VIEWS: 10 PAGES: 27 POSTED ON: 9/14/2012
By bithun jith Done by bithun jith binoy k.v.pattom You must know and memorize the following. Pythagorean Identities: Tangent/Cotangent Identities: sin x sin2 x+ cos2 x=1 tan x cos x 1 + tan2 x = sec2 x cos x 1+ cot2 x= csc2 x cot x sin x Reciprocal Identities: Cofunction Identities: 1 1 sin x csc x sin x cos x csc x sec x csc x sin x 2 2 1 1 cos x sec x cos x sin x sec x csc x sec x cos x 2 2 1 1 tan x cot x cot x tan x tan x cot x 2 2 cot x tan x sin2 x = (sin x)2 11.3 Sum and Difference Formulas Objective: To use the sum and difference formulas for sine and cosine. sin ( + ) = sin cos + sin cos sin ( - ) = sin cos - sin cos 45 60 45 30 1. This can be used to find the sin 105. HOW? 2. Calculate the exact value of sin 375. cos ( + ) = cos cos - sin sin cos ( - ) = cos cos + sin sin Note the similarities and differences to the sine properties. 3. This can be used to find the cos 285. HOW? 4. Calculate the exact value of cos 345. 5. Pr ove : sin(α β) sinα β 2 sinβ cos α We will first look at the special angles called the quadrantal angles. The quadrantal angles are those angles that lie on the axis of the Cartesian coordinate system: 0 , 90, 180, and 270. 90 180 0 270 We also need to be able to recognize these angles when they are given to us in radian measure. Look at the smallest possible positive angle in standard position, other than 0 , yet having the same terminal side as 0 . This is a 360 angle which is equivalent to 2 radians. 90 2 radians If we look at half of that angle, we have 180 or radians . Looking at the angle 180 0 half-way between 0 radians 360 2 radians and 180 or , we have 90 or . 2 3 Looking at the angle half-way 270 2 radians between 180 and 360 , we have Moving all the way around from 0 to 360 270or 3 radians which is 3 of 2 4 completes the circle and and gives the 360 the total (360 or 2 radians). angle which is equal to 2 radians. We can count the quadrantal angles in terms of radians . 2 Notice that after counting these angles based on portions of the full circle, 90 two of these angles reduce to radians radians1.57 radians with which we are familiar, and 2. 2 Add the equivalent degree measure to each of these quadrantal angles. 2 0 180 2 radians 0 radians 4 We can approximate the radians radians 360 2 radian measure of each 3.14 radians 2 radians angle to two decimal places. 6.28 radians One of them, you already know, 3.14 radians. It will probably be a good idea 3 radians 4.71radians to memorize the others. 2 Knowing all of these 270 numbers allows you to quickly identify the location of any angle. We can find the trigonometric functions of the Remember the six quadrantal angles using this definition. We will trigonometric functions defined using a point (x, y) begin with the point (1, 0) on the x axis. on the terminal side of an 90 angle, . radians 2 y r sin csc r y (1, 0) 0 cos x sec r 180 0 radians r x radians or y x 360 tan cot 2 radians x y 270 3 For the angle 0 , we can see that x = 1 and radians 2 y = 0. To visualize the length of r, think about the line of a 1 angle getting closer As this line falls on top of the x axis, and closer to 0 at the point (1, 0). we can see that the length of r is 1. Using the values, x = 1, y = 0, and r = 1, we list the six trig functions of 0. And of course, these values also apply to 0 radians, 360 , 2 radians, etc. sin 0 0 csc0 is undefined 90 radians cos0 1 sec 0 1 2 0 tan 0 0 cot 0 is undefined 1 (1, 0) 0 180 0 radians radians or 360 2 radians It will be just as easy to find the 270 trig functions of the remaining 3 quadrantal angles using the point radians 2 (x, y) and the r value of 1. sin 1 csc 1 2 2 90 cos 0 sec is undefined 2 2 radians 2 tan is undefined cot 0 2 2 (0, 1) sin 0 csc is undefined 0 cos 1 sec 1 180 0 radians tan 0 cot is undefined radians (-1, 0) or (0, -1) 360 2 radians 3 3 sin 1 csc 1 2 2 270 cos 3 0 sec 3 is undefined 3 radians 2 2 2 3 3 tan is undefined cot 0 2 2 Now let’s cut each quadrant in The first angle, half way between 0 half, which basically gives us 8 1 and would be . equal sections. 