Trigonometric Ratios - Mathematics Form 2 Notes

• Introduction
  • Tangent of Acute Angle
  • Sine of an Angle
  • Cosine of an Angle
• Right Angled Triangle Trigonometry
• Sine and Cosines of Complementary Angles 
• Trigonometric Ratios of Special Angles: 30⁰, 45⁰, 60⁰.
  • Tangent, Cosine and Sine of 45^o.
  • Tangent, cosine and sine of 30^o and 60^o.
• Past KCSE Questions on the Topic.

Introduction

Tangent of Acute Angle

The constant ratio between the ^{vertical distance}/_{horizontal distance} is called the tangent. It’s abbreviated as tan

                                           
Tan∅ = opposite side
           adjacent side

Sine of an Angle

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

Sin∅ = opposite side
           hypotenuse

Cosine of an Angle

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

Cosine∅ = Adjacent
                hypotenuse

Right Angled Triangle Trigonometry

Example

In the figure above adjacent length is 4 cm and Angle x= 36⁰. Calculate the opposite length.

Solution

tan 36⁰ = opposite length = PR
                adjacent lengh     4

4tan 36⁰ = PR
Therefore PR = 4 x 0.7265 = 2.9060 cm.

Example

In the above O = 5 cm A = 12 cm calculate angle sin x and cosine x.

Solution

sin x = opp O = 5
           hyp H    H

But H² = 12²×5²
= 169
= √169
H = 13
Therefore sin x= ⁵/₁₃
= 0.3846
Cos x = adj
            hyp
=¹²/₁₃
=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. sin60⁰ = cos 30⁰, Sin30⁰ = cos 60⁰, sin70⁰ = cos 20⁰,

Example

Find acute angles α and β if Sin α = cos 33⁰

Solution

sin α = cos 33
Therefore α + 33 = 90
α = 57⁰

Trigonometric Ratios of Special Angles: 30⁰, 45⁰, 60⁰.

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

Tangent, Cosine and Sine of 45^o.

The triangle should have a base and a height of one unit each, giving hypotenuse of √2. 
Cos 45^o = ¹/_√2 

Sin 45^o= ¹/_√2

Tan 45^o = 1

Tangent, Cosine and Sine of 30^o and 60^o.

The equilateral triangle has a sides of 2 units each

Sin 30⁰ = ¹/₂

Cos 30⁰ = ^√3/₂

Tan 30⁰ = ¹/_√3

Sin 60⁰ =^√3/₂

Cos 60⁰ = ¹/₂

Tan 60⁰= ^√3/₁ = √3

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 θ = ²⁴/₂₅, find without using tables or calculator, the value of
   tan θ − cos θ
   cos θ + sin θ
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 θ, given that ½ sinθ= 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 2Cos 2x⁰ = 0.6000; 0⁰≤ x ≤ 360⁰.
7. Wangechi whose eye level is 1 82cm 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 ;
   1. distance of A from the house
   2. 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 = ⁵/₁₂, find the value of the following without using mathematical tables or calculator:
    1. Cos x
    2. Sin²(90−x)
11. If tan θ =⁸/₁₅, find the value of Sinθ − Cosθ without using a calculator or table
                                                    Cosθ + Sinθ