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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
Using radians, sketch the following 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
Using radians, sketch the following 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°
Ratios and Exact Values
Exact Values for 45°
1
1
45°
xx² = 1² + 1²
x² = 2
x = √2√2
Ratios and Exact Values
Exact Values for 45°
1
1
45°
Sin 45° =
Cos 45° =
Tan 45° =
√2
1√2
1√2
1
Ratios and Exact Values
Exact Values for /4
1
1
/4
Sin /4 =
Cos /4 =
Tan /4 =
√2
1√2
1√2
1
Ratios and Exact Values
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
Ratios and Exact Values
Exact Values for 30° & 60°
x² = 2² - 1²
x² = 3
x = √3√3
x
60°
30°
1
2
Ratios and Exact Values
Exact Values for 30°
√3
Sin 30° =
Cos 30° =
Tan 30° =
1 2
√3 2
1 √3
/3
/6
1
2
Ratios and Exact Values
Exact Values for /6
√3
Sin /6 =
Cos /6 =
Tan /6 =
1 2
√3 2
1 √3
60°
30°
1
2
Ratios and Exact Values
Exact Values for 60°
√3
Sin 60° =
Cos 60° =
Tan 60° =
1 2
√3 2
√3
/3
/6
1
2
Ratios and Exact Values
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
= 1.23 or 5.05 radians
Example 10
= 1.23 or 5.05 radians
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