Trigonometric Functions

Transcription

1 TrigonometricReview.nb Trigonometric Functions The trigonometric (or trig) functions are ver important in our stud of calculus because the are periodic (meaning these functions repeat their values in a regular fashion). This characteristic makes them invaluable as models for phenomena that also repeat; the rising and falling of the tides, length of dalight hours of a region, average temperatures of a cit, etc. Since the trig. functions (especiall sine, cosine, and tangent) are so important it makes sense to make sure we know the pre-calculus characteristics of them eceptionall well. Prior knowledge of the trigonometric functions is assumed in this review but we will hit some of the highlights. What ou ma remember from geometr and algebra are the trigonometric ratios. You probabl defined the ratios in terms of the lengths of the sides of a right triangle. The angles of the triangle were epressed in degrees. In calculus, we talk of the trigonometric functions. There is a difference. Recall that a function is a rule that assigns to each real number in the domain a unique real number in the range. The ke phrase is real number. Degrees are not real numbers; that is the cannot be paired with numbers on the number line. So in calculus we use radians instead. Radians are real numbers and can be located on a number line. Don't panic. What ou learned in geometr and algebra can be applied to the trigonometric functions as long as ou know how to convert angle measures to their equivalent radian measures. First, let's discuss the difference between the two. Think of our old friend the unit circle (a circle with radius with its center at the origin). All of the points on the circle can be described in terms of ordered pairs of numbers; that is, with an and a. If we take the positive -ais as our initial side and rotate a certain number of degrees to our terminal side and then "drop a perpendicular" from the point on the circle where the terminal side crosses the circle we have a right triangle. The point where the terminal side crosses the circle has coordinates (, ) and the hpotenuse of the right triangle is the radius of the circle; in this instance. The angle formed b the initial side and the terminal side, in degrees, is how much we "rotated" from the positive ais. So where do radians come in? Let's look at the prob-

2 TrigonometricReview.nb lem from the viewpoint of two bugs; one at the origin and the other on the circle at the point (, 0). Suppose the bug at the point (, 0) crawls along the circle and stops at the point (, ). Now picture the bug at the origin rotating its bod so it is alwas looking at the bug crawling on the circle. The rotation of the bug at the origin is measured in degrees. The distance the other bug moved on the circle is measured in radians. So radians are actuall lengths! And lengths can be located on a number line and, hence, are real numbers. The two measures are related functionall. The amount of rotation of the bug at the origin has an equivalent measure in radians on the circle. The relationship is etremel eas to comprehend. If the bug on the circle crawls completel around this unit circle its distance traveled is the circumference of the circle, namel p. To follow this bug on its journe the bug at the origin must turn completel around once, or 60. So p is equivalent to 60! This relationship is summarized below. p radians º 60 or p radians º 80 This simple formula gives us an invaluable wa of using what we learned about the trig ratios and appling it to the trigonometric functions. First, let's define the trig. functions. Definitions The sine, cosine, and tangent of a positive acute angle q can be defined as ratios of the side of a right triangle. Using the notation from the figure below, these definitions take the following form: sin q = side opposite q ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ hpotenuse = r ÄÄÄÄÄ = (on a unit circle, since r = ) cos q = side adjacent to q ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ hpotenuse = ÄÄÄÄ r = (on a unit circle) tan q = side opposite q ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ side adjacent to q = ÄÄÄÄ

3 TrigonometricReview.nb Since csc q, sec q, and cot q are defined to be the reciprocals of sin q, cos q, and cot q the are defined as follows; csc q = r ÄÄÄÄ = ÄÄÄÄ (on a unit circle) sec q = r ÄÄÄÄ = ÄÄÄÄ (on a unit circle) cot q = ÄÄÄÄ So how can we find values of the trig. functions? Remember that in geometr and algebra ou used special right triangles to find the values of the trig. ratios. Let's review these triangles now. Special Right Triangles The two special right triangles ou learned in geometr were the triangle and the triangle. The reason the were special was that ou onl needed to know the length of one side of the triangle in order to find the other two sides. Before ou learned the relationships of the sides of these triangles ou had to know two sides of a right triangle before ou could find the length of the third side (using the Pthagorean Theorem). Let's look at the triangle first. Look at the figure below:

4 TrigonometricReview.nb 4 h 60 h ÅÅÅÅÅ 0 h ÅÅÅÅÅ Look familiar? In this special triangle the sides are related in this wa; the side opposite the 0 degree angle is / the hpotenuse and the side opposite the 60 degree angle is / the hpotenuse time the square root of three. If we let the hpotenuse be equal to the triangle looks like the one below: 60 ÅÅÅÅÅ 0 ÅÅÅÅÅ Much easier when we use a unit circle, isn't it? Now, let's look at the triangle. Look at the figure below: s 45 s 45 s Again, let's let the hpotenuse be. Now look at the triangle.

