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Chapter 11
Trigonometric Functions
11.1 Trigonometric Ratios and General Angles
11.2 Trigonometric Ratios of Any Angles
11.3 Graphs of Sine, Cosine and Tangent Functions
Trigonometric Equations
Objectives
11.1 Trigonometric Ratios and General Angles
In this lesson, we will learn how to find the trigonometric ratios for
acute angles, particularly those for 30°, 45° and 60° (or
respectively in radians).
,3
and 4
,6
Trigonometric Ratios of Acute Angles
The three trigonometric ratios
are defined as
OPQ is a right angled triangle
Trigonometric Equations
adjacent
oppositehypotenuse
sin
cos
tan
oppositeoppositehypotenusehypotenuse
adjacentadjacent
PQ
OP
OQ
OP
PQ
OQ
Example 1
In the right-angled triangle ABC, tan θ = 2. Find sin θ and cos θ.
Solution
Trigonometric Equations
A B
C
θ
Since tan θ = ,1
2 2
1
BC = 2 units and AB = 1 unit.
By Pythagoras’ Theorem, AC = . units 5
5
AC
BCsin
5
2
AC
ABcos
5
1
Trigonometric Ratios of Special Angles
Draw a diagonal to the
square.
Draw a unit square.
Trigonometric Equations
The length of the diagonal is √2 and the angle is 45°.
1sin 45
2
1cos 45
2
tan 45 1
Trigonometric Ratios of Special Angles
Draw an equilateral triangle of side 2 cm.
Trigonometric Equations
The altitude bisects the base of the
triangle.
3sin 60
2
1cos 60
2
tan 60 3
Draw an altitude.
The length of the altitude is √3 and the angles are 60° and
30°.
3cos30
2
1sin 30
2
1tan 30
3
Trigonometric Ratios of Complementary Angles
In the right-angled triangle OPQ
but OPQ = 90°– θ
Trigonometric Equations
cosOQ
OP
sinPQ
OP
tanPQ
OQ
sin 90
cos 90
1
tan 90
OPQ = – θ
sin2
cos2
1
tan2
If θ is in radians
2
Trigonometric Equations
Example 2
Using the right-angled triangle in the diagram, show that sin(900 – θ) = cos θ. Hence, deduce the value of
SolutionP Q
R
θ
a
bc
900 – θ.70sin20cos
20cos
.)sin(90 and cos,In c
a
c
aPQR
Thus, sin(900 – θ) = cos θ.
sin 700 = sin (900 – 200)
= cos 200
2
1
20cos2
20cos
70sin20cos
20cos
Trigonometric Equations
O
y
x
P
O
y
x
P
180° –
P '
O
y
x
P
180° +
P '
O
y
x
P
360° –
P '
O
y
x
P
O
y
x
P
–
P '
O
y
x
P
– 180
P '
O
y
x
P
– – 180
P '
Consider angles in the Cartesian plane.
OP is rotated in an anticlockwise direction around the origin O. The basic (reference) angle that OP makes with the positive x–axis is α.
Now OP is rotated in the clockwise direction.
1st quadrant2nd quadrant
4th quadrant3rd quadrant
O
y
x
P
– 360
220
180
Example 3
Given that 00 < θ < 3600 and the basic angle for θ is 400, find the value of θ if it lies in the (a) 3rd quadrant, (b) 4th quadrant.
Solution(a) (b)
320
360
Trigonometric Equations
Using the complementary angle identity.
Substitute for sin θ.
Trigonometric Equations
cos 90 sin
1Given that sin , find the value of sin cos 90
2
sin cos 90 sin sin 1 1 1
2 2 4
1tan 90
tanA
A
Given that tan 2, find the value of 2 tan tan 90A A A
1 12 4
2 22
12 tan tan 90 2 tan
tanA A A
A
Using the complementar
y angle identity.
Substitute for tan A.
Exercise 11.1, qn 2
Trigonometric Equations
1
2sin 45
cos30 sin 60 3 3
2 2
sin 45Without using a calculator, evaluate
cos30 sin 60
1 1
2 3 6
Exercise 11.1, qn 5(a)
Solution
Trigonometric Equations
3 3 1sin cos cos
3 6 3 2 2 2
3 1 11
4 2 4
Without using a calculator, evaluate sin cos cos3 6 3
Exercise 11.1, qn 5(c)
Solution
Trigonometric Equations
70 , 180 70 , 180 70 , 360 70
Find all the angles between 0° and 360° which make a basic angle of 70°.
The angles are as follows:
70 , 110 , 250 , 290
Exercise 11.1, qn 7(b)
Solution
, , , 25 5 5 5
Find all the angles between 0 and 2π which make a basic
angle of
The angles are as follows:
4 6 9, , ,
5 5 5 5
Exercise 11.1, qn 7(b)
Solution
.5
Trigonometric Equations
