Download Exact Trig Values of selected Trig. Functions

Lecture 6–2:
Finding the EXACT VALUE for selected trigonometric functions.
Reducing Complex Fractions
Example 1
Example 2
1
2
2
1. Multiply top and bottom by 2.
2• 1
2
=
2
2
2•
2
2. Multiply top and bopttom by 2.
Reduce
1
1
2
1. Multiply top and bottom by 2.
2 •1
=2
1
2•
2
Reduce
Example 3
1
3
2
1. Multiply top and bottom by 2.
2• 1
2
=
3
3
2•
2
2. Multiply top and bottom by 3.
Reduce
2• 3
3• 3
=
2 3
3
2• 2
2• 2
=
2 2
= 2
2
Example 4
1
2
Reduce
3
2
1. Multiply top and bottom by 2.
1
2•
2 = 1
3
3
2•
2
2. Multiply top and bottom by 3.
1• 3
3• 3
=
3
3
Division by zero
1
0
Division by 0 is undefined
Reduce
Math 370 Section 6 – 2
Page 1
© 2015 Eitel
The Quadrangles and the Reference Angles
(0,1)
S
1
A
(1,0)
(−1,0)
C
T
π
30 6
π
45 4
3
2
(0,−1)
2
2
1
1
2
1
3
2
π
60 3
1
2
2
2
If r = 1 then the six trig functions in terms of x and y are defined to be:
sin θ = y
cos θ = x
tan θ =
y
x
csc θ =
1
y
sec θ =
1
x
cot θ =
x
y
Find the EXACT VALUE for the following trigonometric functions:
Example 1
Example 2
⎛ 5π ⎞
⎝ 3⎠
1. Find the quadrant the angle is in
5 •π
↔ 5(60) = 300o is in the 4th quadrant
3
2. Find the reference angle.
π
RA = 60o or
3
3. Use definition of the function and
and the reference triangles to find
the value of reference angle.
⎛ 11π ⎞
⎝ 6 ⎠
1. Find the quadrant the angle is in
11• π
↔ 11(30) = 330o is in the 4th quadrant
6
2. Find the reference angle.
π
RA = 30o or
6
3. Use definition of the function and
and the reference triangles to find
the value of reference angle.
sin
csc
3
⎛π ⎞
= x=
⎝ 3⎠
2
4. Find the + or − sign for the angle.
sin
csc
(
(1/ 2)
)
3 /2
=
2 3
3
4. Find the + or − sign for the angle.
sinθ is negitive in the 4th quadrant
csc θ is positive in the 4th quadrant
5π
− 3
sin ⎛ ⎞ =
⎝ 3⎠
2
Math 370 Section 6 – 2
⎛π⎞ 1
= =
⎝ 6⎠ y
csc
Page 2
⎛ 11π ⎞ − 2
=
⎝ 6 ⎠
2
© 2015 Eitel
Example 3
Example 4
⎛ 3π ⎞
⎝ 4⎠
1. Find the quadrant the angle is in
2 •π
↔ 3(45) = 135o is in the 2nd quadrant
4
2. Find the reference angle.
π
RA = 45o or
4
3. Use definition of the function and
and the reference triangles to find
the value of reference angle.
cos
⎛ 5π ⎞
⎝ 4⎠
1. Find the quadrant the angle is in
5 •π
↔ 5(45) = 225o is in the 3rd quadrant
4
2. Find the reference angle.
π
RA = 45o or
4
3. Use definition of the function and
and the reference triangles to find
the value of reference angle.
sec
π
1
sec ⎛ ⎞ = =
⎝ 4⎠ x
2
⎛π ⎞
= x=
⎝ 4⎠
2
4. Find the + or − sign for the angle.
4. Find the + or − sign for the angle.
cosθ is negitive in the 2nd quadrant
sec θ is negitive in the 3 rd quadrant
3π
− 2
cos ⎛ ⎞ =
⎝ 4⎠
2
sec
cos
Example 5
⎛ 7π ⎞
⎝ 6 ⎠
1. Find the quadrant the angle is in
7• π
↔ 7(30) = 210o is in the 3rd quadrant
6
2. Find the reference angle.
π
RA = 30o or
6
3. Use definition of the function and
and the reference triangles to find
the value of reference angle.
⎛π⎞ y
= =
⎝ 6⎠ x
(
(1/ 2)
)
3 /2
=
)
⎛ 5π ⎞
=− 2
⎝ 4⎠
Example 6
tan
tan
(
1
= 2
2 /2
⎛ 7π ⎞
⎝ 4 ⎠
1. Find the quadrant the angle is in
7• π
↔ 7(45) = 315o is in the 4th quadrant
4
2. Find the reference angle.
π
RA = 45o or
4
3. Use definition of the function and
and the reference triangles to find
the value of reference angle.
cot
3
3
cot
⎛π⎞ x
= =
⎝ 4⎠ y
(
(
)=1
2 / 2)
2 /2
4. Find the + or − sign for the angle.
4. Find the + or − sign for the angle.
tanθ is positive in the 3 rd quadrant
cot θ is negitive in the 4th quadrant
tan
3
⎛ 7π ⎞
=
⎝ 6 ⎠
3
Math 370 Section 6 – 2
cot
Page 3
⎛ 7π ⎞
= −1
⎝ 4 ⎠
© 2015 Eitel
Example 7
⎛ 15π ⎞
tan
⎝ 4 ⎠
1. Find the quadrant the angle is in
15 • π 8• π 7• π
−
=
↔ 7(45) = 315o
4
4
4
315o is in the 4th quadrant
2. Find the reference angle.
π
RA = 45o or
4
3. Use definition of the function and
and the reference triangles to find
the value of reference angle.
tan
⎛π⎞ y
= =
⎝ 4⎠ x
(
(
)=1
2 / 2)
Example 8
⎛ −5π ⎞
cot
⎝ 4 ⎠
1. Find the quadrant the angle is in
−5 • π
↔ −5(45) = −225 o
4
is 45 in the 2nd quadrant
2. Find the reference angle.
π
RA = 45o or
4
3. Use definition of the function and
and the reference triangles to find
the value of reference angle.