2 2 2 4 We can again count around the circle, but this time we will count 2 90 in terms of 4 radians. Counting 4 2 we say: 2 3 135 4 45 1 2 3 4 5 6 7 8 4 , , , , , , , and . 4 4 4 4 4 4 4 4 Then reduce appropriately. 4 8 2 360 0 4 180 4 Since 0 to 2 radians is 90, we know that is half of 90or 45. Each 4 successive angle is 45 more than the 5 7 previous angle. Now we can name all 4 4 315 225 of these special angles in degrees. 6 3 4 2 270 It is much easier to construct this picture of angles in both degrees and radians than it is to memorize a table involving these angles (45 or 4 reference angles,). Next we will look at two special triangles: the 45 – 45 – 90 triangle and the 30 – 60 – 90 triangle. These triangles will allow us to easily find the trig functions of the special angles, 45 , 30 , and 60 . The lengths of the legs of the 45 45 – 45 – 90 triangle are equal 2 to each other because their 1 corresponding angles are 45 equal. 1 If we let each leg have a length of 1, then we find the hypotenuse You should memorize this to be 2 using the Pythagorean triangle or at least be able theorem. to construct it. These angles will be used frequently. Using the definition of the trigonometric functions as the ratios of the sides of a right triangle, we can now list all six trig functions for a 45 angle. 1 2 45 sin 45 csc45 2 2 2 2 1 2 45 cos 45 sec 45 2 1 2 tan 45 1 cot 45 1 WARM-UP B) occur frequently The expressions sin (A + B) and cos (A + enough in math that it is necessary to find expressions equivalent to them that involve sines and cosines of single angles. So…. Does sin (A + B) = Sin A + Sin B Try letting A = 30 and B = 60 For the 30– 60– 90triangle, we will construct an equilateral triangle (a triangle with 3 equal angles of 60 each, which guarantees 3 equal sides). If we let each side be a length of 2, then cutting the triangle in half will give us a right 30 triangle with a base of 1 and a 2 hypotenuse of 2. This smaller 3 triangle now has angles of 30, 60, and 90 . 60 1 We find the length of the other You should memorize this leg to be 3 , using the triangle or at least be able to Pythagorean theorem. construct it. These angles, also, will be used frequently. Again, using the definition of the trigonometric functions as the ratios of the sides of a right triangle, we can now list all the trig functions for a 30 angle and a 60 angle. 1 sin 30 csc30 2 2 3 2 2 3 cos30 sec 30 2 3 3 30 1 3 tan 30 cot 30 3 2 3 3 3 60 3 2 2 3 1 sin 60 csc60 2 3 3 1 cos 60 sec 60 2 2 1 3 tan 60 3 cot 60 3 3 30 45 2 2 1 3 45 60 1 1 Either memorizing or learning how to construct these triangles is much easier than memorizing tables for the 45 , 30 , and 60 angles. These angles are used frequently and often you need exact function values rather than rounded values. You cannot get exact values on your calculator. 45 30 2 1 2 3 45 1 60 1 Knowing these triangles, understanding the use of reference angles, and remembering how to get the proper sign of a function enables us to find exact values of these special angles. Sine All II A good way to I remember this chart is III IV that ASTC stands for All Students Take Tangent Cosine Calculus. Example 1: Find the six trig functions of 330. First draw the 330 degree angle. Second, find the reference angle, 360 - 330 = 30 To compute the trig functions y of the 30 angle, draw the “special” triangle. 