1. Find the Exact Values of the Trigonometric Functions Using a Point on the Unit Circle 2. Find the Exact Values of the Trigonometric Functions of Quadrantal Angles 3. Find the Exact Values of the Trigonometric Functions of p/4 = 45º 4. Find the Exact Values of the Trigonometric Functions of p/6 = 30º and p/3 = 60º 5. Find the Exact Values of the Trigonometric Functions for integer multiples of p/6 = 30º , p/4 = 45º , and p/3 = 60º 6. Use a Calculator to Approximate the Values of a Trigonometric Function 7. Use a Circle of Radius r to Evaluate the Trigonometric Functions §5.2 Trigonometric Functions: Unit Circle Approach Objectives 11 April 2018 1 Kidoguchi, Kenneth
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1. Find the Exact Values of the Trigonometric Functions Using a
Point on the Unit Circle
2. Find the Exact Values of the Trigonometric Functions of
Quadrantal Angles
3. Find the Exact Values of the Trigonometric Functions of
p/4 = 45º
4. Find the Exact Values of the Trigonometric Functions of
p/6 = 30º and p/3 = 60º
5. Find the Exact Values of the Trigonometric Functions for
integer multiples of p/6 = 30º , p/4 = 45º , and p/3 = 60º
6. Use a Calculator to Approximate the Values of a Trigonometric
Function
7. Use a Circle of Radius r to Evaluate the Trigonometric
Functions
§5.2 Trigonometric Functions: Unit Circle Approach
Objectives
11 April 2018 1 Kidoguchi, Kenneth
11 April 2018 2 Kidoguchi, Kenneth
§5.2 Trigonometric Functions: Unit Circle Approach
The Length of an Arc of a Circle
Angle in radians:
arc length
radius
s
r
(1, 0)
1 1s r
1
2 2s r
2 3
3 3s r
21 revolution 2
r
r
p p
N.B.:
• An angle in radians is
length over length,
hence a dimensionless
quantity.
• For a circle of radius r, a
central angle in
radians subtends an arc
of length s such that:
s = r
x
y
(1, 0)
§5.2 Trigonometric Functions: Unit Circle Approach
Unit Circle and a Point on a Unit Circle
11 April 2018 3Kidoguchi, Kenneth
A unit circle is a circle of radius 1 unit centered at the origin of a
Cartesian (i.e., rectangular) coordinate system.
P = (x, y)
s = t > 0
x
y
(1, 0)
P = (x, y)
s = t < 0
Let t be a real number and let P = (x,y) be the point on the unit circle
that corresponds to t.
§5.2 Trigonometric Functions: Unit Circle Approach
Trigonometric Functions on a Unit Circle
11 April 2018 4 Kidoguchi, Kenneth
Let t be a real number and let P = (x,y) be the point on the unit circle (i.e.,
a circle with r = 1) that corresponds to t.
The sine function associates with t the
y-coordinate of P and is denoted by:
The cosine function associates with t the
x-coordinate of P and is denoted by:
If x ≠ 0, the tangent function associates with t
the ratio of the y-coordinate to the y-
coordinate of P and is denoted by:
If y ≠ 0, the cosecant function defined as:
If x ≠ 0, the secant function defined as:
If y ≠ 0, the cotangent function defined as:
sin( )1
y yt
r
cos( )1
x xt
r
tan( )y
tx
1csc( )
rt
y y
1sec( )
rt
x x
cot( )x
ty
§5.2 Trigonometric Functions: Unit Circle Approach
1. Exact Values of the Trig Functions Using a Point on the Unit Circle
11 April 2018 5 Kidoguchi, Kenneth
Let t be a real number and let be a point on the unit
circle that corresponds to t.
Find the values of sin(t), cos(t), tan(t), csc(t), sec(t), and cot(t).
