# TRIGONOMETRY

**Subject**:

### MATHEMATICS

**Term:**

First Term

**Week:**

Week 6

**Class:**

JSS 3 / BASIC 9

**Previous lesson: **Pupils have previous knowledge of

### EQUATIONS WITH FRACTIONS

that was taught in their previous lesson

**Topic:**

**TRIGONOMETRY**

**Behavioural Objectives:**

At the end of the lesson, learners will be able to

- solve questions on right angle triangle (Revision)
- Explain the relationship between right angle triangles and trigonometric ratios.
- Solve questions on Sine, cosine and tangent of acute angles using tables.
- Applications of trigonometric ratios to finding distances and lengths.

**Instructional Materials:**

- Wall charts
- Pictures
- Related Online Video
- Flash Cards

**Methods of Teaching:**

- Class Discussion
- Group Discussion
- Asking Questions
- Explanation
- Role Modelling
- Role Delegation

**Reference Materials:**

- Scheme of Work
- Online Information
- Textbooks
- Workbooks
- 9 Year Basic Education Curriculum
- Workbooks

**CONTENT:**

**WEEK 6**

**TOPIC: TRIGONOMETRY**

**CONTENT:**

- Right angle triangle (Revision)
- Relationship between right angle triangles and trigonometric ratios.
- Sine, cosine and tangent of acute angles using tables.
- Applications of trigonometric ratios to finding distances and lengths.

** **

**Right Angled Triangle**

A right-angled triangle has one of its angles equal to 90^{O}. That is, if a triangle has one of its three angles equal to 90^{O} then it is a right-angled triangle. A little square at the corner (angle) signifies that the angle is 90^{O}.

Right Angled Triangle, or a right-angled triangle, is a triangle in which one angle is a right angle (that is, a 90-degree angle). The relation between the sides and angles of a right angled triangle is the basis for trigonometry.

A right angled triangle has two other angles apart from the right angle. These are acute angles and are less than 90 degrees. The third angle is always 90 degrees and is called the hypotenuse. It is the longest side in the triangle and is opposite to the right angle.

The other two sides are called the legs of the triangle. They are shorter than the hypotenuse and are adjacent to the right angle.

** **

**Right Angled Triangle and the Trigonometric Ratios**

Opposite |

Hypotenuse |

Adjacent |

A trigonometric ratio is a ratio of the lengths of **any two sides of a right – angled triangle.**The longest side is called the Hypotenuse, the side facing the required angle is called the opposite and the last side is called the adjacent.

A trigonometric ratio is a way of measuring the size of angles. The ratios are based on the lengths of the sides of a right angled triangle. The most common trigonometric ratios are sine, cosine and tangent. These ratios are usually represented by the symbols sin, cos and tan. The sine ratio is defined as the ratio of the length of the side opposite to an angle to the length of the hypotenuse. The cosine ratio is defined as the ratio of the length of the side adjacent to an angle to the length of the hypotenuse. The tangent ratio is defined as the ratio of the length of the side opposite to an angle to the length of the side adjacent to that angle.

These ratios can be used to calculate the sizes of angles in a right angled triangle. The ratios can also be used to solve problems involving triangles that are not right angled. Trigonometric ratios are used in many areas of mathematics, science and engineering. They are particularly useful in fields such as astronomy, physics and calculus.

The three trigonometric ratios are **Sine (sin)**, **Cosine (cos)** and **Tangent (tan)**.

The figure below show triangle XYZ in various positions

O |

H |

A |

X |

Z |

Y |

Z |

H |

A |

O |

Y |

X |

O |

A |

Y |

H |

X |

Z |

** **

Where H, A, O represent the Hypotenuse, Adjacent and Opposite respectively.

The sides of the triangles are as follows.

XZ, the hypotenuse, XY, the side opposite to and YZ, the side adjacent to

Using these abbreviations **hyp**, **opp** and**adj**, the Trigonometric ratios are given as:

**Sin A =** ** Cos A = ** ** Tan A = **

To obtain the value of any trigonometric ratio, the ** four-figure table** is used.

