB, SJHS 2014-2015 n reduced values of 4 45° is one-fourth of a semicircle n By this logic, all 90n° angles take on reduced values of 2 90° is half of a semi-circle By extension, all angles 180n° take on reduced values of n 9 SUMMARY OF EXACT VALUES 0 30 45 60 90 120 135 150 180 Radian 0 π/6 π/4 π/3 π/2 2π/3 3π/4 5π/6 π Degree 210 225 240 270 300 315 330 360 Radian 7π/6 5π/4 4π/3 3π/2 5π/3 7π/4 11π/6 2π AP Calculus AB, SJHS 2014-2015 Degree 10 GRAPHICAL REPRESENTATION OF EXACT VALUES 2 3 4 2 0 5 4 3 2 Fourths 7 4 6 AP Calculus AB, SJHS 2014-2015 5 4 6 3 3 0 7 6 5 3 4 3 Thirds 11 6 11 WHAT SHOULD WE USE: DEGREES OR RADIANS? RADIANS! The only exception is if the problem explicitly uses degrees in its statement, or asks for an answer in degrees AP Calculus AB, SJHS 2014-2015 12 TRIANGLES IN THE COORDINATE PLANE When you draw this triangle, draw an angle arrow on the x-axis, draw a ray starting at the origin in the direction of that angle with length r, and drop a perpendicular to the xaxis Note: r is always positive, but x and y can be positive or negative, depending on the quadrant AP Calculus AB, SJHS 2014-2015 We are given a right triangle of hypotenuse r superimposed onto an x,y-plane; there is an angle θ between the hypotenuse and the x side 13 EX: DRAW A TRIANGLE IN THE X,Y-PLANE WITH HYPOTENUSE 2 AT AN ANGLE AT 50°. 50 AP Calculus AB, SJHS 2014-2015 2 14 EX: DRAW A TRIANGLE IN THE X,Y-PLANE WITH HYPOTENUSE 3 AT AN ANGLE AT 290°. AP Calculus AB, SJHS 2014-2015 290 15 TRIGONOMETRIC FUNCTIONS For a triangle in the x,y-coordinate plane, the three trigonometric functions are defined as: AP Calculus AB, SJHS 2014-2015 y sin r x cos r y tan x 16 THE RECIPROCAL FUNCTIONS Each trig function has an associated reciprocal function: AP Calculus AB, SJHS 2014-2015 r csc y r sec x x cot y 17 RIGHT TRIANGLE GEOMETRY We may use any (x,y) coordinate to solve for the hypotenuse by the Pythagorean Theorem x2 y 2 r 2 24 26 -10 24 12 26 13 10 5 cos 26 13 24 12 tan 10 5 sin 13 12 13 sec 5 5 cot 12 csc AP Calculus AB, SJHS 2014-2015 Ex: Given point (-10,24), find all six trig x functions. 10 2 r x 2 y 2 10 242 26 y 24 18 QUADRANTS AND SIGNS Sine is related to the y-coordinate; thus, any triangle pointing up (above the x-axis) has a positive sine and anything down is a negative Cuz positive is to the right, and negative is to the left for x Cuz up is positive and down is negative for y Tangent is a fraction of the y over x; fractions are positive if top and bottom are both positive or negative So triangles in the (+,+) and (-,-) regions have a positive tangent; this is Quadrants I and III (respectively) Since the others are mixed, that is QII(-,+) and QIV(+,-), then tan is negative in those quadrants AP Calculus AB, SJHS 2014-2015 To analyze what sign to expect, just consider the following: Cosine is related to the x-coordinate; thus, any triangle to the right (of the y-axis) has a positive cosine and anything to the left is negative 19 SUMMARY OF QUADRANT SIGNS I II III IV Sin + + - - Cos + - - + Tan + - + - AP Calculus AB, SJHS 2014-2015 Quadrant 20 DOMAIN AND RANGE The trig functions all have an infinite domain (except some undefined points); their ranges significantly differ, but you can likely see the relationships in this table: Domain Range Sin x 1 y 1 Cos x 1 y 1 Tan 3 5 x , , , y Csc 2 2 2 x y Sec 3 5 y Cot x , , , 2 2 2 x y AP Calculus AB, SJHS 