§5.2 trigonometric functions: unit circle approach...

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


p/4 = 45º


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



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)


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.


Trigonometric Functions on a Unit Circle


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


1. Exact Values of the Trig Functions Using a Point on the Unit Circle


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



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


Trigonometric Functions of Angles - Definition


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


2. Exact Values of Trigonometric Functions of Quadrantal Angles


Find exact values of the six trigonometric functions of:

a) 0 0º b) p/2 90º


2. Exact Values of Trigonometric Functions of Quadrantal Angles


Find exact values of the six trigonometric functions of:

a) p 180º b) 3p/2 270º


Exact Values of Trigonometric Functions of Quadrantal Angles


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


Exact Values of Trigonometric Functions of Quadrantal Angles


(a) sin(5p) (b) cos(–540°)

Find the exact value of:


3. Exact Values of Trigonometric Functions of = p/4


p/4

2/8

3/8

5/8

6/8

7/8

x

y

(1, 0)04/8

(x,y)

Initial Side

Terminal Side


4. Exact Values of Trigonometric Functions of = p/3 & p/6


10 212

2

1

1

2

y

x




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




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


4. Exact Values of a Trigonometric Expression


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


Exact Values of Trigonometric Functions


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


Example Application – Projectile Motion


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


Example Application – Constructing a Rain Gutter


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


Example Application – Constructing a Rain Gutter


A() = 16 sin() (cos() + 1)

(º) A() (in2) Approx A() (in2)

30

45

60



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




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


r

(x,y)



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




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




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. Exact Values of Trig Functions for Integer Multiples of p/4 = 45º


Find the exact values of:

(a) cos 135º

(b) tan4

p

(c) sin 315º


5. Exact Values of Trig Functions for Integer Multiples of p/6 or p/3


Find the exact values of:

7(a) tan

6

p

(b) sin 120º

2(c) cos

3

p

1

(d) sin sin2

p


6. Calculator to Evaluate a Trigonometric Function


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


7. Use a Circle of Radius r to Evaluate a Trigonometric Function


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


Use a Circle of Radius r to Evaluate a Trigonometric Function


Find the exact values of each of the six

trigonometric functions of an angle if

(-3, 3) is a point on its terminal side.

§5.2 trigonometric functions: unit circle approach...

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