2 /2
4. Find the + or − sign for the angle.
cot
π
1
cos ⎛ ⎞ = x =
⎝ 3⎠
2
4. Find the + or − sign for the angle.
cosθ is positive in the 1st quadrant
cos ⎛
⎝
−11π ⎞ 1
=
3 ⎠ 2
Math 370 Section 6 – 2
)=1
2 / 2)
2 /2
cot θ is negitive in the 2nd quadrant
⎛ 15• π ⎞
= −1
⎝
4⎠
Example 9
⎛ −11π ⎞
cos
⎝ 3 ⎠
1. Find the quadrant the angle is in
−11• π −5π π
=
=
3
3
3
π
↔ 60o in the 1st quadrant
3
2. Find the reference angle.
π
RA = 60o or
3
3. Use definition of the function and
and the reference triangles to find
the value of reference angle.
(
(
4. Find the + or − sign for the angle.
tanθ is negitive in the 4th quadrant
tan
⎛π⎞ x
= =
⎝ 4⎠ y
cot
⎛ −5π ⎞
= −1
⎝ 4 ⎠
Example 10
⎛ −11π ⎞
⎝ 6 ⎠
1. Find the quadrant the angle is in
−11• π π
= ↔ 30o in the 1st quadrant
6
6
2. Find the reference angle.
π
RA = 30o or
6
3. Use definition of the function and
and the reference triangles to find
the value of reference angle.
csc
1
⎛ −11π ⎞ 1
= =
=2
⎝ 6 ⎠ y (1/ 2)
4. Find the + or − sign for the angle.
cosθ is positive in the 1st quadrant
csc
−11π ⎞
csc ⎛
=2
⎝ 6 ⎠
Page 4
© 2015 Eitel
Quadrangle Examples
Example 1
Example 2
⎛π ⎞
⎝ 2⎠
1. Find the reference angle.
π
π
is on the y axis at RA = 90o or
2
2
⎛ 3π ⎞
⎝ 2⎠
1. Find the reference angle.
3π
3π
is on the y axis at RA = 270o or
2
2
2. Use definition of the function and
and the reference quadrants to find
the value of the trig. function.
2. Use definition of the function and
and the reference quadrants to find
the value of the trig. function.
sin
sin
cos
⎛π ⎞
= y=1
⎝ 2⎠
Example 3
csc (π )
1. Find the reference angle.
cos
Example 4
sec (3π )
1. Find the reference angle.
π is on the x axis at RA = 180o or π
2. Use definition of the function and
and the reference quadrants to find
the value of the trig. function.
1 1
csc (π ) = = = Undefined (und)
y 0
3π is on the x axis at RA = 180o or π
2. Use definition of the function and
and the reference quadrants to find
the value of the trig. function.
1 1
sec (3π ) = =
= −1
x −1
Example 5
⎛ 3π ⎞
csc
⎝ 2⎠
1. Find the reference angle.
3π
3π
is on the y axis at RA = 270o or
2
2
2. Use definition of the function and
and the reference quadrants to find
the value of the trig. function.
csc
⎛ 3π ⎞ 1 1
= =
= −1
⎝ 2 ⎠ y −1
Math 370 Section 6 – 2
⎛ 3π ⎞
= x = −1
⎝ 2⎠
Example 6
cot (π )
1. Find the reference angle.
π is on the y axis at RA = 180o or π
2. Use definition of the function and
and the reference quadrants to find
the value of the trig. function.
x −1
cot (π ) = =
= undefined
y 0
Page 5
© 2015 Eitel
Example 7
Example 8
⎛ 7π ⎞
⎝ 2⎠
1. Find the reference angle.
7π
3π
is on the y axis at RA = 270o or
2
2
⎛ −11π ⎞
⎝ 2 ⎠
1. Find the reference angle.
−9π
π
is on the y axis at RA = 90o or
2
2
2. Use definition of the function and
and the reference quadrants to find
the value of the trig. function.
2. Use definition of the function and
and the reference quadrants to find
the value of the trig. function.
cos
cos
sin
⎛ 7π ⎞
= x =0
⎝ 2⎠
Example 9
sec (−3π )
1. Find the reference angle.
sin
Example 10
csc (−6π )
1. Find the reference angle.
− 3π is on the x axis at RA = 180o or π
2. Use definition of the function and
and the reference quadrants to find
the value of the trig. function.
1
1
sec (π ) = =
= −1
x −01
Example 11
⎛ −9π ⎞
⎝ 2 ⎠
1. Find the reference angle.
−9π
3π
is on the y axis at RA = 270o or
2
2
sec
2. Use definition of the function and
and the reference quadrants to find
the value of the trig. function.
sec
⎛ −9π ⎞ 1 1
= = = Undefined
⎝ 2 ⎠ x 0
Math 370 Section 6 – 2
⎛ −11π ⎞
= y =1
⎝ 2 ⎠
− 6π is on the x axis at RA = 0o or 0
2. Use definition of the function and
and the reference quadrants to find
the value of the trig. function.
1 1
csc (−6π ) = = = 1
x 1
Example 12
tan (−5π )
1. Find the reference angle.
π is on the y axis at RA = 180o or π
2. Use definition of the function and
and the reference quadrants to find
the value of the trig. function.
y 0
tan (−5π ) = =
= 0
x −1
Page 6
© 2015 Eitel