Higher Mathematics Unit 1 Trigonometric Functions and Graphs.

Higher Mathematics

Unit 1

Trigonometric Functions and

Graphs

So far we have always measured our angles in degrees.

There is another way to measure angles.

It is particularly important in applied mathematics.

Angles can be measured in

RADIANS

Radian Measure

Length AB = radius

AOB subtends an arc equal to a radius

1

2 radians = 360°

radians = 180°

radians = 180°

So 1 radian = 180°

So 1 radian ~ 57° ~

180° = radians

90° = radians 2

Every 90° is radians2

0°

90°

180°

270°

0

2

32

2

60° = radians

(as 180° ⅓ = 60°)

Degrees to radians

Change 60° to radians:

180° = radians3

We can also convert as follows.

Degrees Radians

Convert 150° to radians

(simplifying fraction: divide by 30)

150 180

180

5 6

Radians

Change to Radians:

60° =

120° =

210° =

315° =

60 180

120 180

210 180

315 180

3

Radians

23

Radians

76

Radians

74

Radians

radians =

= 45°

Radians to Degrees:

Change radians to degrees 4

radians = 180°4 4

180°

radians =

= 270°

32 2

3 180°

Change radians to degrees 32

We can also convert as follows.

Radians Degrees

Convert to degrees

180

56

Radians

5 180 6 5 180

6

150°

1804

4

Radians

Change to degrees

23

Radians

34

Radians

53

Radians

2 1803

3 1804

5 1803

45°

120°

135°

300°

The angles in the following table must be known.

They are essential for non-calculator questions.

remember as factors or multiples of 180°

360 180 90 60 45 30

Degrees

Radians

2 2

3

4

6

radians = 180°

120° 135° 210° 270° 315° 360°

56

54

43

53

Most angles in non-calculator work are multiples of those above

Use them to complete the table below

Degrees

Radians

23

120° 135° 150° 210° 225° 240° 270° 300° 315° 360°

23

34

56

76

54

43

32

53

74

11 6

Degrees

Radians

Sketching Trig

Graphs

Trig Graphs

The maximum value for sin x is 1 when x = 90°

The minimum value for sin x is -1 when x = 270°

sin x = 0 (i.e. cuts the x-axis) at:

x = 0°, x = 180°, x = 360

90 180 270 360

-1.5

-1

-0.5

0.5

1

1.5

x

y

0

y = sin x

Trig Graphs

The maximum value for sin x is 1 when x =

The minimum value for sin x is -1 when x =

sin x = 0 (i.e. cuts the x-axis) at:

x = x = x =

/2 3/2 2

-1.5

-1

-0.5

0.5

1

1.5

x

y

0

32

2

0 2

y = asinx

Trig Graphs

When sin x is multiplied by a number, that number gives the maximum and minimum value of the function.

Note the function still cuts the x-axis at: x = 0, & 2

Trig Graphs

The maximum value for cos x is 1 when x = 0° & 360°

The minimum value for cos x is -1 when x = 180°

cos x = 0 (i.e. cuts the x-axis) at:

x = 90°, x = 270°

90 180 270 360

-1.5

-1

-0.5

0.5

1

1.5

x

y

0

y = cos x

Trig Graphs

The maximum value for cos x is 1 when x = 0 & 2

The minimum value for cos x is -1 when x =

cos x = 0 (i.e. cuts the x-axis) at:

x = x =

/2 3/2 2

-1.5

-1

-0.5

0.5

1

1.5

x

y

0

2

32

y = acosx

Trig Graphs

When cos x is multiplied by a number, that number gives the maximum and minimum value of the function.

Note the function still cuts the x-axis at: x =

232

Trig Graphs

Using radians, sketch the following trig graphs:

y = 5sinx

y = 1.5cosx

y = 2cosx

y = 100sinx

When: 0 ≤ x ≤ 2

/2 3/2 2x

y

0

5

-5

y = 5sinx

/2 3/2 2x

y

0

1.5

-1.5

y = 1.5cosx

/2 3/2 2x

y

0

2

-2

y = 2cosx

/2 3/2 2x

y

0

100

-100

y = 100sinx

y = -sinx

/2 3/2 2

-1

1

x

y

0

y = sinx

/2 3/2 2

-1

1

x

y

0

y = sinx

y = -sinx

y = -sinx

/2 3/2 2

-1

1

x

y

0

y = sinx

y = -sinx

The function y = -sinx is a reflection of y = sinx in the x - axis.

y = -cosx

/2 3/2 2

-1

1

x

y

0

y = cosx

/2 3/2 2

-1

1

x

y

0

y = cosx

y = -cosx

y = -cosx

The function y = -cosx is a reflection of y = cosx in the x - axis.

