M5-2 Exact Trig Values

Transcription

1 M5- Exact Trig Values exact values of the trig functions of multiples of and 5 degrees Pre-requisites: M5- (Unit Circle) Estimated Time: hours Summary Learn Solve Revise Answers Summary The es, coes and gents of,, 5, 6 and 9 are worth remembering. They can be obtained ug the unit circle and triangle diagrams. The es, coes and gents of multiples of and 5 outside the first quadrant can be obtained from the first quadrant values with a circle diagram. First Quadrant Learn It is often handy to know the exact values of the es, coes and gents of,, 5, 6 and 9. These are listed in the table to the right. ½ How do we get these? The es and coes of and 9 can be obtained from the unit circle diagram. The s of and 9 can be obtained from the identity =. 5 6 ½ 9 The es, coes and gents for 5 can be obtained by considering a triangle and Pythagoras Theorem as shown here. Make sure you can see how it is done. Black Star Maths M5- Exact Trig Values Page

2 The es, coes and gents of and 6 can be obtained by considering an equilateral triangle with a line down the centre and Pythagoras Theorem as shown here. Make sure you can see how this is done too. Knowing how the values are derived gives you a fall-back in case you forget them. And it s good style. Practice P Learn the values, then check yourself by completing this table An alternative way of writing the values is like this. The patterns make this way is a bit easier to memorise, though it doesn t give them in their most useful form Black Star Maths M5- Exact Trig Values Page

3 Angles outside the first quadrant For these we draw a unit circle. For multiples of 9, we just read the values of the unit circle. For example, 7 = ; 7 = ; 7 is not defined. For other multiples of and 5, we relate the angles to angles in the first quadrant. and read off the trig ratios, taking note of whether it is positive or negative. For example, suppose we wanted to know the, or of 9. First, we mark the angle on a unit circle and write in the angle between it and the x-axis. Then we draw the reflection in the x-axis, then the reflection in the y-axis. 9 9 Then draw right-angle triangles for each point like this: Now all these triangles are congruent. So we can see that the 9 is the same as, but negative (below the x-axis), 9 is the same as, but negative (left of the y-axis), and 9 is the same as (same gradient). 9 So 9 = ½, 9 =, 9 =. In some cases, you can get the answers with fewer steps. For insce, if the angle is in the fourth quadrant (between 7 and 6), you don t need to reflect in the y-axis. Eventually, you might be able to do this in your head without drawing any extra lines. It is worth still drawing a quick circle diagram, though, so you are less likely to make mistakes, particularly in deciding between positive and negative. Black Star Maths M5- Exact Trig Values Page

4 Some people like to use the CAST rule to decide between positive and negative. This provides a way of getting the right answer without really undersding what you are doing. As such, it is not a recommended way to learn. If you want to know it though, a quick Internet search will find it. Practice P Use unit circle diagrams to find the, and of each of the following angles. Check your answers on your calculator. (a) (b) 5 (c) (d) 5 (e) 765 (f) 6 (g) 5 (h) 6 (i) 9 (j) 5 (k) 5 (l) 75 Solve S Find all the angles between and 5 which have a coe of. S Given that (A + B) A B + A B for any angles A and B, find the exact value of 75 without ug a calculator. Revise Revision Set R Complete this table R Use unit circle diagrams to find the, and of each of the following angles. Check your answers on your calculator. (a) 5 (b) (c) (d) 9 (e) 75 (f) 675 Answers P Check against the table on the first page. Black Star Maths M5- Exact Trig Values Page

5 S 5, 5 S + R Check against the table on the first page. Black Star Maths M5- Exact Trig Values Page 5