1 2 6 = ,
5 5P
11 April 2018 7 Kidoguchi, Kenneth
§5.2 Trigonometric Functions: Unit Circle Approach
Trigonometric Functions of Angles
x
y
(1, 0)
sin1
cos1
y
x
(x1,y1)
Quad I
1
1
sin 0
cos 0
1
(x2,y2)
Quad II2
(x3,y3)
Quad III
3
2
2
sin 0
cos 0
3
3
sin 0
cos 0
(x4,y4)
Quad IV4
4
4
sin 0
cos 0
1 = t1
2 = t2
3 = t3 4 = t4
§5.2 Trigonometric Functions: Unit Circle Approach
Trigonometric Functions of Angles - Definition
11 April 2018 8 Kidoguchi, Kenneth
If = t radians, the six trigonometric functions of the angle are
defined as:
sin sin , cos cos , tan tan
csc csc , sec sec , cot cot
t t t
t t t
§5.2 Trigonometric Functions: Unit Circle Approach
2. Exact Values of Trigonometric Functions of Quadrantal Angles
11 April 2018 9 Kidoguchi, Kenneth
Find exact values of the six trigonometric functions of:
a) 0 0º b) p/2 90º
§5.2 Trigonometric Functions: Unit Circle Approach
2. Exact Values of Trigonometric Functions of Quadrantal Angles
11 April 2018 10 Kidoguchi, Kenneth
Find exact values of the six trigonometric functions of:
a) p 180º b) 3p/2 270º
§5.2 Trigonometric Functions: Unit Circle Approach
Exact Values of Trigonometric Functions of Quadrantal Angles
11 April 2018 11 Kidoguchi, Kenneth
Quadrantal Angles
(rad) (º ) sin() cos() tan() csc() sec() cot()
0 0 0 1 0 und 1 und
p/2 90 1 0 und 1 und 0
p 180 0 1 0 und -1 und
3p/2 270 1 0 und 1 und 0
und = undefined
§5.2 Trigonometric Functions: Unit Circle Approach
Exact Values of Trigonometric Functions of Quadrantal Angles
11 April 2018 12 Kidoguchi, Kenneth
(a) sin(5p) (b) cos(–540°)
Find the exact value of:
§5.2 Trigonometric Functions: Unit Circle Approach
3. Exact Values of Trigonometric Functions of = p/4
11 April 2018 13 Kidoguchi, Kenneth
p/4
2/8
3/8
5/8
6/8
7/8
x
y
(1, 0)04/8
(x,y)
Initial Side
Terminal Side
§5.2 Trigonometric Functions: Unit Circle Approach
4. Exact Values of Trigonometric Functions of = p/3 & p/6
11 April 2018 15 Kidoguchi, Kenneth
10 212
2
1
1
2
y
x
11 April 2018 16 Kidoguchi, Kenneth
§5.2 Trigonometric Functions: Unit Circle Approach
4. Exact Values of Trigonometric Functions of = p/3 & p/6
3sin
3 2
1cos
3 2
y
r
x
r
p
p
0
1/12
2/123/12
4/12
5/12
6/12
7/12
8/12
9/12
10/12
11/12
(1, 0)
ry
ry
rry
ryr
ryx
rx
2
3
2/
2/
2
432
2
4122
222
222
(x,y)
r
p/3
r/2 r/2
11 April 2018 17 Kidoguchi, Kenneth
§5.2 Trigonometric Functions: Unit Circle Approach
4. Exact Values of Trigonometric Functions of = p/3 & p/6
0
1/12
2/123/12
4/12
5/12
6/12
7/12
8/12
9/12
10/12
11/12
(1, 0)
(x,y)
r
p/6
1sin cos
6 3 2
3cos sin
6 3 2
y
r
x
r
p p
p p
rx
rx
rrx
rrx
ryx
ry
2
3
2/
2/
2
432
2
4122
222
222
§5.2 Trigonometric Functions: Unit Circle Approach
4. Exact Values of a Trigonometric Expression
11 April 2018 18 Kidoguchi, Kenneth
Find the exact values of each expression.
1(a) sin 180º cos 45ºz
2
3(b) sin cos sin
4 2z
p p p
2
3(c) csc tan4 4
z p p
§5.2 Trigonometric Functions: Unit Circle Approach
Exact Values of Trigonometric Functions
11 April 2018 19 Kidoguchi, Kenneth
Quadrantal Angles
(rad) (º ) sin() cos() tan() csc() sec() cot()
p/6 30 1/2 2
p/4 45 1 1
p/3 60 1/2 2
2 / 2 2 / 2 2 2
3 / 2
3 / 2
1/ 3 2 / 3 3
3 1/ 32 / 3
§5.2 Trigonometric Functions: Unit Circle Approach
Example Application – Projectile Motion
11 April 2018 20 Kidoguchi, Kenneth
R
0v
The path of a projectile with initial velocity v0
and trajectory angle is a parabola. The
horizontal distance travelled by the projectile
is its range R:
2
0 sin 2vR
g
Where g is the acceleration due to gravity. The maximum height H of the
projectile is:
2 2
0 sin
2
vH
g
Find R and H if v0 =150 m/s, = 30°, with g = 9.8 m/s2.
