4.3 TRIGONOMETRY EXTENDED: THE CIRCULAR FUNCTIONS

Transcription

1 4.3 TRIGONOMETRY EXTENDED: THE CIRCULAR FUNCTIONS MR. FORTIER 1. Trig Functions of Any Angle We now extend the definitions of the six basic trig functions beyond triangles so that we do not have to restrict our attention to acute angles, or even to positive angles. In geometry we think of an angle as a union of two rays with a common vertex, Trigonometry takes a more dynamic view by thinking of an angle in terms of a rotating ray. The beginning position of the ray, the initial side, is rotated about its endpoint, called the vertex. The final position is called the terminal side. The measure of an angle is a number that describes the amount of rotation from the initial side to the terminal side of the angle. Positive angles are generated by clockwise rotations. Below shows an angle of measure α, where α is a positive number. To bring the power of the coordinate geometry into the picture(literally), we usually place an angle in standard position in the Cartesian plane, with the vertex of the angle at the origin and its initial side lying along the positive x-axis. Below shows two angles in standard position, one with positive measure α and the other with negative measure β. Two angles in this expanded angle-measurement system can have the same initial side and the same terminal side, yet have different measures. We call such angles coterminal angles. For example, angles of 90, 450, and 70 are all coterminal, as are angles of π radians, 3π radians, and 99π radians. In fact angles are coterminal whenever they differ by an integer multiple of 360 degrees or by an integer multiple of π radians. 1

2 MR. FORTIER EXAMPLE 1: Finding Coterminal Angles Find and draw a positive angle and a negative angle that are coterminal with the given angle. (a.) 30 (b.) 150 (c.) π 3 radians SOLUTION There are infinitely many possible solutions; we will show two for each angle. (a.) Add 360 : = 390 Subtraction 360 : = 330 (b.) Add 360 : = 10 Subtraction 360 : = 870 (c.) π 3 radians Add π : π 3 + π = π 3 + 6π 3 = 8π 3 Subtract π : π 3 π = π 3 6π 3 = 4π 3 Extending the definitions of the six basic trig functions so that they can apply to any angle is surprisingly easy, but first you need to see how our current definitions relate to the (x, y) coordinates in the Cartesian plane. We start in the first quadrant (see Figure 4.3), where the angles are all acute. EXAMPLE : Evaluating Trig Functions Determined by a Point in the First Quadrant (QI) Let θ be the acute angle in standard position whose terminal side contains the point (5,3). Find the six trig functions of θ. SOLUTION The distance from (5,3) to the origin is. So, sin θ = cos θ = tan θ = 3 5 = 0.6 csc θ = sec θ = cot θ =

3 4.3 TRIGONOMETRY EXTENDED: THE CIRCULAR FUNCTIONS 3 EXAMPLE 3: Evaluating Trig Functions Determined by a Point not in QI Let θ be any angle in standard position whose terminal side contains the point (-5,3). Find the six trig functions of θ. SOLUTION The distance from (-5,3) to the origin is. So, sin θ = cos θ = tan θ = 3 5 = 0.6 csc θ = sec θ = cot θ = Notice in Example 3 that θ is any angle in standard position whose terminal side contains the point (-5, 3). There are infinitely many coterminal angles that could play the role of θ, some of them positive and some of them negative. The values of the six trigonometric functions would be the same for all of them. Therefore the definition is as follows: Definition: Trigonometric Functions of any Angle Let θ be any angle in standard position and let P (x, y) be any point on the terminal side of the angle (except the origin). Let r denote the distance from P (x, y) to the origin, i.e., let r = x + y. sin θ = y r cos θ = x r tan θ = y x (x 0) csc θ = r y (y 0) sec θ = r (x 0) cot θ = x y (y 0) Examples and 3 both began with a point P (x, y) rather than an angle θ. Indeed, the point gave us so much information about the trig ratios that we were able to compute them all without ever finding θ. So what do we do if we start with an angle θ in standard position and we want to evaluate the trig functions? We try to find a point (x, y) on its terminal side. We illustrate this process with Example 4.

