Download presentation

1
**Trigonometric Functions and Graphs**

Higher Mathematics Unit 1 Trigonometric Functions and Graphs

2
**RADIANS 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

3
**Radian Measure Length AB = radius**

AOB subtends an arc equal to a radius

4
1 2 radians = 360° radians = 180°

5
** radians = 180° So 1 radian = So 1 radian ~ 57° ~ 180° = radians**

2 2 Every 90° is radians

6
2 90° 180° 2 0° 3 2 270°

7
**180° = radians Degrees to radians Change 60° to radians:**

3 60° = radians (as 180° ⅓ = 60°)

8
** 180 Radians Degrees Convert 150° to radians**

We can also convert as follows. 180 Degrees Radians Convert 150° to radians (simplifying fraction: divide by 30) 150 180 5 6 Radians

9
**Change to Radians: 60° = 120° = Radians 210° = 2 315° = Radians**

60 180 2 3 Radians 120 180 7 6 Radians 210 180 7 4 Radians 315 180

10
** radians = 180° Radians to Degrees: Change radians to degrees**

4 radians = 180° 4 4 180° radians = = 45° 3 2 Change radians to degrees 3 2 radians = = 270° 2 3 180°

11
** 180 Degrees Radians Convert to degrees Radians**

We can also convert as follows. 180 Radians Degrees 5 6 Radians Convert to degrees 5 180 6 5 180 6 150°

12
**Change to degrees 180 45° Radians 2 Radians 120° 3 135°**

4 45° 2 3 Radians 2 180 3 120° 3 4 Radians 3 180 4 135° 5 180 3 5 3 Radians 300°

13
** radians = 180° The angles in the following table must be known.**

They are essential for non-calculator questions. Degrees 360 180 90 60 45 30 2 2 3 4 6 Radians remember as factors or multiples of 180° radians = 180°

14
**Most angles in non-calculator work are multiples of those above**

Use them to complete the table below Degrees 120° Use for Christines Use smartboard document radians to angles doc 135° 210° 270° 315° 360° 2 3 5 6 5 4 4 3 5 3 Radians

15
**Degrees Radians 120° 135° 150° 210° 225° 240° 270° 300° 315° 360° 2 3**

Use for Christines only 135° 150° 210° 225° 240° 270° 300° 315° 360° 2 3 3 4 5 6 7 6 5 4 4 3 3 2 5 3 7 4 11 6 Radians

16
Sketching Trig Graphs For reinforcement and using the graphs go to folder graph transformations

17
**Trig Graphs The maximum value for sin x is 1 when x = 90°**

The minimum value for sin x is when x = 270° sin x = 0 (i.e. cuts the x-axis) at: x = 0°, x = 180°, x = 360

18
**Trig Graphs The maximum value for sin x is 1 when x =**

The minimum value for sin x is when x = 2 3 2 sin x = 0 (i.e. cuts the x-axis) at: x = x = x = 2

19
y = asinx

20
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

21
**Trig Graphs The maximum value for cos x is 1 when x = 0° & 360°**

The minimum value for cos x is when x = 180° cos x = 0 (i.e. cuts the x-axis) at: x = 90°, x = 270°

22
**Trig Graphs The maximum value for cos x is 1 when x = 0 & 2**

The minimum value for cos x is when x = cos x = 0 (i.e. cuts the x-axis) at: x = x = 2 3 2

23
y = acosx

24
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 = 2 3 2

25
**Trig Graphs Using radians, sketch the following trig graphs: y = 5sinx**

y = 1.5cosx y = 2cosx y = 100sinx Use smartboard document graph transformations When: 0 ≤ x ≤ 2p

30
y = -sinx

31
y = -sinx The function y = -sinx is a reflection of y = sinx in the x - axis.

32
y = -cosx

33
y = -cosx The function y = -cosx is a reflection of y = cosx in the x - axis.

34
y = sin nx

35
y = sin nx

36
Trig Graphs PERIOD PERIOD When x is multiplied by a number, that number gives the number of times that the graph “repeats” in 2p. i.e. for y = sin nx: period of graph = 2p n

37
y = cos nx

38
Trig Graphs PERIOD PERIOD When x is multiplied by a number, that number gives the number of times that the graph “repeats” in 2p. i.e. for y = cos nx: period of graph For support of classwork use exercises on smartboard document trig transformations = 2p n

39
**Trig Graphs Using radians, sketch the following trig graphs:**

y = 5sin2x y = 4cos2x y = 6cos3x y = 7sin½x Use smartboard document graph transformations When: 0 ≤ x ≤ 2p