H
§5.2 Trigonometric Functions: Unit Circle Approach
Example Application – Constructing a Rain Gutter
11 April 2018 21 Kidoguchi, Kenneth
A rain gutter is to constructed of aluminum sheets 12 inches wide. After
marking off a length of 4 inches from each edge, this length is bent up at
angle as shown in the figure. The area A of the opening may be
expressed as a function as:
A() = 16 sin() (cos() + 1)
Find the area A of the opening for = 30º, = 45º, and = 60º
§5.2 Trigonometric Functions: Unit Circle Approach
Example Application – Constructing a Rain Gutter
11 April 2018 22 Kidoguchi, Kenneth
A() = 16 sin() (cos() + 1)
(º) A() (in2) Approx A() (in2)
30
45
60
11 April 2018 23 Kidoguchi, Kenneth
§5.2 Trigonometric Functions: Unit Circle Approach
5. Exact Values of Trig Functions for Integer Multiples of p/6, p/4 & p/3
2/8
3/8
5/8
6/8
7/8
x
y
(r, 0)0 2
2/
2
22
22
222
222
ryx
rx
rx
rxx
ryx
4/8
(x,y)
Initial Side
Terminal Sideyx
2
2
2
14/cos
2
2
2
14/sin
p
p
r
x
r
y
/ 4p
11 April 2018 24 Kidoguchi, Kenneth
§5.2 Trigonometric Functions: Unit Circle Approach
5. Exact Values of Trig Functions for Integer Multiples of p/6, p/4 & p/3
2/8
3/8
5/8
6/8
7/8
x
y
(r, 0)0
(x,y)(x,y)
r
2
2
2
14/cos
2
2
2
14/sin
p
p
r
x
r
y
2
2
2
14/cos
2
2
2
14/sin
p
p
r
x
r
y
4/8/ 4p
3 / 4p
11 April 2018 25 Kidoguchi, Kenneth
r
(x,y)
§5.2 Trigonometric Functions: Unit Circle Approach
5. Exact Values of Trig Functions for Integer Multiples of p/6, p/4 & p/3
2/8
3/8
5/8
6/8
7/8
x
y
(r, 0)0
(x,y)
2
2
2
14/cos
2
2
2
14/sin
p
p
r
x
r
y
2
2
2
14/3cos
2
2
2
14/3sin
p
p
r
x
r
y
4/8/ 4p
3 / 4p
11 April 2018 26 Kidoguchi, Kenneth
§5.2 Trigonometric Functions: Unit Circle Approach
5. Exact Values of Trig Functions for Integer Multiples of p/6, p/4 & p/3
2/8
3/8
5/8
6/8
7/8
x
y
(r, 0)0
(x,y)(x,y)
r
r
4/8
2
2
2
14/5cos
2
2
2
14/5sin
p
p
r
x
r
y
2
2
2
14/cos
2
2
2
14/sin
p
p
r
x
r
y
2
2
2
14/3cos
2
2
2
14/3sin
p
p
r
x
r
y
/ 4p
5 / 4p
11 April 2018 27 Kidoguchi, Kenneth
§5.2 Trigonometric Functions: Unit Circle Approach
5. Exact Values of Trig Functions for Integer Multiples of p/6, p/4 & p/3
2/8
3/8
5/8
6/8
7/8
x
y
(r, 0)0
(x,y)
r
4/8
(x,y)
r
2
2
2
14/cos
2
2
2
14/sin
p
p
r
x
r
y
2
2
2
14/5cos
2
2
2
14/5sin
p
p
r
x
r
y
2
2
2
14/cos
2
2
2
14/sin
p
p
r
x
r
y
2
2
2
14/3cos
2
2
2
14/3sin
p
p
r
x
r
y
/ 4p
/ 4p
§5.2 Trigonometric Functions: Unit Circle Approach
5. Exact Values of Trig Functions for Integer Multiples of p/4 = 45º
11 April 2018 28 Kidoguchi, Kenneth
Find the exact values of:
(a) cos 135º
(b) tan4
p
(c) sin 315º
§5.2 Trigonometric Functions: Unit Circle Approach
5. Exact Values of Trig Functions for Integer Multiples of p/6 or p/3
11 April 2018 29 Kidoguchi, Kenneth
Find the exact values of:
7(a) tan
6
p
(b) sin 120º
2(c) cos
3
p
1
(d) sin sin2
p
§5.2 Trigonometric Functions: Unit Circle Approach
6. Calculator to Evaluate a Trigonometric Function
11 April 2018 30 Kidoguchi, Kenneth
Use a calculator to approximate the value of the trigonometric function
rounded to two decimal places:
(a) cos 48º
(b) tan12
p
(c) sec 10
§5.2 Trigonometric Functions: Unit Circle Approach
7. Use a Circle of Radius r to Evaluate a Trigonometric Function
11 April 2018 31 Kidoguchi, Kenneth
For an angle in standard position, let P = (x,y) be the point on the
terminal side of that is also on the circle x2 + y2 = r2. Then
sin
csc , 0
y
r
ry
y
cos
sec , 0
x
r
rx
x
tan , 0
cot , 0
yx
x
xy
y
Theorem
§5.2 Trigonometric Functions: Unit Circle Approach
Use a Circle of Radius r to Evaluate a Trigonometric Function