4 4 MR. FORTIER EXAMPLE 4: Evaluating the Trig Functions of 315 SOLUTION First we draw an angle of 315 in standard position. Without declaring a scale, pick a point P on the terminal side and connet it to the x-axis with a perpendicular segment. Notice that the triangle formed (called a reference triangle) is a triangle. If we arbitrarily choose the horizontal and vertical sides of the reference triangle to be of length 1, then P has coordinates (1,-1). We now use the definitions from above and x = 1, y = 1 and r =. sin 315 = 1 = cos 315 = 1 = tan 315 = 1 1 = 1 csc 315 = 1 = sec 315 = = cot 315 = 1 1 = 1 The fact that the reference triangle in Example 4 was a triangle enabled us to label a point P on the terminal side of the 315 angle and then to find the trig function values. We would also be able to find P if the given angle were to produce a reference triangle.. Evaluating Trig Functions of a Non-quadrantal Angle θ (1) Draw the angle θ in standard position, being careful to place the terminal side in the correct quadrant. () Without declaring a scale on either axis, label a point P (other than the origin) on the terminal side of θ (3) Draw a perpendicular segment from P to the x-axis, determining the reference triangle. If this triangle is one of the triangles whose rations you know, label the sides accordingly. If it is not then you will need to use your calculator. (4) Use the sides of the triangle to determine the coordinates of point P, making them positive or negative according to the signs of x and y in that particular quadrant. (5) Use the coordinates of point P and the definitions to determine the six trig functions.

5 4.3 TRIGONOMETRY EXTENDED: THE CIRCULAR FUNCTIONS 5 EXAMPLE 5: Evaluation More Trig Functions Find the following without a calculator: (a.) sin( 10 ) (b.) tan ( ) 5π 3 (c.) sec ( ) 3π 4 SOLUTION (a.) sin( 10 ) An angle of 10 in standard position determines a reference triangle in the second quadrant. We label the sides accordingly, then use the lengths of the sides to determine the point P ( 3, 1). (Note that the x-coordinate is negative in the second quadrant.) The hypotenuse is r =. Therefore sin( 10 ) = y r = 1. (b.) tan ( ) 5π 3 An angle of 5π 3 radians in standard position determines a reference triangle in the fourth quadrant. We label the sides accordingly, then use the lengths of the sides to determine the point P (1, 3). (Note that the y-coordinate is negative in the fourth quadrant.) The hypotenuse is r =. Therefore tan ( ) 5π 3 = y x = 3 1 = 3. (c.) sec ( ) 4 3π An angle of 4 3π in standard position determines a reference triangle in the third quadrant. We label the sides accordingly, then use the lengths of the sides to determine the point P ( 1, 1). (Note that both coordinates are negative in the third quadrant.) The hypotenuse is r =. Therefore sec ( ) 4 3π = r s = 1 =. Angles whose terminal sides lie along one of the coordinate axes are called quadrantal angles EXAMPLE 6: Evaluating Trig Functions of Quadrantal Angle Find each of the following, if it exists. If the value is undefined, write undefined. (a.) sin( 70 ) (b.) tan 3π (c.) sec ( ) 11π

6 6 MR. FORTIER SOLUTION (a.) sin( 70 ) In standard position, the terminal side of an angle of 70 lies along the positive y-axis. A convenient point P along the positive y-axis is the point for which r = 1, namely (0,1). Therefore sin( 70 ) = y r = 1 1 = 1. (b.) tan 3π In standard position, the terminal side of an angle of 3π lies along the negative x-axis. A convenient point P along the positive x-axis is the point for which r = 1, namely (-1,0). Therefore tan 3π = y x = 0 1 = 0. (c.) sec ( ) 11π In standard position, the terminal side of an angle of 11π lies along the negative y-axis. A convenient point P along the positive y-axis is the point for which r = 1, namely (0,-1). Therefore sec ( ) 11π = r x = 1 0 = undefined. Another good exercise is to use information from one trig ratio to produce the other five. We do not need to know the angle θ, although we do need a hint as correct quadrant (or place a quadrantal angle on the correct side of the origin). Example 7 illustrates how this is done. EXAMPLE 7: Using One Trig Ratio to Find the Others Find cos θ and tan θ by using the given information to construct a reference triangle. (a.) sin θ = 3 7 and tan θ > 0 (b.) sec θ = 3 and sin θ > 0 (c.) cot θ is undefined and sec θ is negative SOLUTION (a.) sin θ = 3 7 and tan θ > 0 Since sin θ is positive, the terminal side is either in QI or QII. The added fact that tan θ is negative means that the terminal side is in QII. We draw a reference triangle in QII with r = 7 and y = 3; then we use the Pythagorean theorem to find that x = = 40. We then use the definitions to get cos θ = and tan θ =