2014-2015 Function 21 SPECIAL RIGHT TRIANGLES 60 30 2 3 45 1 45 1 1 30-60-90 45-45-90 AP Calculus AB, SJHS 2014-2015 2 22 EX: USE THE SPECIAL RIGHT TRIANGLES TO EVALUATE EACH TRIG FUNCTION 60 sin 60 tan30 sec30 2 3 30 1 2 45 1 AP Calculus AB, SJHS 2014-2015 cos45 3 2 1 2 1 3 2 3 45 csc45 2 1 23 WAIT, SO WHAT’S THE POINT OF EXACT VALUES IF WE JUST NEED TO KNOW TRIG FUNCTIONS? The only difference is the sign! Notice that the angles of the exact values apply to special right triangles AP Calculus AB, SJHS 2014-2015 Exact values are great to work with since each class of exact values gives the same answer for a trig function 24 EX: EVALUATE SIN(30°), SIN(150°), SIN(210°), SIN(330°). These seem to be a random collection of angles in degrees AP Calculus AB, SJHS 2014-2015 If we switch to radians we see they fall into the pattern of We already know sin π/6 = ½; so each of these will have the value ½, and the sign depends on the quadrant 25 EX: EVALUATE TAN(45°), TAN(135°), TAN(225°), AND TAN(315°) Switching to radians, we see there all are fractions of fourths, and thus must have the same value We know tan π/4=1 , so all the values are 1 and the signs are dependent on the quadrant AP Calculus AB, SJHS 2014-2015 26 NEGATIVE ANGLES A negative angle means the direction of sweep from 0° is clockwise These follow the same rules as exact values, in that the values of your function is the same but the sign has to do with the quadrant rules AP Calculus AB, SJHS 2014-2015 27 3 sin 120 2 1 cos120 2 1 cos 120 2 120 3 2 AP Calculus AB, SJHS 2014-2015 EX: EVALUATE SIN(120°), SIN(-120°), COS(120°), AND COS(-120°). 2 3 3 sin120 1 120 2 28 “TRIANGLES” OF ONLY ONE LINE A triangle by definition has three lines and three angles, with angles that add up to 180° The situation arises of what happens if two sides have the same length If you are given a point that does not make a proper triangle, then look at the coordinates to help you evaluate the trig functions AP Calculus AB, SJHS 2014-2015 Then it’s not technically a triangle since you would have two angles of 90° This is a purely theoretical case, but let’s just look at it in terms of x and y 29 EX: 90° ANGLE Suppose we want to find the sin 90°, cos90°, and tan90° If we draw a 90° angle and an arbitrary hypotenuse r in that direction, we get a line pointing up (not a triangle) We still have enough information to evaluate, since we know a few things: 1. In this case, there is no x-coordinate 2. In this case, if did have a “triangle”, x=0 and the “hypotenuse” r is the same as our y side r AP Calculus AB, SJHS 2014-2015 x0 y 30 EX: 90° ANGLE Now we can plug in this information: 90 AP Calculus AB, SJHS 2014-2015 y y sin 90 1 r y x 0 cos90 0 r r y y tan 90 undefined x 0 31 EX: SIN, COS, AND TAN OF 180° 2. The “hypotenuse” r has the same length as x, but x is clearly a negative number since the ray points backwards Now we can formulate: y 0 sin180 0 r r x x cos180 1 r x y 0 tan180 0 x x 180 AP Calculus AB, SJHS 2014-2015 This is the same problem, but instead our angle gives the following information: 1. There is no y-coordinate, so y 0 32 TRIG IDENTITIES Since trig functions are a result of triangles, it is worth relating them to the Pythagorean Theorem; we write in terms of x, y, and r 2 2 2 x y r x2 y 2 r 2 2 2 2 r r r x2 y 2 r 2 2 2 2 x x x x2 y 2 r 2 2 2 2 y y y cos sin 1 2 2 1 tan sec 2 2 AP Calculus AB, SJHS 