LESSON 5 Section 6.3

Trig Functions of Real Numbers

UNIT CIRCLE

Remember, the sine of a real number t

(a number that corresponds to radians)

is the y value of a point on a unit circle and the cosine of that real number is the x value of the point on a unit circle.

APPENDIX IV of your textbook shows a good unit circle.

Make a table of

x

and

y

values for the equation

y = sin x.

x

0 π/6 π/4 π/3 π/2 2 π/3 3 π/4 5 π/6 π

y

0 0.5

0.707

0.866

1 0.866

0.707

0.5

0

x

7 π/6 5 π/4 4 π/3 -3 π/2 5 π/3 7 π/4 11 π/6 2 π 13 π/6

y

-0.5

-0.707

-0.866

-1 -0.866

-0.707

-0.5

0 0.5

This is a second revolution around the unit circle. This is another ‘period’ of the curve.

x

2 π 13 π/6 9 π/4 7 π/3 5 π/2 8 π/3 11 π/4 17 π/6 3 π

y

0 0.5

0.707

0.866

1 0.866

0.707

0.5

0

x

19 π/6 13 π/4 10 π/3 7 π/2 11 π/3 15 π/4 23 π/6 4 π 25 π/6

y

-0.5

-0.707

-0.866

-1 -0.866

-0.707

-0.5

0 0.5

y

= sin

x

• This is a

periodic function.

The period 2 π.

is • The domain of the function is all real numbers.

• The range of the function is [-1, 1].

• It is a continuous function. The graph is shown on the next slide.

Graphing the sine curve for -2

π

≤

x

≤ 2

π

.

y



Domain Range

: :    1,1 

Period

: 2  1  ( π/2, 1) -2π -3π 2 -π -π 2 0 (0, 0) π 2 ( π, 0) π 3π 2 (2 π, 0) 2π -1 (3 π/2, - 1)

UNIT CIRCLE

Remember, the sine of a real number t

(a number that corresponds to radians)

is the y value of a point on a unit circle and the cosine of that real number is the x value of the point on a unit circle.

Make a table of

x

and

y

values for

y

= cos x Remember, the

y

value in this table is actually the

x

value on the unit circle.

x

0 π/6 π/4 π/3 π/2 2 π/3 3 π/4 5 π/6 π

y

1 0.866

0.707

0.5

0 -0.5

-0.707

-0.866

-1

x

7 π/6 5 π/4 4 π/3 -3 π/2 5 π/3 7 π/4 11 π/6 2 π 13 π/6

y

-0.866

-0.707

-0.5

0 0.5

0.707

0.866

1 0.866

y

= cos

x

• This is a

periodic function.

The period 2 π.

is • The domain of the function is all real numbers.

• The range of the function is [-1, 1].

• It is a continuous function. The graph is shown on the next slide.

-2π -3π 2 Graphing the cosine curve for -2

π

≤

x

≤ 2

π

.

y



Domain Range

:  :   1,1 

Period

: 2   1 (0, 1) -π -π 2 0 (2 π, 1) ( π/2, 0) π 2 π ( π, - 1) (3 π/2, 0) 3π 2 2π -1

How do the graphs of the sine function and the cosine function compare?

• They are basically the same ‘shape’.

• • They have the same domain and range.

• They have the same period.

If you begin at –π/2 on the cosine curve, you have the sine curve.

y

 sin

x

 cos(

x

  2 )

x

  

, sin

x



___

  The notation above is interpreted as:

‘as x approaches the number π/6 from the right (from values of x larger than π/6), what function value is sin x approaching?’

Since the sine curve is continuous (no breaks or jumps), the answer will be equal to exactly the sin ( π/6) or ½ .

The notation below is interpreted as:

‘as x approaches the number π/6 from the left (from values of x smaller than π/6), what function value is sin x approaching?’

Again, since the sine curve is continuous, the answer will be equal to exactly the sin ( π/6) or ½ .

x

  

, sin

x



_____

Answer the following.

As

x

   , cos

x

 _____ As

x

  , sin

x

 _____

Find all the values x in the interval [0, 2  ) that satisfy the equation. Use the graph to verify these values.

cos

x

 1 2 1 -2π -3π 2 -π -π 2 0 -1 π 2 π 3π 2 2π

Find all the values x in the interval [0, 2  ) that satisfy the equation.

cos

x

 1 2 1 Q I Q IV -2π -3π 2 -π -π 2 0 -1 π 2 π 3π 2 2π

Find all the values x in the interval [0, 2  ) that satisfy the equation.

cos

x



x

 , 1 5 2   3 3 1 -2π -3π 2 -π -π 2 0 -1 π 2 π 3π 2 2π

sin

x

Find all the values x in the interval [0, 2  ) that satisfy the equation.

Use the graph to verify these values.

