TRIGONOMETRIC RATIOS OF GENERAL ANGLES
WEEK 5
SUBJECT: FURTHER MATHEMATICS
CLASS: SS1
TOPIC: TRIGONOMETRIC RATIOS OF GENERAL ANGLES
CONTENT:
- QUADRANTS AND ANGLES
- BASIC TRIGONOMETRIC RATIOS
- RATIOS OF GENERAL ANGLES
- RATIOS FOR SPECIAL ANGLES
45^{0},60^{0}.
QUADRANTS AND ANGLES
The direction to which angles are measured can either be clockwise or anticlockwise. The plane is divided into four partitions by the axes. Each partition is called a quadrant.
The Y – axis and X – axis divides the plane into 4 parts (or 4 quadrants) as shown in the figures below:
Angles are measured relative to the positive x-axis. A positive angle is measured anti-clockwise while a negative angle on the other hand is measured clockwise.
BASIC TRIGONOMETRIC RATIOS
In your Junior School, you learn about angles and how they are measured. Can you still recall how to find the sine of an angle? The below can guide you.
A
B
C
The reciprocal relations corresponding to the three basic ratios are defined as follows:
In summary,
Example 1: If is acute and cos = , find;
- Tan
- Sin
- Cot
- cosec
- sec
Solution
A
5
B
C
Using Pythagoras theorem:
c^{2} = a^{2} + b^{2}
5^{2 = }3^{2} + b^{2 }
b^{2 }= 5^{2} – 3^{2}
= 25 – 9
= 16
b = 4
Thus, a = 3, b = 4 and c = 5
- Tan =
Example 2: Find the complementary angles of the following:
- Sin 36
- Cos 52
- Tan 72
- Sin 15
Solution
A
5
B
C
In the right-angled triangle above where <ACB = 90
90
and
Hence,
- Sin 36 = cos ( )
= cos 54
- Cos 52 = sin ( )
= sin 38
- Tan 72 = cot ( )
= cot 18
- Sin 15 = cos
= cos 75
RATIOS OF GENERAL ANGLES
Consider a situation when is rotated anticlockwise, where then:
1^{st} quadrant ^{00}
Sin
Tan
R
2^{nd} quadrant^{00})
R
3^{rd} quadrant^{00})
R
3^{rd} quadrant^{00})
R
Summarily:
In the first quadrant all the ratios are positive. In the second quadrant, only sine is positive, others are negative.In the third quadrant, only tangent ratio is positive, others are negative. In the fourth quadrant, only cosine is positive while others are negative.
Example 1:
What is the value of the following?
- Sin130^{0} b. c. Tan240^{0} d. Sin320^{0} e. Cos290^{0}
Solution:
- Sin130^{0}
- Tan240^{0}
- Cos290
The final values of the above can be derived from tables.
Ratios for Special Angles
- Angle 45^{0}
P
Q
R
45^{0}
The angle 45^{0 }is an isosceles
^{222}
^{22}
- Angle 30^{0}
60
60
30
30
P
R
Q
Let PQR be equilateral triangle. From the diagram
- Angle 60^{0}
From the same diagram angle 60 gives us the following:
- Angles 0^{0} and 90^{0}
For an angle of a triangle to be zero, then it wil be a situation that the other two angles are right angle. In the same vein the opposite side to the angle is also 0^{0}.
P
Q
R
1
1
^{0} | ^{0} | ^{0} | ^{0} | ^{0} | |
1 | |||||
Example 1: Without using tables, find the values of the following leaving your answer in surd form.
- Sin 135^{0} b. Tan240^{0} c. cos210^{0} d. Sin^{2}330^{0} e. 2Sin120^{0}Cos120^{0}
Solution:
- Sin 135^{0}
- Tan240^{0}
- Cos210^{0}
- Sin^{2}330^{02}
- 2Sin120^{0}Cos120^{0}
Example 2: Find the value of
Solution:
Example 3:A pole leaning against a vertical wall makes an angle of 30^{0} with the wall. The foot of the pole is 5m from the wall. Find the length of the pole.
Solution:
The solution to a problem like this becomes easy using trigonometric ratio.
F
P
30
W
We have FWP where FPW we are to find
We can equally find the area of a triangle, if two side is given and the angle between them.
Example 4:
Calculate the area of ABC given that /AB/=10cm, /BC/= 8cm and <B=30
Solution: Area of a triangle
10cm
8cm
h
30^{0}
A
B
C
Area of triangle ^{2}
Example 5: ABC is an isosceles triangle in which /AB/=/AC/=15cm. Calculate SinC.
Solution:
15cm
15cm
24cm
A
B
C
Note that <ABC = <ACB, Bisector of angle A will bisect /BC/.
^{2} =15^{2} – 12^{2} . h = 9cm. Hence, SinC
Example 6: That angle of elevation of the top T of a vertical pole P on level ground is 60^{0}. The distance from P to the foot of the pole is 55m. calculate the height of the pole.
Solution: Sketch the information.
T
R
60
55m
m
pole
P
CLASS ACTIVITY
- Without using tables, find the values of the following in simplified surd form where necessary.
- b. Sin^{2}225+cos^{2}225
- In the triangle below <A=45^{0}<C=30^{0} and the height is 10cm. find /AC/ and then find the area of the triangle . leave your answer in surd form.
45^{0}
30^{0}
10cm
A
B
C
PRACTICE QUESTIONS
Objective Test:
Choose the correct answer from
- Evaluate Sin60^{0}Cos60^{0} without using table or calculator
- What is the value of Sin30tan^{2}45
- The value of tan135^{0} is
- The value of Sin315^{0 }is
- Calculate Sin^{2}45^{0} + cos^{2}45^{0}