/2 3/2 2

-1

1

x

y

0

y = cosx

y = -cosx

y = sin nx

/2 3/2 2

-1

1

x

y

0

y = sinx

/2 3/2 2

-1

1

x

y

0

y = sinx y = sin2x

y = sin nx

2

-1

1

x

y

0

y = sinx

2

-1

1

x

y

0

y = sin3xy = sinx

/2 3/2 2

-1

1

x

y

0

y = sinx y = sin2x

Trig Graphs

When x is multiplied by a number, that number gives the number of times that the graph “repeats” in 2.

i.e. for y = sin nx: period of graph = 2 n

PERIOD PERIOD

y = cos nx

/2 3/2 2

-1

1

x

y

0

y = cosx

/2 3/2 2

-1

1

x

y

0

y = cosx y = cos2x

/2 3/2 2

-1

1

x

y

0

y = cosx y = cos2x

Trig Graphs

When x is multiplied by a number, that number gives the number of times that the graph “repeats” in 2.

i.e. for y = cos nx: period of graph = 2 n

PERIOD PERIOD

Trig Graphs


y = 5sin2x

y = 4cos2x

y = 6cos3x

y = 7sin½x

When: 0 ≤ x ≤ 2

2x

y

0

5

-5

y = 5sin2x

2x

y

0

4

-4

y = 4cos2x

2x

y

0

6

-6

y = 6cos3x

2x

y

0

7

-7

y = 7sin½x

/2 3/2 2

-3

-2

-1

1

2

3

x

y

0

y = sinx

/2 3/2 2

-3

-2

-1

1

2

3

x

y

0

y = sinx

y = 1 + sinx

/2 3/2 2

-3

-2

-1

1

2

3

x

y

0

y = sinx

y = 1 + sinx

y = 2 + sinx

Adding or subtracting from a Trig Function

/2 3/2 2

-3

-2

-1

1

2

3

x

y

0y = cosx

/2 3/2 2

-3

-2

-1

1

2

3

x

y

0y = cosx

y = 1 + cosx

/2 3/2 2

-3

-2

-1

1

2

3

x

y

0y = cosx

y = 1 + cosx

y = 2 + cosx

/2 3/2 2

-3

-2

-1

1

2

3

x

y

0

y = sinx

/2 3/2 2

-3

-2

-1

1

2

3

x

y

0

y = sinx

y = sinx - 1/2 3/2 2

-3

-2

-1

1

2

3

x

y

0

y = sinx

y = sinx - 1

y = sinx - 2

/2 3/2 2

-3

-2

-1

1

2

3

x

y

0y = cosx

/2 3/2 2

-3

-2

-1

1

2

3

x

y

0y = cosx

y = cosx -1

/2 3/2 2

-3

-2

-1

1

2

3

x

y

0y = cosx

y = cosx -1

y = cosx - 2

Trig Graphs

When a number is added to a trig function the graph “slides” vertically up by that number.

When a number is subtracted from a trig function the graph “slides” vertically down by that number.

Trig Graphs


y = 3 + sin2x

y = cos3x - 4

y = 3sinx + 2

y = 2cos2x - 1

y = 2 - sinx

When: 0 ≤ x ≤ 2

2

-2

-1

1

2

3

4

x

y

0

y = sin2x

y = 3 + sin 2x

2

-5

-4

-3

-2

-1

1

x

y

0

y = cos3x

y = 3 + sin 2x

y = cos3x - 4

2

-3

-2

-1

1

2

3

4

5

x

y

0

y = 3sinx

y = 3sinx + 2

2

-3

-2

-1

1

2

x

y

0

y = 2cos2x

y = 2cos2x - 1

2

-2

-1

1

2

3

x

y

0

y = -sinx

y = 2 - sinx

Adding or subtracting from x

/3 2/3 4/3 5/3 2

-1

1

x

y

0

y = sinx

/3 2/3 4/3 5/3 2

-1

1

x

y

0

y = sinx

y = sin(x -

/3 2/3 4/3 5/3 2

-1

1

x

y

0

y = cosx

/3 2/3 4/3 5/3 2

-1

1

x

y

0

y = cosx

y = cos(x -

/3 2/3 4/3 5/3 2

-1

1

x

y

0

y = cosx

/3 2/3 4/3 5/3 2

-1

1

x

y

0

y = cosx

y = cos(x +

/3 2/3 4/3 5/3 2

-1

1

x

y

0

y = sinx

/3 2/3 4/3 5/3 2

-1

1

x

y

0

y = sinx y = sin(x +

Trig Graphs

When a number is added to x the graph “slides” to the left by that number.