30 S A 2 3 x 30 60 330 1 T C Determine the correct sign for the trig functions of 330 . Only the cosine and the secant are “+”. Example 1 Continued: The six trig functions of 330 are: 1 sin 330 csc 330 2 2 3 2 2 3 cos 330 sec 330 2 3 3 1 3 y tan 330 cot 330 3 3 3 30 S A 2 3 30 x 60 330 1 T C 4 Example 2: Find the six trig functions of 3 . (Slide 1) 4 First determine the location of 3 . With a denominator of 3, the distance from 0 to radians is cut into thirds. Count around the Cartesian coordinate system beginning at 0 until 4 we get to . 3 y We can see that the reference 2 3 angle is , which is the same as 3 3 60. Therefore, we will compute the trig functions of 3 using the 60 angle of the special triangle. 3 3 3 30 3 x 2 3 3 4 60 3 1 4 Example 2: Find the six trig functions of 3 . (Slide 2) Before we write the functions, we need to determine the signs for each function. Remember “All Students Take Calculus”. Since the angle, 43 , is located in the 3rd quadrant, only the tangent and cotangent are positive. All the other functions are negative.. 4 3 4 2 2 3 sin csc 3 2 3 3 3 y 2 4 1 4 3 cos sec 2 3 3 2 3 4 4 1 3 tan 3 3 cot 3 S A 3 3 30 x 2 3 3 T C 4 60 3 1 Example 3: Find the exact value of cos 5 . 4 We will first draw the angle to determine the quadrant. 5 We see that the angle is 5 4 4 A located in the 2nd quadrant S and the cos is negative in the 2nd quadrant. 4 4 0 radians 4 Note that the reference angle is . 4 T C 3 4 We know that is the 4 4 2 same as 45 , so the 4 45 reference angle is 45 . 1 2 cos 54 = 1 2 Using the special triangle 2 2 we can see that the cos of 45 45 or is 12 . 4 1 Key For The Practice Exercises 1. sec 360 = 1 2. tan 420 = 3 5 1 3. sin = 6 2 4. tan 270 is undefined 7 2 2 3 5. csc = 3 3 3 6. cot (-225 ) = -1 13 1 2 7. sin = 4 2 2 11 3 8. cos = 6 2 9. cos(- ) = -1 10. sec 315 = 2 Problems 3 and 7 have solution explanations following this key. Problem 3: Find the sin 5 . 6 We will first draw the angle by counting in a S A negative direction in units of . 0 radians 6 6 5 We can see that 6 is the 6 reference angle and we know 6 T C 4 2 that 6 is the same as 30 . So 6 3 6 we will draw our 30 triangle 6 and see that the sin 30 is 2 . 1 All that’s left is to find the correct sign. 30 And we can see that the correct sign is “-”, since 2 3 the sin is always “-” in the 3rd quadrant. 5 1 60 Answer: sin 6 = 2 1 11.1 - Basic Trigonometry Identities Objective: to be able to verify basic trig identities You must know and memorize the following. Pythagorean Identities: Tangent/Cotangent Identities: sin x sin2 x+ cos2 x=1 tan x cos x 1 + tan2 x = sec2 x cos x 1+ cot2 x= csc2 x cot x sin x Reciprocal Identities: Cofunction Identities: 1 1 sin x csc x sin x cos x csc x sec x csc x sin x 2 2 1 1 cos x sec x cos x sin x sec x csc x sec x cos x 2 2 1 1 tan x cot x cot x tan x tan x cot x 2 2 cot x tan x sin2 x = (sin x)2 13 Problem 7: Find the exact value of cos . 4 We will first draw the angle to determine the quadrant. 2 4 10 13 4 9 We see that the angle is 3 4 4 located in the 3rd quadrant 4 11 S A 4 and the cos is negative in the 4 8 3rd quadrant. 4 4 0 radians 4 12 4 4 Note that the reference angle is . 4 7 5 13 T C 4 4 We know that is the 4 6 4 4 same as 45 , so the 45 reference angle is 45 . 1 2 cos 13 = 1 2 Using the special triangle 2 2 4 we can see that the cos of 45 45 or is 12 . 4 1