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TF.02.4 - Trig Ratios of TF.02.4 - Trig Ratios of Angles in RadiansAngles in Radians

MCR3U - SantowskiMCR3U - Santowski

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(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°

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(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) =

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(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

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(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

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(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 =

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(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

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(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)

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(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

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(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

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(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,