44
**Adding or subtracting from a Trig Function**

48
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. For support of classwork use exercises on smartboard document trig transformations

49
**Trig Graphs Using radians, sketch the following trig graphs:**

y = 3 + sin2x y = cos3x - 4 y = 3sinx + 2 y = 2cos2x - 1 y = 2 - sinx Use smartboard document graph transformations When: 0 ≤ x ≤ 2p

55
**Adding or subtracting from x**

59
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. For support of classwork use exercises on smartboard document trig transformations

60
Example 1 Find the maximum turning point, for 0 ≤ x ≤ p, of the graph y = 5sin(x - p/3). Consider the function y = 5sin x Maximum value is 5 When x = p/2 For y = 5sin(x - p/3) Max occurs at p/2 + p/3 = 5p/6 Turning Point: (5p/6,5)

61
Example 2 Write down the equation of the drawn function and the period of the graph. Write the function as y = asin bx + c

62
Example 2 y = asin bx + c b = 3 (3 wavelengths in 2p) Period of graph 2p 3 = 2p/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

63
**Ratios and Exact Values**

Exact Values for 45° 1 Square 1 45° 1 45°

64
**Ratios and Exact Values**

Exact Values for 45° 1 45° x² = 1² + 1² x² = 2 x = √2 x √2

65
**Ratios and Exact Values**

Exact Values for 45° 1 √2 Sin 45° = Cos 45° = Tan 45° = 1 45° √2 1 √2 1

66
**Ratios and Exact Values**

Exact Values for p/4 1 √2 Sin p/4 = Cos p/4 = Tan p/4 = 1 p/4 √2 1 √2 1

67
**Ratios and Exact Values**

Exact Values for 30° & 60° 60° 30° 2 1 60° 30° 1 2 60° 2 Equilateral Triangle

68
**Ratios and Exact Values**

Exact Values for 30° & 60° 60° 30° 1 2 x² = 2² - 1² x² = 3 x = √3 x √3

69
**Ratios and Exact Values**

Exact Values for 30° 60° 30° 1 2 1 2 Sin 30° = Cos 30° = Tan 30° = √3 2 √3 1 √3

70
**Ratios and Exact Values**

Exact Values for p/6 p/3 p/6 1 2 1 2 Sin p/6 = Cos p/6 = Tan p/6 = √3 2 √3 1 √3

71
**Ratios and Exact Values**

Exact Values for 60° 60° 30° 1 2 Sin 60° = Cos 60° = Tan 60° = √3 2 1 2 √3 √3

72
**Ratios and Exact Values**

Exact Values for p/3 p/3 p/6 1 2 Sin p/3 = Cos p/3 = Tan p/3 = √3 2 1 2 √3 √3

73
**Angles Greater than 90° SIN positive ALL Positive 2nd Quadrant 1st**

Sin A = (+)ve Cos A = (-)ve Tan A = (-)ve 1st Quadrant Sin A = (+)ve Cos A = (+)ve Tan A = (+)ve 180° 0° 3rd Quadrant TAN positive Sin A = (-)ve Cos A = (-)ve Tan A = (+)ve 4th Quadrant Sin A = (-)ve Cos A = (+)ve Tan A = (-)ve COS Positive 270°

74
**Angles Greater than p/2 SIN positive ALL Positive TAN positive COS**

TAN positive COS Positive 3p/2

75
**sin 3p/4 positive cos 7p/6 negative tan 7p/4 cos 5.4 radians TAN**

ALL Positive COS SIN 2p sin 3p/4 cos 7p/6 tan 7p/4 cos 5.4 radians positive negative For class support use smartboard document Four quadrants

76
90° SIN ALL (180 - x)° x° 180° 360° (180 + x)° (360 - x)° COS TAN 270°

77
Example 3 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 For printing only see smatrboard document equations

78
**Since sin x° is positive it is in the 1st and 2nd quadrants**

Example 3 sin x° = ½ Since sin x° is positive it is in the 1st and 2nd quadrants For printing only see smatrboard document equations

79
**Example 3 sin x° = ½ √3 sin x° = ½ sin 30° = ½ x = 30° 30° 2 60° 1**

For printing only see smatrboard document equations 60° 1

80
**Example 3 sin x° = ½ sin 30° = ½ x = 30° x = 30° or x = 180° - 30°**

For printing only see smatrboard document equations

81
Example 3 For printing only see smatrboard document equations

82
p/2 SIN ALL (p - ) p 2p (p + ) (2p - ) COS TAN 3p/2

83
Example 4 Solve √2cos +1 = 0, 0 ≤ ≤ 2 and illustrate the solution in a sketch of y = cos √2cos +1 = 0 √2cos = -1 Cos = √2 -1 For printing only see smatrboard document equations

