1 TF Trig Ratios of Angles in Radians MCR3U - Santowski

2 (A) Review  A radian is another unit for measuring angles, which is based upon the distance that a terminal arm moves around the circumference of a circle  Our “conversion factor” for converting between degrees and radians is the fact that 180° = л radians  1 radian = 57.3° or 180° / л  1° = л /180° radians = radians  We can convert degrees to radians and vice versa using the above conversion factors:  30° x л /180° = л /6 radians which we can leave in л notation  л /4 radians x 180°/ л = 45°

3 (B) Finding Trig Ratios of Angles  ex 1. sin 1.5 rad = ?  Since this is NOT one of our simple, standard angles, I would expect you to use a calculator  To use a calculator, change the mode to radians and simply enter sin(1.5) and we get  ex 2. cos 3.0 rad = cos(3) =  ex 3. tan -2.5 rad = tan(-2.5) =

4 (C) Finding Trig Ratios of Angles  If we are given our standard angles, I would not allow a calculator  ex 4. tan(3л/4)  Let’s work through a couple of steps together.  We can work in either degrees or radians, but we will start with degrees, since we are more familiar with angles in degrees:  So firstly, our angle is 3л/4 = 135°  so we want to know the tan ratio of a 135° angle  (i) draw a diagram and show the principle angle and then the related acute.  (ii) from the related acute, find the trig ratio  (iii) from the quadrant we are in, determine the sign of the trig ratio in that given quadrant  So tan(3л/4) = -1

5 (C) Finding Trig Ratios of Angles  Now we will work in radians, again without a calculator  ex. tan(3л/4)  So firstly, our angle is 3л/4 = which means 3/4 x л  so we move our way around the circumference of a circle, such that we move 3 quarters around half the circle, so we have a л/4 angle in the 2 nd quadrant  (i) draw a diagram and show the principle angle and then the related acute.  (ii) from the related acute, find the trig ratio  (iii) from the quadrant we are in, determine the sign of the trig ratio in that given quadrant  So tan(3л/4) = -1

6 (D) Examples  ex 1. sin л/4 rad =  ex 2. cos 3л/2 rad =  ex 3. sin 11л/6 rad =  ex 4. cos -7л/6 rad =  ex 5. tan 5л/3 rad =

7 (E) Working Backwards – Ratio to Angles  ex 1. sin A =  3/2  Since this is one of our standard ratios, you will not have the use of a calculator  So the angle that goes with  3/2 and the sine ratio is a 60°, or rather a л/3 angle  But we know that we must have a second angle with the same ratio  since the sin ratio is positive, the 2 nd angle must lie in the 2 nd quadrant (due to the positive sine ratio) with a related acute of л/3  So then л - л/3 = 2л/3 as the 2 nd angle

8 (E) Working Backwards – Ratio to Angles  ex. sin A = 0.37  Now this is a “non-standard” ratio, so simply use your calculator (again in radians mode)  Hit sin -1 (0.37) and you get radians (which converts to approximately 21.7°)  This angle of radians is only the 1 st quadrant angle, though  there is also a 2 nd quadrant angle whose sin ratio is a positive 0.37  and that would be л – = 2.76 radians (since is the related acute)

9 (F) Further Examples  ex 1. cos A = 0.54  ex 2. tan A = 2.49  ex 3. sin B =  ex 4. cos B =  ex 5. tan B = -1.85

10 (G) Internet Links  Try the following on-line quiz:  Trigonometry Review from Jerry L. Stanbrough Trigonometry Review from Jerry L. Stanbrough Trigonometry Review from Jerry L. Stanbrough  Go to the second quiz

11 (H) Homework  AW, p300, Q1-12  Handouts  Nelson text, p532, Q5,7,10bd,11cd,