7 4.3 TRIGONOMETRY EXTENDED: THE CIRCULAR FUNCTIONS 7 (b.) sec θ = 3 and sin θ > 0 Since sec θ is positive, the terminal side is either in QI or QIV. The added fact that sin θ is positive means that the terminal side is in QI. We draw a reference triangle in QI with r = 3 and x = 1; then we use the Pythagorean theorem to find that y = = 8. We then use the definitions to get cos θ = and tan θ = (c.) cot θ is undefined and sec θ is negative Since cot θ is undefined, we conclude that y = 0 and that θ is a quadrantal angle on the x-axis. The added fact that sec θ is negative means that the terminal side is along the negative x-axis. We choose the point (-1,0) on the terminal side and use the definitions to get. cos θ = 1 and tan θ = 0 1 = Trig Functions of Real Numbers Now that we have extended the six basic trig functions to apply any angle, we are ready to appreciate them as functions of real numbers and to study their behavior. First, for reasons discussed in the first section of this chapter, we must agree to measure θ in radian mode so that the real number units of the input will match the real numbers units of the output. When considering the trig functions as functions of real numbers, the angles will be measured in radians. Definition: Unit Circle The unit circle is a circle of radius 1 centered at the origin. The unit circle provides an ideal connection between triangle trigonometry and the trig functions. Because arc length along the unit circle corresponds exactly to radian measure, we can use the circle itself as a sort of number line for the input values of our functions. This involves the wrapping function, which associates points on the number line with points on the circle. The figure below shows how the wrapping function works. The real line is placed tangent to the unit circle at the point (1,0), the point from which we measure angles in standard position. When the line is wrapped around the unit circle in both the positive (counter clockwise) and negative (clockwise) directions, each point t on the real line will fall on a point of the circle that lies on the terminal side of an angle of t radians in standard position. Using the coordinates (x, y) of this point, we can find the six trigonometric ratios for the angle t just as we did in Example 7-except even more easily, since r = 1.

8 8 MR. FORTIER DEFINITION: Trig Functions of Real Numbers Let t be any real number, and let P (x, y) be the point corresponding to t when the number line is wrapped onto the unit circle as described above. Then sin t = y cos t = x tan t = y x csc t = 1 y (y 0) sec t = 1 x (x 0) cot t = x y (y 0) Although it is still helpful to draw reference triangles inside the unit circle to see the ratios geometrically, this latest round of definitions does not invoke triangles at all. The real number t determines a point on the unit circle, and the (x, y) coordinates of the point determine the six trigonometric ratios. For this reason, the trig functions when applied to real numbers are usually called the circular functions. 4. Periodic Functions Statements 5 and 7 in Exploration exercise above reveals an important property of the circular functions that we need to define for future reference. DEFINITION: Periodic Function A function y = f(t) is periodic if there is a positive number c such that f(t + c) = f(t) for all values of t in the domain of f. The smallest such number c is called the period of the function.

9 4.3 TRIGONOMETRY EXTENDED: THE CIRCULAR FUNCTIONS 9 The Exploration suggests that the sine and cosine functions have period π and that the tangent function has period π. We use this periodicity later to model predictably repetitive behavior in the real world, but meanwhile we can also use it to solve little non-calculator training problems like in some of the previous examples in this section. EXAMPLE 8: Using Periodicity Find each of the following numbers without a calculator. (a) sin ( 57,801π ) (b) cos(88.45π) cos(80.45π) (c) tan ( π 4 99, 999π) SOLUTION ( ) ( ) (a) sin 57,801π = sin π + 57,801 = sin ( π + 8, 900π) = sin ( ) π = 1 Notice that 8, 900π is just a large multiple of π so π and ( π + 8, 900π) wrap to the same point on the unit circle, namely (0,1). (b) cos(88.45π) cos(80.45π) = cos(88.45π + 8π) cos(80.45π) = 0 Notice that 80.45π and 80.45π + 8π wrap to the same point on the unit circle, so the cosine of one is the same as the cosine of the other. (c) tan ( π 4 99, 999π) Since the period of the tangent functions is π rather than π, 99, 999π is a large multiple of the period of the tangent function. Therefore, tan ( π 4 99, 999π) = tan ( ) π 4 = 1 The 16 Point Unit Circle At this point you should be able to use reference triangles and quadrantal angles to evaluate trig functions for all integer multiples of 30 or 45 (equivalently π 6 radians or π 4 radians). All of these special values wrap to the 16 special points shown on the unit circle.