84
**Since cos is negative it is in the 2nd and 3rd quadrants**

Example 4 Cos = Since cos is negative it is in the 2nd and 3rd quadrants √2 -1 For printing only see smatrboard document equations

85
**Example 4 Cos = √2 cos = cos /4 = = /4 1 p/4 1 -1 √2 1 √2 1 √2**

For printing only see smatrboard document equations 1 √2 1

86
**Example 4 Cos = cos = cos /4 = = /4 So = - /4 or + /4**

√2 -1 √2 1 cos = cos /4 = = /4 So = - /4 or + /4 = 3/4 or 5/4 √2 1 For printing only see smatrboard document equations

87
Example 4 For printing only see smatrboard document equations

88
**Consider if the equation was cos x = ½**

Example 5 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 As cos x is positive it must be in the 1st and 4th quadrants. For printing only see smatrboard document equations

89
**Example 5 √3 cos x = ½ Cos 60° = ½ x = 60° or 360° - 60°**

30° 2 √3 cos x = ½ Cos 60° = ½ x = 60° or 360° - 60° x = 60° or 300° For printing only see smatrboard document equations 60° 1

90
**However the function we are using is cos 3x **

Example 5 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° For printing only see smatrboard document equations

91
Example 5 For printing only see smatrboard document equations

92
**and negative, x will be in all four quadrants**

Example 6 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 √2 1 For printing only see smatrboard document equations

93
**Example 6 sin x = sin 45° = x = 45° √2 1 45° 1 1 √2 1 √2**

For printing only see smatrboard document equations 45° 1

94
**Example 6 sin x = x = 45° or 180° - 45° or 180° + 45° or 360° - 45°**

√2 1 For printing only see smatrboard document equations

95
**Factorise the equation Consider the equation as: 4x² + 11x + 6 = **

Example 7 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 For printing only see smatrboard document equations

96
**sin = or sin = -2 (no solution)**

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 4 -3 For printing only see smatrboard document equations

97
**As sin is negative answer must be in 3rd and 4th quadrants**

Example 7 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 For printing only see smatrboard document equations

98
**Example 7 = 0.85 radians = + 0.85 or = 2 - 0.85**

= 3.99 or 5.43 radians For printing only see smatrboard document equations

99
Reminders: sin² x° + cos² x° = 1 sin² x° = 1 - cos² x° cos² x° = 1 - sin² x°

100
**Solve cos² x° + sin x° = 1, for 0 ≤ x ≤ 360 **

Example 8 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° For printing only see smatrboard document equations

101
**Consider if the equation was sin x = 0.6 x = 36.87 or 180 - 36.87 **

Example 9 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 = or x = 36.87° or ° For printing only see smatrboard document equations

102
**The function we are considering is sin (2x - 20) **

Example 9 x = 36.87° or ° The function we are considering is sin (2x - 20) Therefore 2x - 20 = or 2x - 20 = , 2x = or 2x = x = 28.4° or x = 81.6° For printing only see smatrboard document equations

103
**The function repeats itself twice in 360° **

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 = ° or x = ° x = 28.4° or 81.6° or 208.4° or 261.6° For printing only see smatrboard document equations

104
**Solve 3cos(2 + /4) = 1, correct to 1 decimal place, for 0 ≤ ≤ **

Example 10 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 (remember to put calculator in radians) = 1.23 or = 1.23 or 5.05 radians For printing only see smatrboard document equations

105
**The function we are considering is cos(2 + /4)**

Example 10 = 1.23 or 5.05 radians The function we are considering is cos(2 + /4) 2 + /4 = or 2 + /4 = 5.05 2 = or 2 = 2 = or 2 = 4.26 = or = 2.1 (to 1dp) Do not need to add on a wave length of as ≤ ≤ For printing only see smatrboard document equations

Similar presentations

OK

Slide 6-1 COMPLEX NUMBERS AND POLAR COORDINATES 8.1 Complex Numbers 8.2 Trigonometric Form for Complex Numbers Chapter 8.

Slide 6-1 COMPLEX NUMBERS AND POLAR COORDINATES 8.1 Complex Numbers 8.2 Trigonometric Form for Complex Numbers Chapter 8.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on time division switching pdf Ppt on bugatti veyron engine Ppt on building brand equity Training ppt on quality Ppt on series and parallel circuits lessons Ppt on phonetic transcription generator Ppt on acid base balance Ppt on chromosomes and genes and locus Ppt on ozone depletion Ppt on conservation agriculture in india