Form 2 Mathematics – TRIGONOMETRY

Trigonometric ratio

Introduction: TRI – is the Greek word which means three.

-Trigonometry is the branch of mathematics which deals with measurement.

-A Trigonometric ratio consists of three parts that is – Hypotenuse, Adjacent and opposite.

Consider the diagram below which is the right angled triangle

For example

Assume angle B

For trigonometrical ratios we have sine, cosine and tangents which used to find the length of any side and angles.

Refer to the right angled triangle below:-

For sine(sin)

Sin  =

For cosine (cos)

Consider the diagram

Cos  =

For Tangents (tan)

Tan  = where

AC is called Adjacent

BC is called Opposite

Tan  =

In summary

Where: S – sine, T – tan, C – cos, O –opposite, A – adjacent, H – opposite

Example 1.

Find a) Sin Â

b) Cos  and (c) tan Â

Solution:

a) Sin  =

Sin  =

Sin  =

b) Cos  =

Cos  =

Cos  =

c) Tan  =

Tan  =

Tan  =

Example 2

Find the value of sin Â, cos  and tan Â

Solution:

Where Opposite = 4, Adjacent = 3, Hypotenuse =?

Use Pythagoras theorem

a² + b² = c²

4² + 3² = c²

=

c = 5

Sin  = =

Cos  = =

Tan  = =

Example 3

i) If the sin  = find the value of

(a) Cos  , (b) Tan Â

Solution:

given sin  = where sin  =

By using Pythagoras theorem

a² + b² = c²

(45)² + b² = (53)²

2025 + b² = 2809

b² = 2809 – 2055

=

b = 28

a) Cos  = =

b) Tan  = =

Solution:

from Pythagoras theorem

a² + b² = c²

m² + n² = c²

(m + n)² – 2mn = c²

=

c =

a. Sin B= =

b. Cos  = =

Example 4

If cos y = Find (a) sin y (b)tan y

Solution:

Given that Cos y =

But cos y =

a² + b² = c²

a² + (21)² = (29)²

a² + 441 = 841

a² = 841 – 441

=

a = 20

a. Sin y = =

b. Tan y = =

Special angles

The trigonometrical ratios has the special angles which are 0°, 30°,45°, 60°, and 90°.The special angles does not need table or calculator to find their ratios.

To prove the value of trigonometric ratio for special angles

Consider the diagram below

Note:

when the point B move toward the point c the angle of A = 0° and the length of AB = AC and BC = 0

Sin 0° =

Sin =

Sin 0° =

Sin 0°= 0

Tan 0°=

Tan 0° =

Tan 0°= 0

Cos 0° =

Cos 0° =

But AC = AB

Cos 0°= 1

Consider the diagram

If the point A moves towards point C the difference from A to C becomes zero. The angle between A and C become 90°

Sin 90° = but opp = hyp

Sin 90° = 1

Cos 90° =

Cos 90° =

Cos 90° = 0

Tan 90° =

Tan 90° =

Tan 90° = (undefined)

Example

Find sin 45°, cos 45, and Tan 45° consider a right angled triangle ABC with 1 unit in length

The length Of AC = CB = 1 unit But use Pythagoras theorem to find the length AB.

From Pythagoras theorem

a² + b² = c²

1² + 1² = c²

=

c=

Sin 45 ° =

Sin 45° =

Rationalize the denominator

Sine 45° =

= =

Cos 45° =

Cos 45° =

Cos 45° =

= =

Tan 45° =

Tan 45° =

Tan 45°= 1

Find Sin, Cos and Tan of (30° and 60˚)

Consider the equilateral triangle ΔABC

Use Pythagoras theorem

a² + b² = c²

1² + b² = 2²

b² = 2² – 1²

b² = 4 – 1

=

b =

Sin 60 =

Sin 60° =

Sin 60° =

Cos 60 =

Cos 60° = or 0.5

Cos 60° = or 0.5

Tan 60° =

Tan 60° =

Tan 60° =

Sin 30° =

Sin 30° = or 0.5

Tan 30° =

Tan 30°=

=

Tan 30° =

Cos 30° =

Cos 30° =

In summary:

Sin 0 =

Sin 0 =

Sin 0 = 0

Example 1.

Find the value of 4 sin 45° + 2 tan 60 without using table

Solution:

4 sin 45° + 2 Tan 60°

= 4 () + 2)

= 2+ 2

Note:

Trigonometry: Is the branch of mathematics that deals with the properties of angles and sides of right angled triangle

TRIGONOMETRICAL RATIONS FOR SPECIAL ANGLES

Trigonometrical rations for special angles deal with 0°, 30°, 45°, 60°, 90°.

A	0°	30°	45°	60°	90°
Sin A	0				1
Cos A	1				0
Tan A	0		1

Example 1.

Without using mathematical table evaluate

(i) 4 sin 45° + cos 30°

(ii) 4 tan 60° – Sin 90°

Solution:

i) 4 sin 45° + Cos 30°

= 4 x +

=2+

=4

(ii) 4 tan 60° – sin 90

Solution:

4 tan 60° – sin 90= 43 – 1

(iii) 3(cos60° + Tan 60°)

Solution:

= 3(+ 3)

= 3

=

=

Example 2

Find the value of without using mathematical table

Solution:

2 + 33

EXERCISE

1. Evaluate the following without using mathematical table

(a). 3 sin 45° + 7tan 30°

(b).

Solution:

3sin 45° + 7tan 30°

= 3 x + 7 x 3

= 3 x + 7 x 3

=

TRIGONOMETRIC TABLES

– Trigonometric tables deals with readings of the value of angles of sine, cosine and tangent from mathematical table when they are already prepared into four decimal

HOW TO READ THE VALUE OF TRIGONOMETRIC ANGLES FROM MATHEMATICAL TABLES

Example 1

Find the value of the following by using mathematical table

(i) Sin 43°= 0.6820

(ii) Sin 58º = 0.8480

(iii) Sin 24°42′ = 0.4179

(iv) Sin 52°26′ = 0.7923 + 4 = 0.7927

Example 2

By using mathematical tables evaluate the following

(a)Cos 37° =0.7986

(b)Cos82° =0.1392

(c)Cos 71°34′ =0.3162

(d)Tan 20° = 0.3640

(e)Tan 68° =2.4751

(f)Tan 54°22′ = 1.3950

Example3.

Find the value of the following letter from trigonometric rations

(i) Sin p = 0.6820

Sin p = 0.6820

P = Sin^-1 (0.6820)

P = 43°

ii) Sin Q = 0.7291

Q = Sin^-1 (0.7291)

Q = 46°48’

iii) Tan R = 5.42°45

R = 5. 42°45

R = tan^-1 (5.42°45)

R = 79°33

APPLICATION OF SINE, COSINE AND TANGENT RATIOS IN SOLVING A TRIANGLE

Sine, cosine and triangle of angles are used to solve the length of unknown sides of triangles

Example

From

Sine C =

Sin 55˚=

20 x Sin 55˚= x

20 x 0.8198 = x

x = 16.4 cm

The value of x = 16.4cm

Example 2

Find the length XY in a XYZ

Solution:

From

tan 62˚ =

tan 62° =

13 x tan 62° = xy

13 x 1.8807 = xy

24.4cm = xy

xy = 24.4cm

Example

Evaluate the value of m and give your answer into 3 decimal places

From

Cos R =

Cos 36° =

m =

m =

m = 14.833 cm

The value of m is 14.833 cm

EXERCISE

Solution:

From

tan 43° =

tan 43°=

17 x tan 43° = y

y = 17

y =15.85cm