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Special Angles and their Trig Functions By Jeannie Taylor Through Funding Provided by a VCCS LearningWare Grant 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 half-way between 0 and 180 or , we have 90 or . 2 Looking at the angle half-way between 180 and 360 , we have 270or 32 radians which is 34 of the total (360 or 2 radians). 180 radians 0 360 2 radians 270 3 radians 2 Moving all the way around from 0 to 360 completes the circle and and gives the 360 angle which is equal to 2 radians. We can count the quadrantal angles in terms of Notice that after counting these angles based on portions of the full circle, two of these angles reduce to radians with which we are familiar, and 2. Add the equivalent degree measure to each of these quadrantal angles. 180 We can approximate the radian measure of each angle to two decimal places. One of them, you already know, 3.14 radians . It will probably be a good idea to memorize the others. Knowing all of these numbers allows you to quickly identify the location of any angle. 2 radians. 90 2 radians 1.57 radians 0 2 radians 2 radians 3.14 radians 0 radians 4 2 radians 360 2 radians 6.28 radians 3 radians 4.71 radians 2 270 We can find the trigonometric functions of the quadrantal angles using this definition. We will begin with the point (1, 0) on the x axis. Remember the six trigonometric functions defined using a point (x, y) on the terminal side of an angle, . 90 2 y sin r r csc y x cos r r sec x tan y x cot radians (1, 0) 180 0 0 radians radians or 360 x y 2 radians 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 csc 0 is undefined cos 0 1 sec 0 1 tan 0 90 2 radians 0 0 cot 0 is undefined 1 (1, 0) 180 0 0 radians radians or 360 2 radians It will be just as easy to find the trig functions of the remaining quadrantal angles using the point (x, y) and the r value of 1. 270 3 radians 2 sin 1 2 csc 1 2 cos 0 2 sec is undefined 2 tan is undefined 2 cot 0 2 90 2 radians (0, 1) sin 0 cos 1 csc is undefined sec 1 tan 0 cot is undefined 0 180 radians (-1, 0) 0 radians (0, -1) 3 1 2 csc 3 1 2 270 cos 3 0 2 sec 3 is undefined 2 3 radians 2 tan 3 is undefined 2 3 0 2 360 2 radians sin cot or Now let’s cut each quadrant in half, which basically gives us 8 equal sections. The first angle, half way between 0 1 and 2 would be 2 2 4 . We can again count around the circle, but this time we will count in terms of 4 radians. Counting we say: 2 90 4 2 3 135 4 1 2 3 4 5 6 7 8 , , , , , , , and . 4 4 4 4 4 4 4 4 Then reduce appropriately. Since 0 to 2 radians is 90, we know that 4 is half of 90or 45. Each successive angle is 45 more than the previous angle. Now we can name all of these special angles in degrees. 2 4 45 4 4 180 0 5 4 225 7 4 315 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,). 8 2 360 4 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 – 90 triangle are equal to each other because their corresponding angles are equal. If we let each leg have a length of 1, then we find the hypotenuse to be 2 using the Pythagorean theorem. 45 2 1 45 1 You should memorize this triangle or at least be able 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. sin 45 1 2 2 2 45 csc 45 2 2 1 2 cos 45 2 sec 45 2 tan 45 1 cot 45 1 45 1 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 triangle with a base of 1 and a hypotenuse of 2. This smaller triangle now has angles of 30, 60, and 90 . 30 2 3 60 1 You should memorize this triangle or at least be able to construct it. These angles, also, will be used frequently. We find the length of the other leg to be 3 , using the Pythagorean theorem. 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. sin 30 1 2 csc 30 2 cos 30 3 2 tan 30 1 30 2 3 2 sec 30 3 3 3 cot 30 2 3 3 2 3 3 3 3 3 3 60 1 sin 60 3 2 csc 60 cos 60 1 2 sec 60 2 tan 60 3 cot 60 2 3 1 3 45 30 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 2 1 30 2 45 3 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 A good way to II I remember this chart is that ASTC stands for III IV 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 of the 30 angle, draw the “special” triangle. y 30 S 2 A 3 60 1 Determine the correct sign for the trig functions of 330 . Only the cosine and the secant are “+”. 330 T C 30 x Example 1 Continued: The six trig functions of 330 are: 1 2 3 cos 330 2 sin 330 csc 330 2 sec 330 1 3 3 3 tan 330 2 2 3 3 3 y cot 330 3 30 S 2 A 3 30 60 1 330 T C x Example 2: Find the six trig functions of First determine the location of 4 3 4 3. (Slide 1) . 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 We can see that the reference angle is , which is the same as 3 60. Therefore, we will compute the trig functions of 3 using the 60 angle of the special triangle. 3 3 3 30 3 3 60 1 3 3 2 y 2 3 4 3 x 4 3. Example 2: Find the six trig functions of (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.. sin 4 3 3 2 csc 4 2 2 3 3 3 3 cos 4 1 3 2 sec 4 2 3 tan 4 3 3 cot 4 1 3 3 3 3 3 y 2 3 S A 30 x 3 2 3 T 60 1 4 3 C 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 4 located in the 2nd quadrant and the cos is negative in the 2nd quadrant. Note that the reference angle is 5 4 4 4 4 0 radians . 4 T 3 4 4 We know that is the same as 45 , so the reference angle is 45 . Using the special triangle we can see that the cos of 45 or 4 is 12 . A S 45 C 4 2 4 2 1 45 1 cos 54 = 1 2 2 2 Practice Exercises 1. Find the value of the sec 360 without using a calculator. 2. Find the exact value of the tan 420 . 5 . 6 3. Find the exact value of sin 4. Find the tan 270 without using a calculator. 5. Find the exact value of the csc 73 . 6. Find the exact value of the cot (-225 ). 13 4 11 6 7. Find the exact value of the sin . 8. Find the exact value of the cos . 9. Find the value of the cos(- ) without using a calculator. 10. Find the exact value of the sec 315 . Key For The Practice Exercises 1. sec 360 = 1 3 2. tan 420 = 3. sin 5 6 = 1 2 4. tan 270 is undefined 5. csc 7 3 2 2 3 = 3 3 6. cot (-225 ) = -1 7. sin 8. cos 13 4 11 6 = = 3 2 1 2 2 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 negative direction in units of . 6 A S 0 radians 6 5 We can see that 6 is the 6 6 T C reference angle and we know 2 4 that 6 is the same as 30 . So 6 3 6 we will draw our 30 triangle 6 1 and see that the sin 30 is 2 . All that’s left is to find the correct sign. 30 3 And we can see that the correct sign is “-”, since the sin is always “-” in the 3rd quadrant. 2 60 1 Answer: 5 sin 6 = 1 2 13 . 4 Problem 7: Find the exact value of cos We will first draw the angle to determine the quadrant. 13 4 We see that the angle is located in the 3rd quadrant and the cos is negative in the 3rd quadrant. Note that the reference angle is We know that 4 is the same as 45 , so the reference angle is 45 . Using the special triangle we can see that the cos of 45 or 4 is 12 . 2 4 10 4 3 4 4 4 . 4 45 A S 11 4 9 4 4 8 4 0 radians 4 12 4 5 13 4 4 T C 7 4 6 4 2 1 cos 13 = 45 1 4 1 2 2 2