Trigonometry - PP1

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    RADIAN - DEGREE

    A radian is the angle measure of a perfectly cut piece of pie.

    r

    r

    r

    As soon as we see

    that the arc length

    equals the radius,

    we know the central

    angle is 1 radian.

    Since circumference C of

    a circle with radius r is

    calculated using 2 r,

    It follows that a complete

    rotation will produce an

    angle of 2 radians

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    n rad180 rad

    =

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    RADIAN MEASURE

    When we construct a circle graph, we assume that the area of a sectorof a

    circle is proportional to the sector angle.

    The length of the arc bounding the sector is proportional to the sector angle and is

    called the arc length.

    O

    Arc Length

    Sector Angle

    Sector

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    Arc Length of Sector Sector Angle

    Circumference Full-turn Angle=

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    EX 1:

    Calculate the arc

    length of a sector of a

    circle of radius 20 cmif the sector angle is

    140.

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    EX 1: CALCULATE THE ARC LENGTH OF A

    SECTOR OF A CIRCLE OF RADIUS 20 CM

    IF THE SECTOR ANGLE IS 140.

    Solution Create a visual

    What do we know? r = 20 cm angle = 140

    L = 2r 360

    L = 2r L = (140)(2) (20) = 49cm

    360 360

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    EX 2:

    Find the measure of theangle, to the nearest

    tenth of a degree,

    subtended at the center

    of a circle, radius R, by

    the arc of each length

    a) R b) 2R c) 3R

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    SOLUTIONS

    Go back to the original proportion:

    Arc Length = Sector Angle

    Circumference Full-turn Angle

    What do we know?

    a) R = = 360 = 57.3

    2R 360 2

    b) 2R = = 360 = 114.6

    2R 360

    c) 3R = = (3)(360) = 171.9

    2R 360 2

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    In Example 2 we discovered that an angle of 180/ {approximately 57} is subtended at

    the center of a circle by an arc length of R, where R is the radius.

    Definition: One RADIAN is the measure of an angle which is subtended at the center of acircle by an arc equal in length to the radius of the circle.

    From this definition: 1 radian = 180/

    Multiply both sides by to get the following results:

    radians = 180

    Therefore, a full-tern angle, 360 is equal to 2 radians.

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    DOES THIS LOOK FAMILIAR?

    Arc Length of Sector Sector Angle

    Circumference Full-turn Angle=

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    =

    In DEGREES

    r180

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    =

    In RADIANS

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    EX 3:

    A circle has radius 6.5 cm.Calculate the length of an

    arc of this circle

    subtended by each angle.

    a) 2.4 radians b) 75

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    SOLUTION

    a) L = r= (6.5)(2.4)

    = 15.6 cm

    b) L = r/180

    = (75) (6.5)/180

    = 8.5 cm

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    QUESTIONS

    1. Convert from degrees to radians. Express the answer interms of .

    a) 30 b) 60 c) 225 d) 300

    e) 330 f) 405 g) 120 h) 270

    2. Covert from radians to degrees.

    a) /2rads b) -2 /3rads c) 3/4rads d)2rads

    e) -3/2rads f) 7/4rads g) -11/6rads h) /6rads

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    3. Find the length of the arc which subtends each angle at the

    center of a circle of radius 5cm. Give answers to 1 decimal place.

    a) 2.0 rads b) 3.0 rads c) 1.8 radsd) 6.1 rads e) 4.2 rads f) 0.6 rads

    4. Find the length of the arc of a circle with radius 12 cm that

    subtents each sector angle. Give answers to 1 decimal place wherenecessary

    a) 135 b) 75 c) 105 d) 165

    e) 240 f) 180 g) 310 h) 200

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    5. Find the arc length to the nearest centimetre of the sector

    of a circle with radius:

    a) 7m, if the sector angle is:

    i) 120 ii) 210

    b) 90cm, if the sector angle is:

    i) 30 ii) 225c) 216mm, if the sector angle is:

    i) 135 ii) 300

    6. How many radians are there in:

    a) A full turn b) a half turn c) a quarter turn

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    7. Calculate the arc length to the nearest metre of a

    sector of a circle with radius 6m if the sector

    angle is 140

    8. Two sectors of the same circle have sector angles

    of 35 and 105 respectively. The arc length ofthe smaller sector is 17cm. What is the arc

    length of the larger sector?

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    9. The Earth travels in a nearly circular orbit around the sun.

