6.3 2nd april 2013
TRANSCRIPT
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6.3 Radian Measure
A 2nd way to measure the rotation of an angle is to use RADIANS.When the terminal arm rotates around a circle the same distance as the radius of the arm, then the angle created has the measure of 1 RADIAN.
r s When s=r in length then the anglehas the measure of 1 radian.
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What is the formula for the circumference of a circle.
C= ________This can be thought of as C=_____radians.
This is equivalent to 360°, as both represent a complete rotationof a circle.
Therefore 180° is equivalent to _____radians.
What would 90° be in radians? _____
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If 360° = 2 r Divide both sides by 2 to find the degree equivalent for 1 radian.Divide both sides by 360° to find the radian equivalentfor 1 degree.
Since every radian is equivalent to we therefore have a formula to change radians to degrees.
Ex) Change the following radian measures into degrees:
a)
b)
c)
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Since every degree has a radian equivalent of thenwe have a formula to change degrees into radians.
Change the following degree measures into radians:
a) 45°
b) 160°
c) 360° (This should be NO suprise!)
d) 500°
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More examples:
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In Grade 11, we learned cosine (x) , sine (y), and tangent valuesfor 30°, 45°, 60°, and 90° positions around the unit circle.
We now switch these positions to radian measure. These angles needto be memorized. (i.e: You need to know that 150° is without using the conversion formula)
Often written
Notice that the tangentvalues are not includedon this diagram but needto be learned as well!
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Practice Questions:
1) What is a positive coterminal angle [0 , 2 ] of:
a)
b)
2) What is the sine value of:
a)
b)
3) What is the cosine value of:
a)
b)
4) What is the tangent value of:
a)
b)
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More Practice:
1.
In question b) since we do NOT know exact values for thedenominator 7, we use our calculator. How will we doCosecant?
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Now let's practice the reverse concept where we FIND the angle,GIVEN the ratio:
Find from [0,2 ] in each of the following given that:
a)
b)
c)
d)
Find from in each of the following:
e)
f)
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Use your calculator to help with these:
a) Find from for
b) Find from for
c)
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Now: given point locations, find trig ratios and then findangle values, in radians.
1) Using P(1,2) find:
a)
b) find from
2) Using P(2,3) find:
a) find and
b) find from
c) find from
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If find :
a)the values for the other 5 trigonometric ratios.
b) find the possible values of if the domain is
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Now try these:
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Now try these:
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The final topic involves the ARC length around the outside of a circle (denoted as "s")
r s
Remember from the beginningof this lesson that if s=r, then anglehas the measure 1 radian.Therefore if s=10cm and r=5cmwhat would be the angle size?,,,if s=20cm and r=5 then =____
We arrive at the following ratio:
This ratio is called the Arc Length ratio and is often denoted
This ratio requires the angle to be in RADIANS!
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Example Questions:
1) If a wheel of radius 40cm rolls 3m, whatis the angle of rotation in DEGREES?
2) A ferris wheel of radius 40m, rotates through an angle of 200°. What is the arc length ofthis rotation?
3) 2 cities are at the same longitude. City A is 29°Nwhile city B is at 43°S. If the earth has a diameterof approximately 12 800 kms, find the distancebetween the 2 cities.
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HOMEWORK:Page494 #4,5,6,7,8
9a),11,12,1414 Mult. Ch. #1,2BONUS: #10