# Lesson Notes By Weeks and Term - Junior Secondary School 3

TRIGNOMETRICAL RATIO

SUBJECT: MATHEMATICS

CLASS:  JSS 3

DATE:

TERM: 2nd TERM

WEEK SEVEN                                                                             DATE------------

TOPIC: TRIGNOMETRICAL RATIO

Trignometrical ratio is a ratio of the lengths of two sides of a right-angle triangle. The three trignometrical ratios are sine (sin) cosine (cos) and tangent (tan). The word tri- means three, thus trignometrical ratio deals with three sided figure (triangle).

In a right-angled triangle, the longest side is called the hypotenuse (opp the right angle), the side adjacent (next) to the given angle is called the Adjacent while the side opposite to the given angle is called the opposite.

B                                                                   B

Fig 1                                                                          Fig 2    A        opp             C

Note: To be able to know the ratio easily take note of the acronym SOHCAHTOA. Where S stands for sine, C stands for cosine, and T for tangent.

Degree and Minutes

Angles are often measured to the nearest degree. In some state, degree may be subdivided into minutes and.

Note:

10 =60 minutes. This is written as 60/.

To change from minutes to degree, we divide the number by 60.

Example: convert 100 to minutes

Solution: 10 x 60 = 600mins

EVALUATION:

1. Convert the following to minutes: A. 160   B. 500
2. Rewrite and give your answer in degree to 1.dp A. 460 151  B. 390 251  C. 1400 4

SINE OF ANGLE

In a right-angle triangle, the ratio of opposite to hypotenuse is defined as the sine of the angle under consideration.

From fig 1, sin ø = AB/BC. The ratio does not depend on the size of the triangle but depends only on the size of the angle (ø).

To find sine of the angles, we use of either calculator or the sine table. In use of sine table, since the sine of angle increases as the angle increases, thus the differences will be added.

EVALUATION

use mathematical table to find

1. sin 43
2. sin 14.58
3. sin 30.6

USE OF SINE IN SOLVING TRIANGLES

Example:

Find the marked side or the angle in each of the following. Give your answer to 2.s.f.g.

4cm

Fig 1                         x

15cm              46                                   ø   9cm

Fig 2

Solution:                                            From Fig 2

From fig 1                                                sinø = opp/hyp

Sin46 = 15/x                                             sinø =4cm/9cm

X=15/sin46                                              sinø = 0.444

X = 15/0.7193                                          ø = sin1 0.4444

X=20.85,     X=21 (2.s.f.g)                      ø = 26.49, ø= 26 (2.s.f.g)

EVALUATION

What is value of X and ø in the below triangle

28                   x

ø

10

COSINE OF ANGLES

In a right angled triangle, the ratio of adj/hyp is defined as the of angle under consideration.

Using diagram:

ø

opp

Thus Ó¨ = AB/AC

This value of the ratio does not depend on the size of triangle but on the size of angle.

CALCULATIONS OF COSINE OF ANGLES

Find the unknown side or angle in the below triangles

2                                                                             5cm

ø

ø                                              3.5cm

1

solution:

2                                                                                      5cm

ø                                                                  ø

1.     3.5cm

cosø = ½                                                     cosø = 3.5cm/ 5cm

ø = cos1 0.5                                                     ø = cos1 0.7

ø = 600                                                        ø = 45.67

TANGENT OF ANGLES

The tangent of  any angle is the  ratio opp/adjacent. In short form, tanÓ¨ = opp/adj

CALCULATING TANGENT OF TRIANGLES

Examples: find the side of the triangle marked  x. correct to 2 S.F.G in the figure below.

40

8cm          hyp

opp   xcm

solution:

tan ø = opp/adj = xcm/8cm

tan ø = x/8,  ø = tan 40 x 8

ø = 0.8391x8

ø = 6.7128;  ø = 6.7( 2 s.f.g)

EVALUATION: calculate the side of the triangle marked x

60                                                   19cm

(i)                                         (ii)                                        75

5                                            x

x

Essential mathematics for J.S.S 3 Pg 101-116

Exam Focus for J.S.C.E. for J.S.S 3    Pg 224-235

WEEKEND ASSIGNMENT

1. Convert 32.40  to degree and minutes. A. 32 42 1 B. 32 441 C. 32 241
2. Cos 60 is equal to ------- A. 0.5 B. 0.49 C. 1/25

Calculate the side marked P,Q, and α

Draw:

x

5                           q

30

p

1. the value of P is     A. 9.6 B. 8.7 C. 10
2. the value of q is --- A. 10 B. 8 C. 13
3. the value of α is ---A. 45 B.60 C. 30

THEORY

x

1.                                                                                           calculate x

16cm

7

A

1.                           ø       œ                               calculate the unknown angles.

20cm                                  60cm

B                    45                                      C

35cm