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 complementary 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 of a right-angled triangle.
4. Sine and cosine of complementary 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 opposite side and adjacent side of a right-angled triangle is called the tangent. It is abbreviated as tan.

Tan = opposite / adjacent

Sine of an Angle

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

Sin = opposite / hypotenuse

Cosine of an Angle

The ratio of the side adjacent to the angle and the hypotenuse is called the cosine.

Cosine = adjacent / hypotenuse

Example

In the figure above, the adjacent length is 4 cm and angle x is given. Calculate the opposite length.

Solution

Opposite length = 4 × tan x cm.

Example

In the figure above, opposite (o) = 5 cm, adjacent (a) = 12 cm. Calculate sin x and cos x.

Solution

sin x = opposite / hypotenuse = 5 / 13 = 0.3846

cos x = adjacent / hypotenuse = 12 / 13 = 0.9231

Sine and Cosine of Complementary Angles

For any two complementary angles x and y, sin x = cos y and cos x = sin y. For example, sin 30° = cos 60°.

Example

Find the acute angles given sin x = cos y.

Solution

Therefore, x and y are complementary angles such that x + y = 90°.

Trigonometric Ratios of Special Angles

These trigonometric ratios can be deduced by using isosceles right-angled triangles and equilateral triangles as follows.

Tangent, Cosine, and Sine of 45°

The triangle should have a base and a height of one unit each, giving a hypotenuse of √2.

Cos 45° = sin 45° = 1 / √2, tan 45° = 1

Tangent, Cosine, and Sine of 30° and 60°

The equilateral triangle has sides of 2 units each.

Sin 30° = 1/2, Cos 30° = √3/2, Tan 30° = 1/√3

Sin 60° = √3/2, Cos 60° = 1/2, Tan 60° = √3

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. (2 marks)

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°. After walking 100 meters towards the foot of the building, he stopped at point B where he observed it again at an angle of 60°. Find the height of the building.

4. Find the value of θ, given that ½ sin θ = 0.35 for 0° ≤ θ ≤ 360°.

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

6. Solve for x in 2 Cos 2x° = 0.6000, 0° ≤ x ≤ 360°.

7. Wangechi, whose eye level is 182 cm tall, observed the angle of elevation to the top of her house to be 32° from her eye level at point A. She walks 20 m towards the house on a straight line to point B, at which point she observes the angle of elevation to the top of the building to be 40°. Calculate, correct to 2 decimal places:

• 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:

• (a) Cos x
• (b) Sin²(90° – x)

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

Cos θ + Sin θ