**N.B.:** Teacher should teach students how to use the four-figure table to find the values of trigonometric ratios.

**Application of Trigonometric Ratios to Finding Distances and Lengths.**

Trigonometric ratios can be used to solve real life problems such as angles of elevation and depressions and bearings and other real life situations that result in right-angled triangles.

**Use of Sine **

Sine and cosine of angles are used to find the lengths of unknown side in triangles. The table below gives the sine of some chosen angles.

Angle A |
Sin A |

30⁰ | 0.5000 |

35⁰ | 0.5736 |

40⁰ | 0.6428 |

45⁰ | 0.7071 |

50⁰ | 0.7660 |

55⁰ | 0.8192 |

60⁰ | 0.8660 |

The values in the table are given to 4 significant figures.

**Example**

- Calculate the value of x in the figure below.

20cm

55⁰

** **

** **

**Solution**

In the figure above, the hypotenuse (hyp) is given and x is opposite (opp) the given angle. Thus, we can only use:

**Sin A =**

Sin 55⁰ =

Thus, = 20 x sin 55⁰

= 20 x 0.8192

= 16.384 cm

= 16 cm to 2 s.f.

- Use tables to find the angles:
- whose sine is = ,
- whose sin is 0.649 2,

**Solution**

- Let the angle be A, then sin A = .

Express as a decimal fraction correct to 4 d.p.

. = 0.285 7

Sin A = 0.285 7

Looking within the sine table entries,

0.2857 is opposite 16⁰ and under 0.6⁰.

Thus, A = 16.6⁰.

- Let the angle be x, then sin x = 0.649 2.

Looking within the sin table entries, the nearest value to 0.649 2 is 0.648 1. 0.648 1 is opposite 40⁰ and under 0.4⁰.

The difference between 6 492 and 6 481 is 11. Look for 11 in the difference column along the row of 40⁰. 11 is under 8.

Thus, x = 40.40⁰ + 0.08⁰

X = 40.48⁰

**Class Activity:**

Using the four-figure table, find the values of x, y and z in each of the triangles below.Give all answers correct to 2 s.f.

- 3.

45⁰

10 cm x y 6 km Z 3 km

30⁰ 50⁰

** **

**Use of Cosine**

Angle A |
Cos A |

30⁰ | 0.8660 |

35⁰ | 0.8192 |

40⁰ | 0.7660 |

45⁰ | 0.7071 |

50⁰ |
0.6428 |

55⁰ | 0.5736 |

60⁰ | 0.5000 |

** **

**Example**

- Calculate the value of y in the figure below.

Y |

** **

17 cm

50^{0}

** **

**Solution **

The hypotenuse (hyp) is 17 cm and y is adjacent (adj) to the given angle. Thus use the cosine of the given angle.

**Cos A = **

Cos 50⁰ =

Thus, = 17 x cos 50⁰

= 17 x 0.642 8

= 10.93 cm

= 11 cm to 2 s.f

- Use tables to find the angles:
- whose cosine is 0.4478,
- whose cosine is 0.568 2,

**Solution**

- Let the angle be B, then cos B = 0.4478.

Looking within the cosine table entries,

0.447 8 is opposite 63⁰ and under 0.4.

Thus, B = 63.4⁰.

- Let the angle be y, then cos y = 0.568 2.

Looking within the cosine table entries, the nearest value to 0.568 2 is 0.567 8. 0.567 8 is opposite 55⁰ and under 0.4⁰.

The difference between 5 682 and 5 678 is 4.

Look for 4 in the difference column along the row of 55⁰.4 is under 3. As the angles increase, their cosines decrease, therefore subtract the difference.

Thus, y = 55.40⁰ – 0.03⁰

Y = 55.37⁰

z |

**Class Activity:**

Using the four-figure table, find the values of x, y and z in each of the triangles below.Give all answers correct to 2 s.f.

