8. INTRODUCTION TO TRIGONOMETRY

There is perhaps nothing which so occupies the middle position of mathematics as trigonometry. – J.F. Herbart (1890)

You have already studied about triangles, and in particular, right triangles, in your earlier classes. Let us take some examples from our surroundings where right triangles can be imagined to be formed. For instance:

1. Suppose the students of a school are visiting Qutub Minar. Now, if a student is looking at the top of the Minar, a right triangle can be imagined to be made. Can the student find out the height of the Minar, without actually measuring it?
2. Suppose a girl is sitting on the balcony of her house located on the bank of a river. She is looking down at a flower pot placed on a stair of a temple situated nearby on the other bank of the river. A right triangle is imagined to be made in this situation. If you know the height at which the person is sitting, can you find the width of the river?
3. Suppose a hot air balloon is flying in the air. A girl happens to spot the balloon in the sky and runs to her mother to tell her about it. Her mother rushes out of the house to look at the balloon. Now, when the girl had spotted the balloon initially it was at point A. When both the mother and daughter came out to see it, it had already travelled to another point B. Can you find the altitude of B from the ground? In all the situations given above, the distances or heights can be found by using some mathematical techniques, which come under a branch of mathematics called ‘trigonometry’. The word ‘trigonometry’ is derived from the Greek words ‘tri’ (meaning three), ‘gon’ (meaning sides) and ‘metron’ (meaning measure). In fact, trigonometry is the study of relationships between the sides and angles of a triangle.

Trigonometry is a crucial part of mathematics that focuses on the relationships between the angles and sides of triangles, particularly right triangles. The section introduces real-life scenarios where such triangles appear, which emphasizes the practicality of trigonometry. The historical context explains that the term comes from Greek roots, indicating its foundation in measuring triangles.

Imagine you're at a park looking at a tall slide, and you want to know how high it is without climbing it. By standing a little distance away and looking up at the top, you can form a right triangle with the ground, your line of sight, and the height of the slide. Using trigonometric principles, you can calculate the height of the slide based on your distance from it and the angle of your gaze.

In this chapter, we will study some ratios of the sides of a right triangle with respect to its acute angles, called trigonometric ratios of the angle. We will restrict our discussion to acute angles only. However, these ratios can be extended to other angles also. We will also define the trigonometric ratios for angles of measure 0° and 90°. We will calculate trigonometric ratios for some specific angles and establish some identities involving these ratios, called trigonometric identities.

The main focus will be on understanding trigonometric ratios related to acute angles in right triangles, which are essential for solving various mathematical problems. Trigonometric ratios include sine, cosine, and tangent, and their relationships can be extended beyond acute angles. This section emphasizes learning how to define and compute these ratios for specific angles.

Consider a ladder leaning against a wall. The height the ladder reaches, the distance from the wall to the base of the ladder (the ground), and the length of the ladder itself can all be represented as a right triangle. Here, trigonometric ratios can help determine the angle at which the ladder is placed, and understanding these angles can help ensure safety and effectiveness in scenarios like construction.

The trigonometric ratios of the angle A in right triangle ABC are defined as follows:

• sine of ∠A = side opposite to angle A / hypotenuse
• cosine of ∠A = side adjacent to angle A / hypotenuse
• tangent of ∠A = side opposite to angle A / side adjacent to angle A Additionally, we have the reciprocal ratios:
• cosecant of ∠A = 1/sine of ∠A
• secant of ∠A = 1/cosine of ∠A
• cotangent of ∠A = 1/tangent of ∠A.

Trigonometric ratios are essential tools in trigonometry that express the relationships between angles and side lengths in right triangles. For example, the sine, cosine, and tangent functions allow us to find unknown side lengths and angles when certain values are known. These ratios are also reciprocal relationships, which means knowing one ratio can help calculate the others.

Think of a tree and your position on the ground. If you look up at the top of the tree (forming a right triangle), the height of the tree can be found using the sine function if you know your distance from the base of the tree and the angle you look up. This illustrates how trigonometrical ratios help measure heights indirectly in real life.

We shall now calculate the trigonometric ratios of some specific angles: 0°, 30°, 45°, 60°, and 90°. These values are:

• sin 0° = 0, cos 0° = 1, tan 0° = 0
• sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3
• sin 45° = √2/2, cos 45° = √2/2, tan 45° = 1
• sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3
• sin 90° = 1, cos 90° = 0, tan 90° is undefined.

Knowing specific trigonometric ratios allows us to solve problems involving angles of 0°, 30°, 45°, 60°, and 90° effectively. For example, sine values can be used to find opposite sides when a hypotenuse is known, and these fundamental ratios serve as building blocks for more complex trigonometric calculations.

Imagine you’re building a ramp for a wheelchair. Knowing how steep the ramp needs to be translates into specific angles. By using the sine of those angles, you can determine the height you need for the ramp to ensure it is practical and safe. This real-world application showcases the usefulness of knowing specific angle ratios.

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