lesson 5 section 6.3

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 y x y

0 0 7π/6 -0.5

π/6 0.5 5π/4 -0.707

π/4 0.707 4π/3 -0.866

π/3 0.866 -3π/2 -1

π/2 1 5π/3 -0.866

2π/3 0.866 7π/4 -0.707

3π/4 0.707 11π/6 -0.5

5π/6 0.5 2π 0

π 0 13π/6 0.5

x y x y

2π 0 19π/6 -0.5

13π/6 0.5 13π/4 -0.707

9π/4 0.707 10π/3 -0.866

7π/3 0.866 7π/2 -1

5π/2 1 11π/3 -0.866

8π/3 0.866 15π/4 -0.707

11π/4 0.707 23π/6 -0.5

17π/6 0.5 4π 0

3π 0 25π/6 0.5

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

y = sin x

• This is a periodic function. The period is 2π.

• 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 -2π ≤ x ≤ 2π.

-1

0

1

-π 2

-π-3π 2

-2π π2

π 3π 2

2π

sin( )

: ,

: 1,1

: 2

y x

Domain

Range

Period

(0, 0)

(π/2, 1)

(π, 0)

(3π/2, - 1)

(2π, 0)

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

x y x y

0 1 7π/6 -0.866

π/6 0.866 5π/4 -0.707

π/4 0.707 4π/3 -0.5

π/3 0.5 -3π/2 0

π/2 0 5π/3 0.5

2π/3 -0.5 7π/4 0.707

3π/4 -0.707 11π/6 0.866

5π/6 -0.866 2π 1

π -1 13π/6 0.866

Remember, the y value in this table is actually the x value on the unit circle.

y = cos x

• This is a periodic function. The period is 2π.

• 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 cosine curve for -2π ≤ x ≤ 2π.

-1

0

1

-π 2

-π-3π 2

-2π π2

π 3π 2

2π

cos( )

: ,

: 1,1

: 2

y x

Domain

Range

Period

(0, 1)

(π/2, 0)

(π, - 1)

(3π/2, 0)

(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.

sin cos( )

2y x x

, sin ___6

, sin _____6

x x

x 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 .

Answer the following.

As ,cos _____

As , sin _____4

x x

x x

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

Use the graph to verify these values.

1cos

2x

-1

0

1

-π 2

-π-3π 2

-2π π2

π 3π 2

2π


1cos

2x

-1

0

1

-π 2

-π-3π 2

-2π π2

π 3π 2

2π

Q I Q IV


1cos

25

,3 3

x

x

-1

0

1

-π 2

-π-3π 2

-2π π2

π 3π 2

2π


Use the graph to verify these values.

1sin

2x

-1

0

1

-π 2

-π-3π 2

-2π π2

π 3π 2

2π


1sin

2x

-1

0

1

-π 2

-π-3π 2

-2π π2

π 3π 2

2π

Q III Q IV


1sin

25 7

,4 4

x

x

-1

0

1

-π 2

-π-3π 2

-2π π2

π 3π 2

2π

Make a table of x and y values for y = tan x

x y x y

-π/2 undefined 0.49π 31.821

-0.49π -31.821 π/2 undefined

-π/3 -1.732 0.51π -31.821

-π/4 -1 2π/3 -1.732

-π/6 -0.577 3π/4 -1

0 0 5π/6 -0.577

π/6 0.577 π 0

π/4 1 7π/6 0.577

π/3 1.732 5π/4 1

Remember, tan x is (sinx / cosx).

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 -3π/2, -π/2, π/2, 3π/2, etc. There are vertical asymptotes at these values. The graph is shown on the next slide.

Graphing the tangent curve for -2π ≤ x ≤ 2π.

-10

-8

-6

-4

-2

0

2

4

6

8

10

-π 2

-π-3π 2

-2π π2

π 3π 2

2π

tan

:{ | / 2 where n is an integer}

: ,

:

y x

Domain x x n

Range

Period

(-π/4, -1)

(π/4, 1)

As , tan _____2

As , tan _____2

x x

x 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 , tan _____

As , tan _____2

As , tan _____2

x x

x x

x x


tan x = 1

-10

-8

-6

-4

-2

0

2

4

6

8

10

-π 2

-π-3π 2

-2π π2

π 3π 2

2π

Q I Q III


-10

-8

-6

-4

-2

0

2

4

6

8

10

-π 2

-π-3π 2

-2π π2

π 3π 2

2π

tan 1

5,

4 4

x

x

Find all the values x in the interval

that satisfy the equation.3

,2 2

1tan

3x

-10

-8

-6

-4

-2

0

2

4

6

8

10

-π 2

-π-3π 2

-2π π2

π 3π 2

2π



,2 2

1tan

3x

-10

-8

-6

-4

-2

0

2

4

6

8

10

-π 2

-π-3π 2

-2π π2

π 3π 2

2π

Q I Q III



,2 2

1tan

37

,6 6

x

x

-10

-8

-6

-4

-2

0

2

4

6

8

10

-π 2

-π-3π 2

-2π π2

π 3π 2

2π



,2 2

tan 1x

-10

-8

-6

-4

-2

0

2

4

6

8

10

-π 2

-π-3π 2

-2π π2

π 3π 2

2π



,2 2

tan 1x

-10

-8

-6

-4

-2

0

2

4

6

8

10

-π 2

-π-3π 2

-2π π2

π 3π 2

2π

Q IV Q II



,2 2

tan 1

3,

4 4

x

x

-10

-8

-6

-4

-2

0

2

4

6

8

10

-π 2

-π-3π 2

-2π π2

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

-1

1

3

-π 2

-π-3π 2

-2π π2

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

-4

-3

-2

-1

0

1

-π 2

-π-3π 2

-2π π2

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

(-3π/2, -π/2)

(-π/2, π/2)

(π/2, 3π/2)

(3π/2, 2π]