    The radius of the orbit is about 149 000 000 km.

    a) What is the measure in radians of the angle subtended atthe sun by the position of the Earth at two different times

    24h apart?

    b) About how far does the Earth travel in one day in its orbit

    around the sun?Earth now

    Earth 24 hrs later

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    10. The angular velocityof an object is the angle per unit

    time through which an object rotates about a rotation

    center.

    a) What is the angular velocity in radians per second of a

    car tire of diameter 64cm when the car is travelling at

    100km/h?

    b) Write an expression for the angular velocity in radians

    per second for a car tire of diameter d centimeters

    when the car is travelling at x kilmeters per hour.

    1 Centimeter per Second = 0.036 Kilometers per Hour

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    QUESTIONS

    1. Convert from degrees to radians. Express the answer in terms of .

    a) 30 b) 60 c) 225 d) 300

    e) 330 f) 405 g) 120 h) 270

    2. Covert from radians to degrees.

    a) /2 b) -2 /3 c) 3/4 d)2

    e) -3/2 f) 7/4 g) -11/6 h) /6

    /6 /3 5/4 5/3

    11/6 9/4 2/3 3/2

    90 120 135 360

    -270 315 - 330 30

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    3. Find the length of the arc which subtends each angle at the

    center of a circle of radius 5cm. Give answers to 1 decimal place.

    a) 2.0 rads b) 3.0 rads c) 1.8 rads

    d) 6.1 rads e) 4.2 rads f) 0.6 rads

    4. Find the length of the arc of a circle with radius 12 cm that

    subtents each sector angle. Give answers to 1 decimal place wherenecessary

    a) 135 b) 75 c) 105 d) 165

    e) 240 f) 180 g) 310 h) 200

    10 cm 15cm 9cm

    30.5cm 21cm 3cm

    28.3cm 15.7cm 22cm 34.5cm

    50.2cm 37.7cm 64.9cm 41.9cm

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    5. Find the arc length to the nearest centimetre of the sector

    of a circle with radius:

    a) 7m, if the sector angle is:

    i) 120 ii) 210

    b) 90cm, if the sector angle is:

    i) 30 ii) 225c) 216mm, if the sector angle is:

    i) 135 ii) 300

    6. How many radians are there in:

    a) A full turn b) a half turn c) a quarter turn

    14.7 m 25.6 m

    47.1cm 353.3 cm

    508.7mm 1130.4mm

    2 /2

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    7. Calculate the arc length to the nearest meter of a

    sector of a circle with radius 6m if the sector

    angle is 140

    8. Two sectors of the same circle have sector angles of 35

    and 105 respectively. The arc length of the smaller sectoris 17cm. What is the arc length of the larger sector?

    140 6 180 = 14.7m

    Radius small circle = radius big circle

    LS = 17 = 35 r 180 r = 27.8435

    Using found r and LB equation LB = 51cm

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    9. The Earth travels in a nearly circular orbit around the sun.

    The radius of the orbit is about 149 000 000 km.

    a) About how far does the Earth travel in one day in its orbit around the sun?

    b) What is the measure in radians of the angle subtended at the sun by the position

    of the Earth at two different times 24h apart? Earth now

    Earth 24 hrs later

    The Earth circles the sun once a year, how far does it go?

    FIND CIRCUMFERENCE: 935720000km per year 365 days

    Therefore it travels 2563616.43 km per day

    = L/r = 00.0172 rads

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    10. The angular velocityof an object is the angle per unit

    time through which an object rotates about a rotation

    center.a) What is the angular velocity in radians per second of a car tire of diameter 64cm

    when the car is travelling at 100km/h?

    1 Centimeter per Second = 0.036 Kilometers per Hour

    Angular Velocity = 360/t (seconds) = 2/t(seconds)

    Ctire = 200.96

    How many rotations per second?

    100km/hr = 2777.7cm/s Vang= 360 2

    0.0723476 0.07

    2777.7 cm = 200.96 = 4975/s = 86.8 rads

    s ?

    = 0.0723476 seconds per rotation

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    Write an expression for the angular velocity in radians per second for a car tire of

    diameter d centimeters when the car is travelling at x kilometers per hour.

    Vang= 360 = 2 where t is measured in seconds

    t t

    Speed (cm/s) = Circumference time = Circumference = 2r

    time Speed y

    y (cm/s) = 0.036x (km/h)

    we need d not r

    Vang= 2

    d 2 1

    0.036x d 0.036X