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0.2: Trignometric FunctionsObjective:•Understand
relationships between angles and sides of right triangles•Review trig functions of special & quadrantal angles
©2008 Roy L. Gover (www.mrgover.com)
Calculus Prerequisite Knowledge
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DefinitionThe special angles are:
•30°,60°, and 45° in degrees
• and in radians , ,
6 3 4
In calculus, we use radians.
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Important Idea
These angles are special because they have exact value trig functions.
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Consider the first two special angles…
Long Side
Short
Sid
e
Hypotenus
e 6
3
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Lon
g Sid
e
Hypotenuse 6
3
Short Side
…orientation does not change the relationships between sides and angles
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Lon
g Sid
eHyp
oten
use 6
3
Short Side
…orientation does not change the relationships between sides and angles
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Analysis
Consider an equilateral triangle with sides 2 units long
2
2
2
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Analysis...equilater
al triangles are equiangular and each angle must be
2
2
2
3
3
3
3
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Analysis
…the angle bisector intersects the base at the midpoint
2 2
1 13
3
6
6
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Analysis
2 2
1 1
1
2?
?= 3
6
6
6
3
3
3
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Try ThisFind the height of an equilateral triangle with each side 4 inches long.
4 4
4
?
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Important IdeaIn a rad triangle:, ,
6 3 2
•the short side is one-half the hypotenuse.
•the long side is times the short side.
3Memoriz
e
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Try ThisFind the length of the missing sides:
6
10
5
5 3
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Try This
Find the length of the missing sides:
3
5
5 3
3
10 3
3
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Try ThisWhat is wrong with this picture? 6
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Consider the last special angle…
6
3
4
4
4
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Hypot
enus
e
…orientation does not change the relationships between sides and angles
4
4
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Hypotenuse
…orientation does not change the relationships between sides and angles
4
4
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…and sides opposite equal angles are equal... 4
4
x
x
and by the2x
pythagorean theorem, the hypotenuse is...
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Important IdeaIn a triangle:
•The legs of the triangle are equal.
•the hypotenuse is times the length of the leg.
Memoriz
e2
, ,4 4 2
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Try ThisFind the length of the missing sides
4
4
2
22 2
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Try ThisFind the length of the missing sides
2
2
24
4
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Review
•
-1 1
-1
1
x
yr
oppositesin
hypotenuse
y
r
adjacentcos
hypotenuse
x
r
oppositetan
adjacent
y
x
•
• Memorize
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Review
•
-1 1
-1
1
x
yr
hypotenusecsc
opposite
r
y
hypotenusesec
adjacent
r
x
adjacentcot
opposite
x
y
•
• Memorize
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Find the exact values of the six trig
3
-1 1
-1
1
Example
functions of the given special angle in standard position.
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-1 1
-1
1
Analysis
1. Make a right triangle.
rads3
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-1 1
-1
1
Analysis
2. Let the short side be 1, then use your knowledge of special angles
32
1
to find the measures of each side of the right triangle.
rads3
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-1 1
-1
1
Analysis
32
1
3. Use the definitions of the trig functions to find each trig function exact value.
rads3
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-1 1
-1
1
Analysis
32
1
opp 3sin
hyp 2
adj 1cos
hyp 2
opp 3tan 3
adj 1
rads3
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-1 1
-1
1
Analysis
32
1
hyp 2 3csc
opp 3
hypsec 2
adj
adj 3cot
opp 3
rads3
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Find the exact values of the six trig
-1 1
-1
1Try This
functions of the given special angle in standard position.
4
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Solution2
sin4 2
2cos
4 2
tan 14
csc 24
sec 24
cot 14
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Important IdeaIf the given angle is other than a quadrant I angle, the exact values of the trig functions are calculated using the reference angle of the given angle.
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Review
Reference Angle: the angle between a given angle and the nearest x axis. (Note: x axis; not y axis). Reference angles are always positive.
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Review
How you find the reference angle depends on which quadrant contains the given angle.
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Review
2
1
-1
-2
-2 -1 1 2 3
ReferenceAngle
GivenAngle &
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Review2
1
-1
-2
-3 -2 -1 1 2
GivenAngleReference
Angle
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Review
-3 -2 -1 1 2
2
1
-1
-2
ReferenceAngle
GivenAngle
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Review
-3 -2 -1 1 2
2
1
-1
-2
ReferenceAngle
GivenAngle
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ExampleDraw a diagram showing the relevant angles and sides and find the sin, cos & tan of
5
6
Given Angle
Ref. Angle6
5
6
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Example
5
6
6
Short Side
Long Side
Given Angle
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Example
5
6
6
2
1
3
Why is 1 & 2 pos. & neg.?
3
5 1sin
6 2
5 3cos
6 2
5 3tan
6 3
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Try ThisDraw a diagram showing the relevant angles and sides and find the csc, sec & cot of
5
6
Given Angle
Ref. Angle6
5
6
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Solution
5
6
6
2
1
3 5csc 2
6
5 2 3sec
6 3
5cot 3
6
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-1 1
-1
1Find the values of the six trig functions of the given angle in standard position.
2
sincostan
cscseccot
Quadrantal Angles
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-1 1
-1
1Find the values of the six trig functions of the given angle in standard position.
2
sincostan
cscseccot
Quadrantal Angles
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-1 1
-1
1Find the values of the six trig functions of the given angle in standard position.
sincostan
cscseccot
Quadrantal Angles
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-1 1
-1
1Find the values of the six trig functions of the given angle in standard position.
3
2
sincostan
cscseccot
Quadrantal Angles
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Important Ideas•Trig functions of quadrantal angles have exact values.
•Trig functions of all other angles have approximate values.
•Trig functions of special angles have exact values.
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Example
Use a calculator to approximate cos to 4 decimal places.
You did not forget to check “mode”, did you???
4
9
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Example
Use a calculator to approximate csc(11.55) to 4 decimal places.
Check “mode” every time.
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Lesson Close
You will frequently use the trig knowledge you learned in Precalculus in Calculus
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Assignment
Worksheet 0-2