Trigonometry/ProOneSoftware-Trigonometry-Win31.iso/trig/chapter0.5r

Text File | 1995-04-09 | 6KB | 203 lines

 201
à 0.5ïMeasurenent of Angles
äïPlease convert the following angle measurements from
êêone unit of measure to another.
âSïChange 38°45'22" toêëConvert 45° to radian measure.
êèdegree-decimal form.êêë πêê π
#êêêêêë xè=è──── ∙ 45°ï=è──
êè38°45'22" = 38.756°êêë 180°êë 4
êêêêêë Thus 45° is equal to π/4.
êêêè (Please see Details)

éSêêèAn angle is defined to be two half-lines with
@fig0501.bmp,25,15
êêêthe same initial point called the vertex.ïThe
êêêfirst side is called the initial side, and the
êêêsecond side is called the terminal side of the
êêêangle.ïThe angle is considered to be positive if
êêêit is found by rotating the terminal side in the
êêêcounter-clockwise direction from the initial side.
êêêA clockwise rotation is said to be a negative angle.
è The size or measure of an angle can be given in several different
units of measure.ïWe will measure angles in degrees, radians, or revo-
lutions.ïBasically ç three measures go from the smallest unit (the
degree) to the largest unit (the revolution).ïSometimes it will be more
convenient to use one measure instead of the other two.
è Degree measurement can be traced back to the ancient Sumerians
(4000-2000 BC).ïIt is possible that they used 360° based on their cal-
endar having 360 days.ïEach degree is divided into 60 minutes, and each
minute is broken into 60 seconds.ïThe use of 60 could be attributed
to their use of 60 as a base for their number system.ïAn angle such as
43°23'42" is read 43 degrees, 23 minutes, and 42 seconds.ïA more useful
form is degree-decimal form.ïYou can change this angle to degree-deci-
mal form by first dividing 60 into the seconds and then into the min-
utes.
êêï43°23'42"è=è43°23.7'è=è43.395°
You can also do this on your built-in calculator.ïJust enter the angle,
in the form, 43.2342, press the "dd" button and you will see the degree-
decimal form.ïTo change this back to degree, minute, second form, just
press the "dms" botton.
êêêèThe second unit of measure of an angle is call-
êêêed radian measure.ïThe radian measure of an angle
êêêis defined to be the ratio of the arc length "s"
êêêto the radius "r" (Θï=ïs/r).ïWhile there are
êêê360 equal parts of a circle in degree measure,
@fig0502.bmp,15,175
êêêthere are only just over 6 equal parts of a circle
êêêin radian measure.ïLater, when we use formulas for
êêêarc length, linear velocity, and angular velocity,
êêêit will be necessary to use only radian measure in
êêêç formulas. The third and largest unit of
êêêmeasure of an angle is the number of revolutions.
êêêSince one revolution equals the whole circle, it
êêêwill be useful only for very large angles.ïFor
êêêexample, it will be useful when describing the
êêênumber of revolutions of an engine crankshaft.
êêêRemember, an angle can be given in any of the three
êêêunits of measure.ïAlso, the measures go from
êêêsmallest to largest.ïA degree is 1/360 of a
êêêcircle, a radian is just under 1/6 of a circle,
êêêand a revolution is the whole circle.
 1ê Change 65°42'26" to degree-decimal form.


êêè A)ï65.832°êëB)ï65.36°

êêè C)ï65.707°êëD)ïå
üë Divide 60 first into seconds and then into minutes.

êê65°42'26"è≈è65°42.4333'è≈è65.707°
è You can also do this on your built-in calculator.ïJust enter the
angle in the form (65.4226), and press the "dd" button.ïYou will see
angle displayed in decimal form.
Ç C
 2ê Change 83.069° to degree, minute, second form.


êêè A)ï83°12'55"êèB)ï83°4'8"

êêè C)ï83°22'14"êèD)ïå
üë Multiply 60 times the decimal part to get the minutes,
then multiply 60 times the decimal part of the minutes to get the sec-
onds.
êê83.069°è=è83°4.14'è≈è83°4'8"

è You can also do this on your built-in calculator.ïJust enter the
angle in the form (83.069), and press the "dms" button.ïYour answer
will appear as 83.04084, but you should interpret this as approximately
83°4'8".
Ç B
 3êëChange 45° to radian measure.


êêè A)ïπ/4êêïB)ïπ/12

êêè C)ïπ/6êêïD)ïå
üèSince the circumference of a circle is 2πr, the radian measure
for the whole circle is Θ = 2πr/r = 2π.ïThus 360° equals 2π radians.
A convenient halfway mark is 180° equals π radians.ïSince degree and
radian measures of angles are proportional, we can set up the following
equation.
êëxêïπêêëπêê π
#êè ───è=è────èorèxè=è──── ∙ 45°è=è─
êè 45°ê180°êêï180°êë 4
Thus 45° = π/4 radians.ïWe can change any angle from degree measure
to radian measure by multiplying by π/180°.ïYou can also do this on
your built-in calculator.ïJust enter 45° and click on the "Rad" button.
You will see .785398∙∙∙, which is the decimal form of π/4.
Ç A
 4êëChange 2π/3 to degree measure.


êêè A)ï60°êêïB)ï120°

êêè C)ï135°êê D)ïå
üèSince 180° equals π radians, we can set up a proportion to find
2π/3 in degree measure.
êêêê xêï180°
#êêêê────è=è────
êêêê2π/3ê π

êêêêè 180°è2π
#êêêèxè=è──── ∙ ──è=è120°
êêêêëπë 3
Thus 2π/3 radians equals 120°.ïTo change any angle from radian measure
to degree measure, just multiply by 180°/π.ïTo use your calculator,
first click on the "Rad" button, then enter "2" times "pi" divided
by "3", and finally press enter.ïNext, click on the "Deg" button and
you will see 120°.
Ç B
 5êëChange 760° to revolutions.


êêëA)ï2.5 rev.êè B)ï2 rev.

êêëC)ï2.11 revêè D)ïå

ü
êWe can set up a proportion using the fact that 360° equals one
êrevolution.
êêêêxêï1 rev
#êêêë ────è=è─────
êêêë 760°ê 360°

êêêê1 rev
#êêë xè=è───── ∙ 760°è≈è2.11 rev
êêêê 360°
Ç C
 6êè Change 23 revolutions to degrees.


êêëA)ï156°êê B)ï389°

êêëC)ï8,280°êë D)ïå

ü
êWe can set up a proportion using the fact that 360° equals one
êrevolution.
êêêêxêë360°
#êêêë ──────è=è─────
êêêë 23 revê1 rev

êêêê 360°
#êêë xè=è───── ∙ 23 revè≈è8,280°
êêêê1 rev
Ç C
 7êèChange 6.4π radians to revolutions.


êêëA)ï3.2 revêëB)ï5.7 rev

êêëC)ï208 revêëD)ïå

ü
êWe can set up a proportion using the fact that 2π radians equals
êone revolution.
êêêêxêï1 rev
#êêêë ────è=è─────
êêêë 6.4πê 2π

êêêê1 rev
#êêë xè=è───── ∙ 6.4πè=è3.2 rev
êêêêï2π
Ç A
 8êèChange 150 revolutions to radians.


êêëA)ï486êêïB)ï125π

êêëC)ï300πêê D)ïå

ü
êWe can set up a proportion using the fact that 2π radians equals
êone revolution.
êêêêxêè 2π
#êêêè ───────è=è─────
êêêè 150 revê1 rev

êêêêï2π
#êêë xè=è───── ∙ 150 revè=è300π radians
êêêê1 rev
Ç C