  1 2 1 -2π -3π 2 -π -π 2 0 -1 π 2 π 3π 2 2π

Find all the values x in the interval [0, 2  ) that satisfy the equation.

sin

x

  1 2 1 -2π -3π 2 -π -π 2 0 -1 π 2 π Q III 3π 2 2π Q IV

Find all the values x in the interval [0, 2  ) that satisfy the equation.

sin

x



x

  1 5   , 4 4 2 1 -2π -3π 2 -π -π 2 0 -1 π 2 π 3π 2 2π

Make a table of

x

and

y

values for

y

= tan x Remember, tan

x

is (sin

x

/ cos

x

).

x y x

π/2 undefined 0.49

π -0.49

π -31.821

π/2 π/3 π/4 π/6 -1.732

-1 -0.577

0.51

2 3 π π/3 π/4 0 π/6 π/4 π/3 0 0.577

1 1.732

5 π/6 π 7 π/6 5 π/4

y

31.821

undefined -31.821

-1.732

-1 -0.577

0 0.577

1

y

= tan x

• This is a

periodic function.

The period is π.

• The domain of the function is all real numbers, except those of the form π/2 +nπ.

• The range of the function is all real numbers.

• It is not a continuous function. The function is undefined at -3 π/2, -π/2, π/2, 3π/2, etc. There are

vertical asymptotes

at these values. The graph is shown on the next slide.

-2π -3π 2 Graphing the tangent curve for -2

π

≤

x

≤ 2

π

.

y

 tan

Domain Range

: 

Period

:  -π   10 8 6 4 2 ( π/4, -1) -π -2 2 -4 -6 -8 -10 

n

 where n is an integer} ( π/4, 1) π 2 π 3π 2 2π

As

x

    2   , tan

x

 _____ As

x

    2   , tan

x

 _____ For all

x

values where the tangent curve is continuous, approaching from the left or the right will equal the value of the tangent at

x

. However, the two cases above are different; because there is a vertical asymptote when

x

= π/2. If approaching from the left (the smaller side), the answer is infinity. If approaching from the right (the larger side), the answer is negative infinity.

Find the answers.

As

x

  , tan

x

 _____ As

x

  , tan

x

 _____ As

x

  , tan

x

 _____

Find all the values x in the interval [0, 2  ) that satisfy the equation.

-2π tan

x

= 1 -3π 2 -π -π 2 10 8 6 4 2 0 -2 -4 -6 -8 -10 Q I π 2 π Q III 3π 2 2π

-2π -3π 2 Find all the values x in the interval [0, 2  ) that satisfy the equation.

tan

x

 1

x

 4 , 4 -π -π 2 10 8 6 4 2 0 -2 -4 -6 -8 -10 π 2 π 3π 2 2π

-2π tan

x

 -3π 2 Find all the values x in the interval    2 , 2  that satisfy the equation.

1 3 -π -π 2 10 8 6 4 2 0 -2 -4 -6 -8 -10 π 2 π 3π 2 2π

-2π tan

x

 -3π 2 Find all the values x in the interval    2 , 2  that satisfy the equation.

1 3 -π -π 2 10 8 6 4 2 0 -2 -4 -6 -8 -10 Q I π 2 π Q III 3π 2 2π

-2π -3π 2 Find all the values x in the interval    2 , 2  tan

x

 1 3

x

 6 , 6 that satisfy the equation.

-π -π 2 10 8 6 4 2 0 -2 -4 -6 -8 -10 π 2 π 3π 2 2π

-2π Find all the values x in the interval    2 , 2    that satisfy the equation.

tan

x

 

1

-3π 2 -π -π 2 10 8 6 4 2 0 -2 -4 -6 -8 -10 π 2 π 3π 2 2π

-2π Find all the values x in the interval    2 , 2    that satisfy the equation.

tan

x

 

1

10 8 6 4 -3π 2 -π -π 2 0 -2 -4 -6 -8 -10 π 2 Q II π 3π 2 2π

-2π Find all the values x in the interval    2 , 2    that satisfy the equation.

-3π 2 tan

x

  1

x

  4 , 4 -π -π 2 10 8 6 4 2 0 -2 -4 -6 -8 -10 π 2 π 3π 2 2π

Sketch the graph of

y

= sin

x

+ 1

This will be a graph of the basic sine function, but shifted one unit up.

The domain will be all real numbers. What would be the range?

Since the range of a basic sine function is [-1, 1], the domain of the function above would be [0, 2].

Sketch the graph of

y

= sin

x

+ 1 3 1 -2π -3π 2 -π -π 2 -1 π 2 π 3π 2 2π

Sketch the graph of

y

= cos

x

- 2

This would be the graph of a basic cosine function shifted 2 units down.

The domain is still all real numbers. What is the range?

The basic cosine function has a range of [-1, 1]. The range of the function above would be [-3, -1].

Sketch the graph of

y

= cos

x

- 2 -2π -3π 2 -π 1 0 -π 2 -1 -2 -3 -4 π 2 π 3π 2 2π

Find the intervals from –2π to 2π where the graph of

y

= tan

x

is: a) Increasing b) Decreasing Remember: No brackets should be used on values of

x

where the function is not defined.

a) Increasing: [-2 π, -3π/2) b) The function never decreases.

(-3 π/2, -π/2) ( π/2, π/2) ( π/2, 3π/2) (3 π/2, 2π]