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TF.02.4 - Trig Ratios of Angles in Radians

TF.02.4 - Trig Ratios of Angles in Radians. MCR3U - Santowski. (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

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TF.02.4 - Trig Ratios of Angles in Radians

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  1. TF.02.4 - 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 = 0.017 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 0.9975 • 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 2nd 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 2nd angle must lie in the 2nd quadrant (due to the positive sine ratio) with a related acute of л/3 • So then л - л/3 = 2л/3 as the 2nd 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 0.379 radians (which converts to approximately 21.7°) • This angle of 0.379 radians is only the 1st quadrant angle, though  there is also a 2nd quadrant angle whose sin ratio is a positive 0.37  and that would be л – 0.379 = 2.76 radians (since 0.379 is the related acute)

  9. (F) Further Examples • ex 1. cos A = 0.54 • ex 2. tan A = 2.49 • ex 3. sin B = -0.68 • ex 4. cos B = -0.42 • ex 5. tan B = -1.85

  10. (G) Internet Links • Try the following on-line quiz: • 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,

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