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