When a number is subtracted from x the graph “slides” to the right by that number by that number.

Example 1

Find the maximum turning point, for 0 ≤ x ≤ , of the graph y = 5sin(x - /3).

Consider the function y = 5sin x

Maximum value is 5

When x = /2

For y = 5sin(x - /3)

Max occurs at

/2 + /3 = 5/6

Turning Point: (5/6,5)

/2 3/2 2x

y

0

5

-5

y = 5sinx

Example 2

Write down the equation of the drawn function and the period of the graph.

Write the function as y = asin bx + c

2x

y

0

8

-4

2x

y

0

8

-4

Example 2

y = asin bx + c

b = 3 (3 wavelengths in 2)

Period of graph 2 3

= 2/3

Difference between max and min = 12

a = 12 2 = 6

y = 6sin 3x + c (graph then shifts up 2)

c = +2 y = 6sin 3x + 2

Ratios and Exact Values

Exact Values for 45°

1

1

Square 1

1

45°

1

1

45°



1

1

45°

xx² = 1² + 1²

x² = 2

x = √2√2



1

1

45°

Sin 45° =

Cos 45° =

Tan 45° =

√2

1√2

1√2

1


Exact Values for /4

1

1

/4

Sin /4 =

Cos /4 =

Tan /4 =

√2

1√2

1√2

1


Exact Values for 30° & 60°

60°

30°

2

1 1

60°

30°

1

2

60°

60°

60°

2

2

2

Equilateral Triangle

60°

30°

1

2


Exact Values for 30° & 60°

x² = 2² - 1²

x² = 3

x = √3√3

x

60°

30°

1

2



√3

Sin 30° =

Cos 30° =

Tan 30° =

1 2

√3 2

1 √3

/3

/6

1

2


Exact Values for /6

√3

Sin /6 =

Cos /6 =

Tan /6 =

1 2

√3 2

1 √3

60°

30°

1

2



√3

Sin 60° =

Cos 60° =

Tan 60° =

1 2

√3 2

√3

/3

/6

1

2


Exact Values for /3

√3

Sin /3 =

Cos /3 =

Tan /3 =

1 2

√3 2

√3

COS

Positive

SIN

positive

1st

Quadrant

Sin A = (+)ve

Cos A = (+)ve

Tan A = (+)ve

ALL

Positive

4th

Quadrant

Angles Greater than 90°

0°

90°

180°

270°

2nd

Quadrant

3rd

Quadrant

Sin A = (+)ve

Cos A = (-)ve

Tan A = (-)ve

Sin A = (-)ve

Cos A = (-)ve

Tan A = (+)ve

TAN

positive

Sin A = (-)ve

Cos A = (+)ve

Tan A = (-)ve

TAN

positive

ALL

Positive

COS

Positive

SIN

positive

Angles Greater than /2

0

/2

3/2

2

2

/2

3/2

TAN

positive

ALL

Positive

COS

Positive

SIN

positive

sin 3p/4

cos 7p/6

tan 7p/4

cos 5.4 radians

positive

negative

negative

positive

ALLSIN

TAN COS

°

°

270°

360°x°(180 - x)°

(180 + x)° (360 - x)°

Solve 2sin x° = 1, 0° ≤ x ≤ 360° and illustrate the solution in a sketch of y = sin x

2 sin x° = 1

sin x° = ½

Example 3

Example 3

sin x° = ½

Since sin x° is positive it is in the 1st and 2nd quadrants

Example 3

sin x° = ½

60°

30°

1

2

√3sin x° = ½

sin 30° = ½

x = 30°

Example 3

sin x° = ½

sin 30° = ½

x = 30°

x = 30° or x = 180° - 30°

x = 30° or x = 150°

Example 3

180 360

-1

-0.5

0.5

1

x

y

30 150

ALLSIN

TAN COS

/2

3/2

2( - )

( + ) (2 - )

Solve √2cos +1 = 0, 0 ≤ ≤ 2 and illustrate the solution in a sketch of y = cos

√2cos +1 = 0

√2cos = -1

Cos =

Example 4

√2-1

Cos =

Since cos is negative it is in the 2nd and 3rd quadrants

Example 4

√2-1

Cos =

Example 4

√2-1

1

1

/4

√2

cos =

cos /4 =

= /4

√21

√2 1

Cos =

Example 4

√2-1

cos =

cos /4 =

= /4

So = - /4 or + /4

= 3/4 or 5/4

√21

√2 1

2

-1

-0.5

0.5

1

x

y

03/4 5/4

Example 4

Solve cos 3x° = ½, 0° ≤ x ≤ 360° and illustrate the solution in a sketch of y = cos 3x

