Lesson 22
Ma 15800
March 8th , 2017
This lesson is one of many to come of the lessons in trigonometry. Today we start off by
working with angles. Recall there are a few ways to represents angles, in degrees, in radians,
and later we’ll see in degrees ,minutes, and seconds (DMS). We start of with a definition.
We say two angles are Coterminal if their terminal sides are the same. Pictorally, we see
that the following angles are coterminal,
β
θ
α
Now it becomes clear that two angles are coterminal if and only if they differ by an integer
multiple of 360˝ or 2π radians (depending on the angle system we’re in).
Example 1: List some coterminal angles for
(a) 45˝
(b)
π
3
Solution: We just need to add integer multiples of 360˝ and 2π radians to each angle.
(a) 405˝ , 765˝ , ´315˝ , ´675˝ , . . . etc.
(b)
7π 13π ´5π ´11π
,
,
,
, . . . etc.
3 3
3
3
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Now we should be familiar with degrees and radians already. Naturally we might ask is there
a way to convert between the two? The answer is yes, and below are the formulas to convert
degrees to radians and vice-versa.
Degrees ñ Radians:
pangle in degreesq ¨
π
“ pangle in radiansq
180˝
180˝
“ pangle in degreesq
π
Now that we are comfortable with both degrees and radians, let us introduce a new notation for write angles. This isn’t so much of a ”new” angle system but rather and ”old”
one as it was mostly used for navigation before the use of GPS. This is what’s known as
Degrees, Minutes, and Seconds notation (which we write DMS for short). The goal is to
eliminate the messiness and uncertainty of long annoying decimals. The analogy is to think
of one degree as one hour on a clock. You would never hear somebody say 1.5 hours, but
rather one hour thirty minutes (and however many seconds). Let us do an example.
Radians ñ Degrees:
pangle in radiansq ¨
Example 2: Convert 57.5992˝ to degrees, minutes, seconds (DMS) notation. Round to the
nearest second.
Solution: We split this into degrees, minutes, and seconds (hence the name)
. Clearly there 57˝ full degrees
. Now we wish to consider the decimal 0.5992 and ask how much of an hour (one degree)
does this represent. The next step down from hours is minutes. So we ask how many full
minutes is 0.5992 hours? 0.5992 ¨ 60 “ 35.952 ùñ 35 minutes
. Now we repeat the process with the new decimal 0.952. How many seconds are there in
0.952 minutes? 0.952 ¨ 60 “ 57.12 ùñ 57 seconds
Now we stop as going to seconds is precise enough. Thus we round to the nearest second
and write
57.5992˝ “ 57˝ , 351 , 572
Example 3: Convert 13˝ , 501 , 152 to decimal degrees. Round to 4 decimal places.
Solution: Now we try to unravel the process above. Luckily this is a little easier as we know
there are 60 minutes in a hour (hence a degree) and 60 ¨ 60 “ 3600 seconds in an hour. Then
converting each part to hours(degrees) and summing them up, we obtain
13 `
50
15
`
« 13.8375˝
60 3600
Note that we have only illustrated the process to go between decimal degrees and DMS
notation. Unfortunately, there is no direct way to convert between radians and DMS or the
other way around. In order to this, we must first convert to degrees and then we may convert
to the format we desire.
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Let us continue with a few definitions. We define the complement of an angle θ (which we
shall denote θc ), to be the angle such that
θc ` θ “ 90˝ or
π
2
Pictorally,
θc
θ
Clearly, if θc ` θ “ 90˝ then θc “ 90˝ ´ θ (alternatively
this gives an easy way to find the complement.
π
2
if we’re working in radians) and
For some angle θ we define the suppplement of θ (which we shall denote as θs ) to be the
angle such that
θs ` θ “ 180˝ or π
Pictorally,
θ
θs
Clearly, if θs ` θ “ 180˝ then θs “ 180˝ ´ θ (alternatively π if we’re working in radians) and
this gives an easy way to find the supplement.
Note that it is very straight forward to subtract angles from 90˝ or 180˝ ( or π2 and π if in
radians) if the given angle is in degrees or radians. What about DMS?
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Example 4: Given the angle θ “ 57˝ , 61 , 582 find the following
(a) The complement of θ
(b) The supplement of θ
Solution:
(a) Somehow we need to evaluate 90˝ ´ 57˝ , 61 , 582 . To do this we write 90˝ in a clever
way to put it into DMS notation. We write 90˝ “ 89˝ , 591 , 602 . Then we see that
subtraction follows piece by piece,
89˝ 591 602
´ 57˝ 61 582
32˝ 531
22
Thus the complement of θ is 32˝ , 531 , 22
(b) Similarly, we write 180˝ “ 179˝ , 591 , 602 . Then
179˝ 591 602
´ 57˝ 61 582
122˝ 531
Thus the supplement of θ is 122˝ , 531 , 22
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Next we define, for some angle θ, the acute reference angle, which we denote θR to be the
smallest acute angle that can be made with any part of the x-axis and the terminal side of
θ.
In general, the best way to find an acute reference angle is to draw a picture where the angle
θ lies and then determine which part of the x-axis to use. In some cases the acute reference
angle will be the supplement, but not always.
Example 5: Find the acute reference angle for the following angles
(a)
5π
4
(b) 300˝
(c) 45˝
Solution: As stated, the best way to precede is to draw a picture where the angle lies.
(a)
5π
4
θR
Now with θR the smallest positive angle made with the axis, we only need to compute
π
5π
´ π. Thus θR “
4
4
(b)
300˝
θR
5
Now we only need to compute 360˝ ´ 330˝ . Thus θR “ 60˝
(c)
45˝
θR
In this case, because θ is acute, it is equal to its acute reference angle. Thus θR “ 45˝
Lastly we define the quadrants as the four regions broken up by the x and y axes. They are
labeled as follows
II
I
II
IV
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