Mathematics Form 1-4 : CHAPTER THIRTY TWO – TRIGONOMETRIC RATIOS

Specific Objectives

By the end of the topic the learner should be able to:

1. Define tangent, sine and cosine ratios from a right angled triangle
2. Read and use tables of trigonometric ratios
3. Use sine, cosine and tangent in calculating lengths and angles
4. Establish and use the relationship of sine and cosine of complimentary angles
5. Relate the three trigonometric ratios
6. Determine the trigonometric ratios of special angles 30°, 45°, 60° and 90°without using tables
7. Read and use tables of logarithms of sine, cosine and tangent
8. Apply the knowledge of trigonometry to real life situations.

Content

1. Tangent, sine and cosine of angles
2. Trigonometric tables
3. Angles and sides o f a right angled triangle
4. Sine and cosine of complimentary angles
5. Relationship between tangent, sine and cosine
6. Trigonometric ratios of special angles 30°, 45°, 60° and 90°
7. Logarithms of sines, cosines and tangents
8. Application of trigonometry to real life situations.

Introduction

Tangent of Acute Angle

The constant ratio between the is called the tangent. It’s abbreviated as tan

Tan =

Sine of an Angle

The ratio of the side of angle x to the hypotenuse side is called the sine.

Sin

Cosine of an Angle

The ratio of the side adjacent to the angle and hypotenuse.

Cosine

Example

In the figure above adjacent length is 4 cm and Angle x. Calculate the opposite length.

Solution

cm.

Example

In the above o = 5 cm a = 12 cm calculate angle sin x and cosine x.

Solution

But

Therefore sin x

= 0.3846

Cos x =

=

=0.9231

Sine and cosines of complementary angles

For any two complementary angles x and y, sin x = cos y cos x = sin y e.g. sin,

Sin, sin,

Example

Find acute angles

Sin

Solution

Therefore

Trigonometric ratios of special Angles .

These trigonometric ratios can be deducted by the use of isosceles right – angled triangle and equilateral triangles as follows.

Tangent cosine and sine of.

The triangle should have a base and a height of one unit each, giving hypotenuse of.

Cos sin tan

Tangent cosine and sine of

The equilateral triangle has a sides of 2 units each

Sin

Sin

End of topic  

Past KCSE Questions on the topic.

1.  Given sin (90 – a) = ½ , find without using trigonometric tables the value of cos a (2mks)

2.  If ,find without using tables or calculator, the value of

(3 marks)

3. At point A, David observed the top of a tall building at an angle of 30^o. After walking for 100meters towards the foot of the building he stopped at point B where he observed it again at an angle of 60^o. Find the height of the building

4.  Find the value of q, given that ½ sinq = 0.35 for 0^o ≤ θ ≤ 360^o  

5.  A man walks from point A towards the foot of a tall building 240 m away. After covering 180m, he observes that the angle of elevation of the top of the building is 45^o. Determine the angle of elevation of the top of the building from A

6.  Solve for x in 2 Cos2x⁰ = 0.6000 0⁰≤ x ≤ 360⁰.

7.  Wangechi whose eye level is 182cm tall observed the angle of elevation to the top of her house to be 32º from her eye level at point A. she walks 20m towards the house

on a straight line to a point B at which point she observes the angle of elevation to the

top of the building to the 40º. Calculate, correct to 2 decimal places the

 a)distance of A from the house

 b) The height of the house

8.  Given that cos A = ⁵/₁₃ and angle A is acute, find the value of:-

2 tan A + 3 sin A  

9.  Given that tan 5° = 3 + 5, without using tables or a calculator, determine tan 25°, leaving your answer in the form a + b c

10.  Given that tan x = 5, find the value of the following without using mathematical tables or calculator: 12

 (a) Cos x

 (b) Sin²(90-x)

11.  If tan θ =⁸/₁₅, find the value of Sinθ – Cosθ without using a calculator or table

 Cosθ + Sinθ