2014-2015 If we rewrite this by dividing by x, y, or r, we get our three trig identities: cot 2 1 csc2 Do not memorize these! Just remember how to get them if you need them! 33 PROOFS INVOLVING TRIG FUNCTIONS Since trig functions are related to one another by x, y, and r, it is an instructive exercise to rearrange a given expression of trig functions into another equivalent expression To do this, use your basic definitions and your exact values AP Calculus AB, SJHS 2014-2015 34 EX: PROVE THE FOLLOWING STATEMENT BY REARRANGING THE LEFT SIDE ONLY. 1 sin x sin x csc x 1 sin x 1 sin x sin x 1 sin x 1 1 sin x sin x sin x sin x AP Calculus AB, SJHS 2014-2015 sin x 1 sin x 1 sin x 1 sin x sin x csc x 1 csc x 1 sin x csc x sin x sin x sin x 35 TRIGONOMETRIC EQUATIONS AND INVERSE TRIGONOMETRY A trigonometric equation is an equation containing one or more trig functions Simple trig equations can be solved analytically Equations containing several different trig functions are often only possible to solve with a computer Once the equation is in the form that the trig function has a value, you may use the “inverse trig function” to get the answer The inverse trig function simply means the operation (not function) used to convert your output value to your input angle Algebraically, it is more correct to use an inverse trig function to solve an equation; logically, you may just infer your answer from the information These are sometimes referred to as “arc” (like arcsine) AP Calculus AB, SJHS 2014-2015 36 DEFINITION OF AN INVERSE TRIG FUNCTION Because inverse trig functions “undo” the function, we get the property: sin 1 sin Thus, the function and inverse switch their domain (x) and range (y) coordinates 1 sin sin A A This works both ways! Ex: Cosine vs Arccosine cos 0 1 Cos 1 1 0 AP Calculus AB, SJHS 2014-2015 By definition, an inverse trig function is defined at the points its parent function has a value at; its value is the angle input of the parent function 37 DOMAIN OF INVERSE TRIG FUNCTIONS Ex: sin 0 1, sin 2 1, sin 4 arcsin 1 ? Which value do we pick? Is the answer 0, 2π, 4π, or something else 1 Actual answer is Sin 1 0 AP Calculus AB, SJHS 2014-2015 Since these functions must have well defined values, it its essential to restrict the domains Why? Because otherwise we wouldn’t know which value to choose! 38 DOMAIN AND RANGE OF INVERSE TRIG FUNCTIONS Domain Sin 1 x 1 x 1 Cos 1 x 1 x 1 Tan 1 x x Range y 2 2 0 y 2 y 2 AP Calculus AB, SJHS 2014-2015 y f x 39 SIMPLE EXAMPLE OF AN INVERSE TRIG FUNCTION For what value of θ does sin θ = 1? We know that sin π/2 = 1, so logically θ= π/2 Algebraically, the process goes: sin sin 1 sin 1 1 sin 1 2 1 AP Calculus AB, SJHS 2014-2015 sin 1 40 IS IT NECESSARY TO USE INVERSE TRIG FUNCTIONS No, not if you can logically deduce the answer yourself Because of this tedious process, and the fact that inverse functions are difficult to conceptualize, most people prefer to use the logical approach You do need to use inverse trig functions if you are solving for angles not included in the exact values set In that case, you would need to use your calculator’s inverse trig button AP Calculus AB, SJHS 2014-2015 41 NOT SO SIMPLE INVERSE TRIG PROBLEM Simplify : 2 cos Sin 1 5 So the answer is just the cosine of this triangle! We need to solve for x, but other than that it’s just writing the cosine. 