Consider if the equation was cos x = ½

Example 5

As cos x is positive it must be in the 1st and 4th quadrants.

cos x = ½

Cos 60° = ½

x = 60° or 360° - 60°

x = 60° or 300°

Example 5

60°

30°

1

2

√3

However the function we are using is cos 3x

Therefore if x = 60° or 300° for cos x = ½

3x = 60° or 3x = 300°: x = 20° or 100°

the graph repeats itself 3 times in 360°

with a wavelength of 120°

as the function has a wavelength of 120°

x = 20° or 100° or 140° or 220° or 260° or 340°

Example 5

Example 5

-1

-0.5

0.5

1

x

y

20 100 140 220 260 340

Solve 2sin² x° = 1

sin² x = ½

sin x = √½

sin x =

As sin x is positive

and negative, x will be in

all four quadrants

Example 6

√21

sin x =

sin 45° =

x = 45°

Example 6

√21

1

1

°

√2

√21

sin x =

x = 45° or 180° - 45° or 180° + 45° or 360° - 45°

x = 45° or 135° or 225° or 315°

Example 6

√21

Solve 4sin² + 11sin + 6 = 0, correct to 2 decimal places, for 0 ≤ ≤ 2

Factorise the equation

Consider the equation as: 4x² + 11x + 6 =

(4x + 3)(x + 2) = 0

4sin² + 11sin + 6 = 0

(4sin + 3)(sin + 2) = 0

Example 7

4sin² + 11sin + 6 = 0

(4sin + 3)(sin + 2) = 0

4sin + 3 = 0 or sin+ 2 = 0

4sin = -3 or sin = -2

sin = or sin = -2 (no solution)

Therefore we have to solve sin = -0.75

Example 7

4-3

sin = -0.75

As sin is negative answer must be in 3rd and 4th quadrants

sin = 0.75

= sin-¹0.75 (radians)

= 0.85 radians

Example 7

= 0.85 radians

= + 0.85 or = 2 - 0.85

= 3.14 + 0.85 or = 6.28 - 0.85

= 3.99 or 5.43 radians

Example 7

Reminders:

sin² x° + cos² x° = 1

sin² x° = 1 - cos² x°

cos² x° = 1 - sin² x°

Solve cos² x° + sin x° = 1, for 0 ≤ x ≤ 360

(substitute cos² x° = 1 - sin²x° into the equation)

1 - sin² x° + sin x° = 1

1 - sin² x° + sin x° -1 = 0

sin x° - sin² x° = 0

sin x°(1 - sin x°) = 0

sin x° = 0 or 1 - sin x° = 0

sin x° = 1

x = 0°or 180° or 360° or x = 90°

Example 8

Solve sin (2x - 20)° = 0.6, correct to 1 decimal place, for 0 ≤ x ≤ 360

Consider if the equation was sin x = 0.6

x = 36.87 or 180 - 36.87

x = 36.87° or 143.13°

Example 9

x = 36.87° or 143.13°

The function we are considering is sin (2x - 20)

Therefore 2x - 20 = 36.87 or 2x - 20 = 143.13 ,

2x = 56.87 or 2x = 163.13

x = 28.4° or x = 81.6°

Example 9

The function repeats itself twice in 360°

i.e. it has a wavelength of 180°

x = 28.4° or x = 81.6°

or x = 180 + 28.4° or x = 180 + 81.6°

x = 28.4° or 81.6° or 208.4° or 261.6°

Example 9

Solve 3cos(2 + /4) = 1, correct to 1 decimal place, for 0 ≤ ≤

Consider if the equation was 3cos x = 1

cos x = ⅓

= 1.23 or 2 - 1.23 (remember to put calculator in radians)

= 1.23 or 6.28 - 1.23


Example 10


The function we are considering is cos(2 + /4)

2 + /4 = 1.23 or 2 + /4 = 5.05

2 = 1.23 - 0.79 or 2 = 5.05 - 0.79

2 = 0.44 or 2 = 4.26

= 0.2 or = 2.1 (to 1dp)

Do not need to add on a wave length of as 0 ≤ ≤

Example 10

Higher Mathematics Unit 1 Trigonometric Functions and Graphs.

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Transcript of Higher Mathematics Unit 1 Trigonometric Functions and Graphs.