1 TF.02.4 - Trig Ratios of Angles in Radians MCR3U - Santowski.
-
Upload
kerry-evans -
Category
Documents
-
view
215 -
download
2
Transcript of 1 TF.02.4 - Trig Ratios of Angles in Radians MCR3U - Santowski.
11
TF.02.4 - Trig Ratios of TF.02.4 - Trig Ratios of Angles in RadiansAngles in Radians
MCR3U - SantowskiMCR3U - Santowski
22
(A) Review(A) Review A radian is another unit for measuring angles, which is based upon A radian is another unit for measuring angles, which is based upon
the distance that a terminal arm moves around the circumference of the distance that a terminal arm moves around the circumference of a circlea circle
Our “conversion factor” for converting between degrees and radians Our “conversion factor” for converting between degrees and radians is the fact that 180° = is the fact that 180° = л radiansл radians
1 radian = 57.3° or 180° / 1 radian = 57.3° or 180° / л л 1° = 1° = лл /180° radians = 0.017 radians /180° radians = 0.017 radians
We can convert degrees to radians and vice versa using the above We can convert degrees to radians and vice versa using the above conversion factors:conversion factors:
30° x 30° x лл /180° = /180° = лл /6 radians which we can leave in /6 radians which we can leave in лл notation notation лл /4 radians x 180°/ /4 radians x 180°/ лл = 45° = 45°
33
(B) Finding Trig Ratios of Angles(B) Finding Trig Ratios of Angles
ex 1. sin 1.5 rad = ?ex 1. sin 1.5 rad = ?
Since this is NOT one of our simple, standard Since this is NOT one of our simple, standard angles, I would expect you to use a calculatorangles, I would expect you to use a calculator
To use a calculator, change the mode to radians To use a calculator, change the mode to radians and simply enter sin(1.5) and we get 0.9975and simply enter sin(1.5) and we get 0.9975
ex 2. cos 3.0 rad = cos(3) = ex 2. cos 3.0 rad = cos(3) = ex 3. tan -2.5 rad = tan(-2.5) = ex 3. tan -2.5 rad = tan(-2.5) =
44
(C) Finding Trig Ratios of Angles(C) Finding Trig Ratios of Angles
If we are given our standard angles, I would not allow a calculatorIf we are given our standard angles, I would not allow a calculator
ex 4. tan(3ex 4. tan(3лл/4)/4)
Let’s work through a couple of steps together. 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 We can work in either degrees or radians, but we will start with degrees, since we are
more familiar with angles in degrees:more familiar with angles in degrees:
So firstly, our angle is 3So firstly, our angle is 3л/4 = 135° л/4 = 135° so we want to know the tan ratio of a 135° angle 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. (i) draw a diagram and show the principle angle and then the related acute. (ii) from the related acute, find the trig ratio(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 (iii) from the quadrant we are in, determine the sign of the trig ratio in that given
quadrantquadrant
So tan(3So tan(3л/4) = -1л/4) = -1
55
(C) Finding Trig Ratios of Angles(C) Finding Trig Ratios of Angles
Now we will work in radians, again without a calculatorNow we will work in radians, again without a calculator
ex . tan(3ex . tan(3лл/4)/4)
So firstly, our angle is 3So firstly, our angle is 3л/4 = which means 3/4 x л л/4 = which means 3/4 x л so we move our way so we move our way around the circumference of a circle, such that we move 3 quarters around 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 2half the circle, so we have a л/4 angle in the 2ndnd quadrant quadrant
(i) draw a diagram and show the principle angle and then the related acute. (i) draw a diagram and show the principle angle and then the related acute. (ii) from the related acute, find the trig ratio(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 (iii) from the quadrant we are in, determine the sign of the trig ratio in that
given quadrantgiven quadrant
So tan(3So tan(3л/4) = -1л/4) = -1
66
(D) Examples(D) Examples
ex 1. sin ex 1. sin лл/4 rad = /4 rad = ex 2. cos 3ex 2. cos 3лл/2 rad = /2 rad = ex 3. sin 11ex 3. sin 11лл/6 rad = /6 rad = ex 4. cos -7ex 4. cos -7лл/6 rad = /6 rad = ex 5. tan 5ex 5. tan 5лл/3 rad = /3 rad =
77
(E) Working Backwards – Ratio to Angles(E) Working Backwards – Ratio to Angles
ex 1. sin A = ex 1. sin A = 3/23/2
Since this is one of our standard ratios, you will not have the use of Since this is one of our standard ratios, you will not have the use of a calculatora calculator
So the angle that goes with So the angle that goes with 3/2 and the sine ratio is a 60°, or rather 3/2 and the sine ratio is a 60°, or rather a a л/3 angle л/3 angle
But we know that we must have a second angle with the same ratio But we know that we must have a second angle with the same ratio since the sin ratio is positive, the 2 since the sin ratio is positive, the 2ndnd angle must lie in the 2 angle must lie in the 2ndnd quadrant (due to the positive sine ratio) with a related acute of quadrant (due to the positive sine ratio) with a related acute of л/3 л/3
So then So then л - л - л/3 = л/3 = 2 2л/3 as the 2л/3 as the 2ndnd angle angle
88
(E) Working Backwards – Ratio to Angles(E) Working Backwards – Ratio to Angles
ex. sin A = 0.37ex. sin A = 0.37
Now this is a “non-standard” ratio, so simply use your calculator Now this is a “non-standard” ratio, so simply use your calculator (again in radians mode)(again in radians mode)
Hit sinHit sin-1-1(0.37) and you get 0.379 radians (which converts to (0.37) and you get 0.379 radians (which converts to approximately 21.7°)approximately 21.7°)
This angle of 0.379 radians is only the 1This angle of 0.379 radians is only the 1stst quadrant angle, though quadrant angle, though there is also a 2there is also a 2ndnd quadrant angle whose sin ratio is a positive 0.37 quadrant angle whose sin ratio is a positive 0.37 and that would be and that would be л – 0.379 = 2.76 radians (since 0.379 is the л – 0.379 = 2.76 radians (since 0.379 is the related acute)related acute)
99
(F) Further Examples(F) Further Examples
ex 1. cos A = 0.54ex 1. cos A = 0.54 ex 2. tan A = 2.49ex 2. tan A = 2.49 ex 3. sin B = -0.68ex 3. sin B = -0.68 ex 4. cos B = -0.42ex 4. cos B = -0.42 ex 5. tan B = -1.85ex 5. tan B = -1.85
1010
(G) Internet Links(G) Internet Links
Try the following on-line quiz:Try the following on-line quiz:
Trigonometry Review from Jerry L. Trigonometry Review from Jerry L. StanbroughStanbrough
Go to the second quizGo to the second quiz
1111
(H) Homework(H) Homework
AW, p300, Q1-12AW, p300, Q1-12 HandoutsHandouts
Nelson text, p532, Q5,7,10bd,11cd,Nelson text, p532, Q5,7,10bd,11cd,