2 x 2 2 52 21 5 -2 AP Calculus AB, SJHS 2014-2015 This is tricky! Arcsine is defined only in Quadrants I and IV. Since the arcsine is negative, we assume it’s a fourth quadrant angle. The question asks: what is the cosine of the angle defined by the inverse sine of (-2/5). x 25 4 21 21 2 cos Sin1 5 5 42 EX: 11 SIN X = 2 Let’s get this into a useable form: 2 sin x 11 2 sin sin x sin 11 x 0.183 1 1 AP Calculus AB, SJHS 2014-2015 11sin x 2 So our answer is 0.183 radians, or 10.5° 43 EX: 2cos x 1 12 1 1 Cos cos x Cos 12 2 1 Cos 1 undoes cos; both go away x x 12 3 this is an exact value ! 3 12 x AP Calculus AB, SJHS 2014-2015 1 cos x 12 2 4 44 EX: 4sin 3x 1 5 First step is to isolate the sine: This doesn’t make sense. No matter what, sine is between -1 and 1. Since 5/4 is out of our range for sine, this doesn’t have a solution. AP Calculus AB, SJHS 2014-2015 5 sin 3x 1 4 45 1 2 EX: tan x sec x 1 2 tan 2 x sec 2 x 1 1 2 sec x 1 sec x 1 2 AP Calculus AB, SJHS 2014-2015 Anytime we mix trig functions, the equation is more difficult. Always check which functions they are, because some are related by the trig identities. In this case, we can see that tan and sec are related (1+tan2x=sec2x). You should replace the squared term and see where that takes you. 46 EXAMPLE CONTINUED sec x 1 AP Calculus AB, SJHS 2014-2015 This is a difference of two squares now, so separate into factors. You’ll see it simplifies quickly: 1 sec x 1 sec x 1 sec x 1 2 1 sec x 1 sec x 1 sec x 1 2 sec x 1 sec x 1 1 sec x 1 1 2 sec x 1 2 1 1 cos x 1 cos x Cos 1 cos x Cos 1 1 x0 47 EX: 2sin sin 1 0 2 As stated this is difficult to see a sin Let’s make a simple variable substitution 2a 2 a 1 0 Our new equation is easy; just factor and solve 2a 1 a 1 0 AP Calculus AB, SJHS 2014-2015 1 a ,1 2 This solves “a” but we need to solve for our original variable θ Let’s replace with our substituted variable 1 sin 2 sin 1 7 11 , 6 6 2 7 11 , 2 6 , 6 48 EX: 3cos x 6sin x cos x 0 Consider each trig function to be its own variable! a sin x b cos x 3b 1 2a 0 b 0 1 a 2 x 3b 0 1 2a 0 cos x 0 1 sin x 2 5 , , 6 2 6 x 2 x , 5 6 6 AP Calculus AB, SJHS 2014-2015 3b 6ab 0 49 SOLVE THE TRIANGLE This means to solve all parts, typically given only two parts In this class, we will focus on right triangles Given two sides Solve third side using Pythagorean Theorem Solve angles using inverse functions Given a side and angle Given right triangles, the last angle is obvious Use trig functions to solve another side Solve for last side with: AP Calculus AB, SJHS 2014-2015 No Law of Sines or Cosines! Pythagorean Theorem (long way) Any other trig function (much faster) 50 SOLVE THE TRIANGLE 37 14.3 11.4 53 sin 53 11.4 r r 11.4 14.3 sin 53 We can use Pythagorean Theorem for the last side, or another trig function - let’s use cosine cos53 x x 14.3cos53 8.6 14.3 AP Calculus AB, SJHS 2014-2015 The third angle completes the triangle: 180 ° – 90 ° – 53 ° = 37° To solve the sides, choose a trig function that includes your given side - here, you are given a y-coordinate 51 SOLVE THE TRIANGLE 54.2° 54.6 31.9 44.3 We can use the Pythagorean Theorem for the last side: x 2 31.92 54.62 x 54.62 31.92 44.3 For the angles, choose any trig function and take its inverse 31.9 tan 44.3 31.9 Tan 35.8 44.3 1 AP Calculus AB, SJHS 2014-2015 35.8° The last angle is simple: 180° - 90° - 35.8° = 54.2 ° 52 END AP Calculus AB, SJHS 2014-2015 53 toothpastefordinner.com