Trigonometric Ratios - Mathematics Form 2 Notes

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Tangent of Acute Angle

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

trigonmetry triangle
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

Cosine of an Angle

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

Cosine∅ = Adjacent

Right Angled Triangle Trigonometry

right angle trigonom rvCTD

right angle trigonometry


trigonmetry triangle

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


tan 360 = opposite length = PR
                adjacent lengh     4

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


trigonmetry triangle

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


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 =

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,


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


sin α = cos 33
Therefore α + 33 = 90
α = 57

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.

trigonometric ratios special angles

Tangent, Cosine and Sine of 45o.

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

Sin 45o= 1/√2

Tan 45o = 1

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

The equilateral triangle has a sides of 2 units each

Sin 300 = 1/2

Cos 300√3/2

Tan 30= 1/√3

Sin 600 =√3/2

Cos 6001/2

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 30o. After walking for 100meters towards the foot of the building he stopped at point B where he observed it again at an angle of 60o. Find the height of the building
  4. Find the value of θ, given that ½ sinθ= 0.35 for 0o ≤ θ ≤ 360o
  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 45o. 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
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