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:

Image result for quadrant and angles

http://images.tutorcircle.com/cms/images/tcimages/graphing-quadrantsSHR.jpg

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

52 = 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:

1st quadrant 00

Sin

Tan

R

2nd quadrant00)

R

 

 

3rd quadrant00)

R

 

 

 

 

 

3rd 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

 

 

0 0 0 0 0
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
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