5 TrigonometricReview.nb 5 45 Å 45 Å If we position these triangles in the first quadrant with the bottom leg of the triangle coincident with the positive ais we have a triangle in standard position. The length of the two legs will then correspond to the and coordinates of a point on the circle. q Let's investigate finding eact values of the trig function. Eact Values of the Trigonometric Functions Let's sa we wanted to find the eact value of = sin p/6. At first (at least until we get used to working with radians) we ma need to convert the radian measure into its equivalent degree measure. We can use conversion formulas derived from the fact that p radians is equivalent to 80 but it reall is simpler to do the math in our head. It's

6 TrigonometricReview.nb 6 alwas easier to use relationships than memorizing formulas. Think of it this wa. The radian measure is p/6. Another wa to sa this is one-sith p. So the equivalent degree measure is one-sith 80 or 0. So we are tring to find sin 0. Look at the figure below. We have a triangle situated in standard position. 0! ÅÅ ÅÅÅÅÅ According to our definition sin q = ÅÅÅÅ r = (in a unit circle) so sin p/6 = sin 0 = /. So using the two special right triangles, finding equivalent degree measures for radian measures, and drawing right triangles in standard position we can find the eact values of the trig functions for radian measures of p/6/ p/4, and p/ (these are equivalent to 0, 45, and 60 ). Let's construct a table of these values below. q (in radians) q (in degrees) sin q cos q tan q p/6 0 ÅÅÅÅ p/4 45 p/ 60 ÅÅÅÅ

7 TrigonometricReview.nb 7 All we need to do now is find function values at q = 0 and q = p/. Since we can't actuall draw triangles for these angles we just use the definitions. At q = 0, = and = 0. So sin 0 = 0 and cos 0 =. Since tangent is defined to be / then tan 0 = 0/ = 0. At q = p/, = 0 and =. So sin p/ = and cos p/ = 0. In this case tan p/ = /0 = undefined. So we now have a complete table for the sine, cosine, and tangent functions in the first quadrant. q (in radians) q (in degrees) sin q cos q tan q p/6 0 ÅÅÅÅ p/4 45 p/ 60 ÅÅÅÅ p/ 90 0 und. The big difference between the trigonometric functions and the trigonometric ratios is that using the functions we can find function values for angles of rotation greater than p/ (90 ). In geometr we were confined to angles less than 90. However, due to the periodicit of the trig functions we can easil find the function values of sine, cosine, and tangent functions in the II, III, and IV quadrants. What we need to keep in mind is that since sin q = then the sign of the sine function depends on the sign of the -coordinate. Similarl, since cos q =, then the sign of the cosine function depends on the sign of. Finall, since tan q = /, then the sign of the tangent function depends on both

8 TrigonometricReview.nb 8 and. Let's look at an eample. Eample : Find the eact value of sin 5p/6, cos 5p/6, and tan 5p/6. Solution: First notice that 5p/6 is in the second quadrant (its almost p). In the second quadrant the -coordinates are negative and the -coordinates are positive. Now let's find an equivalent degree measure for 5p/6. Notice that this is five-siths p. Five-siths of 80 is 50. If we rotate 50 and "drop a perpendicular" we get a triangle in the second quadrant. Look at the figure below. ÅÅÅÅÅ -! ÅÅ 0 5 p Å 6 So, sin 5p/6 = / cos 5p/6 = - tan 5p/6 = - The 0 angle measured from the terminal side of our angle of rotation to the -ais is known as the reference angle. The reference angle is the angle formed b the terminal side of the angle of rotation and the -ais. If the terminal side of the angle is in the third or fourth quadrant we must actuall "drop a perpendicular" upward to the -ais! Knowing this angle and the eact values of the trig functions in the first quadrant will give us the values of the trig functions in all of the quadrants. We need onl find out the

9 TrigonometricReview.nb 9 sign of the function in the quadrant in question and use the and triangle relationships to determine the magnitude of the lengths.