- 3.

22mm x 50^{0}

40^{0} 12 km 25cm

** Y **60^{0}

**Use of Tangent**

Tangents of some chosen angles

Angle A |
Tan A |

25⁰ | 0.4663 |

30⁰ | 0.5774 |

35⁰ | 0.7002 |

40⁰ | 0.8391 |

45⁰ | 1.000 |

50⁰ | 1.192 |

55⁰ | 1.428 |

60⁰ | 1.732 |

65⁰ | 2.145 |

70⁰ | 2.747 |

* *

**Example**

- The angle of elevation of the top of a building is 25⁰ from a point 70 m away on a level ground. Calculate the height of the building.

**Solution**

The angle of elevation from a point 70 m away on a level ground to the top of a building is 25⁰. The height of the building can be found using the following equation:

Height = Distance * Tan (Angle)

Therefore, the height of the building is:

Height = 70 * Tan (25)

Height = 70 * 0.466

Height = 32.62 m

HK represents the height of the building; AK is on level ground.

H

A 25° K

70 m

**Tan A = **

Tan 25^{0} =

Let HK be x cm. KA = 70 cm and, from table above, tan 25⁰ = 0.4663.

Hence, = 0.4663

X = 0.4663 x70=4.663 x 7

= 32.641

∴The height of the building is 33 m to 2 s.f.

- Use tables to find the tangents of angles
- 32
^{0}59.6^{0}

Solution

- Looking within the table entries, the number opposite 32
^{0}and under 0.0 is 0.6249.

Thus, tan 32^{0} = 0.6249.

- The number opposite 59
^{0}and under 0.6^{0}is 1.704. Thus, tan 59.6^{0}is 1.704. Thus, tan 59.6^{0}= 1.704.

**Class Activity:**

Z |

10km |

Using the four-figure table, find the values of x, y and z in each of the triangles below.Give all answers correct to 2 s.f.

- 3.

50^{0}

x 12cm 4km Y 20km

15^{0}

5cm 7km

** **

** 30 ^{0}**

** **

**NOTE: Educators are to solve more examples for students.**

**Assignment:**

- A village is 8 km on a bearing of 040
^{0}from a point O. Calculate how far the village is north of O. - Use tables to find the angles whose tangents are:
- 9556 b.
- Calculate the length of the hypotenuse of the triangle below.

** **

8 cm

43^{0}

- A car travels 120 m along a straight road which is inclined at 8⁰ to the horizontal. Calculate the vertical distance through which the car rises.

120

8⁰ h

- A cone is 6 cm high and its vertical angle is 54⁰. Calculate the radius of its base.

**Practice Questions:**

- Calculate the lengths a, b, c, d, in the diagrams below. All lengths being in cm.

**i ii **

5 b c d

37^{0} 40.2^{0}

- Calculate the angles ∝, β in the triangles below.

i ii

0.9 m 3m 7 m 5 m

∝ β

- Use table to find the value of each triangles below.

7

35^{0}

65^{0}

- Use 4 figure table to find the Sine, Cosine and Tangent of the following:
- 35
^{0}23.1^{0}c. 19.5^{0} - Use 4 figure table to find the angles whose Sine, Cosine and Tangent are as follows:
- 9325 b. 0.8847 c.
- A cone is 8 cm high and its vertical angle is 62°. Find the diameter of its base.
- An isosceles triangle has a vertical angle of 116°, and its base is 8 cm long. Calculate its height.
- Find the angle of elevation of the top of a flagpole 31.9 m high from a point 55 m away on level ground.

**PRESENTATION:**

Step 1:

The subject teacher revises the previous topic

Step 2:

He or she introduces the new topic

Step 3:

The class teacher allows the pupils to give their own examples and he corrects them when the needs arise

**CONCLUSION:**

The subject goes round to mark the pupil’s notes. He does the necessary corrections