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

45⁰,60⁰.

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;

1. Tan
2. Sin
3. Cot
4. cosec
5. sec

Solution

A

5

B

C

Using Pythagoras theorem:

c² = a² + b²

5^{2 =} 3² + b²

b² = 5² – 3²

= 25 – 9

= 16

b = 4

Thus, a = 3, b = 4 and c = 5

1. Tan =

Example 2: Find the complementary angles of the following:

1. Sin 36
2. Cos 52
3. Tan 72
4. Sin 15

Solution

A

5

B

C

In the right-angled triangle above where <ACB = 90

90

and

Hence,

1. Sin 36 = cos ( )

= cos 54

1. Cos 52 = sin ( )

= sin 38

1. Tan 72 = cot ( )

= cot 18

1. 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 quadrant⁰⁰)

R

3^rd quadrant⁰⁰)

R

3^rd quadrant⁰⁰)

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?

1. Sin130⁰ b. c. Tan240⁰ d. Sin320⁰ e. Cos290⁰

Solution:

1. Sin130⁰
2. Tan240⁰
3. Cos290

The final values of the above can be derived from tables.

Ratios for Special Angles

• Angle 45⁰

P

Q

R

45⁰

The angle 45⁰ is an isosceles

²²²

²²

• Angle 30⁰

60

60

30

30

P

R

Q

Let PQR be equilateral triangle. From the diagram

• Angle 60⁰

From the same diagram angle 60 gives us the following:

• Angles 0⁰ and 90⁰

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 0⁰.

P

Q

R

1

1

Example 1: Without using tables, find the values of the following leaving your answer in surd form.

1. Sin 135⁰ b. Tan240⁰ c. cos210⁰ d. Sin²330⁰ e. 2Sin120⁰Cos120⁰

Solution:

1. Sin 135⁰
2. Tan240⁰
3. Cos210⁰
4. Sin²330⁰²
5. 2Sin120⁰Cos120⁰

Example 2: Find the value of

Solution:

Example 3:A pole leaning against a vertical wall makes an angle of 30⁰ 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

30⁰

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 =15² – 12² . h = 9cm. Hence, SinC

Example 6: That angle of elevation of the top T of a vertical pole P on level ground is 60⁰. 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

1. Without using tables, find the values of the following in simplified surd form where necessary.
2. b. Sin²225+cos²225
3. In the triangle below <A=45⁰<C=30⁰ and the height is 10cm. find /AC/ and then find the area of the triangle . leave your answer in surd form.

45⁰

30⁰

10cm

A

B

C

PRACTICE QUESTIONS

Objective Test:

Choose the correct answer from

1. Evaluate Sin60⁰Cos60⁰ without using table or calculator
2. What is the value of Sin30tan²45
3. The value of tan135⁰ is
4. The value of Sin315⁰ is
5. Calculate Sin²45⁰ + cos²45⁰