8.2. Trigonometric Ratios

In Section 8.1, you have seen some right triangles imagined to be formed in different situations.

Let us take a right triangle ABC as shown in Fig. 8.4.

Here, ∠CAB (or, in brief, angle A) is an acute angle. Note the position of the side BC with respect to angle A. It faces ∠A. We call it the side opposite to angle A. AC is the hypotenuse of the right triangle and the side AB is a part of ∠A. So, we call it the side adjacent to angle A.

In a right triangle, the trigonometric ratios are based on the relationships between the lengths of the sides and the angles of the triangle. Each angle has an opposite side, an adjacent side, and a hypotenuse.

• The opposite side is the one opposite to the angle we are considering (in this case, BC is opposite angle A).
• The adjacent side is the one next to the angle we are considering (AB is adjacent to angle A).
• The hypotenuse is always the longest side of the triangle and is opposite the right angle (AC in this case).

Imagine you're standing on the ground looking up at a tall tree (triangle ABC). The height of the tree is the 'opposite side' (BC), the distance from you to the base of the tree is the 'adjacent side' (AB), and a line drawn from the top of the tree to where you are standing (the hypotenuse) represents the 'hypotenuse' (AC).

You have studied the concept of ‘ratio’ in your earlier classes. We now define certain ratios involving the sides of a right triangle, and call them trigonometric ratios.

The trigonometric ratios of the angle A in right triangle ABC (see Fig. 8.4) are defined as follows:

• sin A = (side opposite to angle A) / (hypotenuse)
• cos A = (side adjacent to angle A) / (hypotenuse)
• tan A = (side opposite to angle A) / (side adjacent to angle A)
• cosec A = 1 / sin A
• sec A = 1 / cos A
• cot A = 1 / tan A

Trigonometric ratios are based on the relationships of the sides of the triangle to its angles. For angle A in triangle ABC:

1. Sine (sin A) measures the ratio of the length of the opposite side to the hypotenuse.
2. Cosine (cos A) measures the ratio of the length of the adjacent side to the hypotenuse.
3. Tangent (tan A) measures the ratio of the length of the opposite side to the adjacent side.

The ratios cosecant (cosec A), secant (sec A), and cotangent (cot A) are the reciprocals of the sine, cosine, and tangent, respectively. This means:

• cosec A = 1/sin A
• sec A = 1/cos A
• cot A = 1/tan A

Think of trigonometric ratios as statistics for measuring the 'steepness' or 'height' of a hill you're climbing. If you know the height of the hill (opposite) and the distance you walked on the ground (adjacent), you can determine how steep it is (tangent). For a more adventurous example, you can measure how high up you can climb (sine) relative to the steepest possible height (hypotenuse).

Note that the ratios cosec A, sec A, and cot A are respectively the reciprocals of the ratios sin A, cos A, and tan A.

Reciprocals in mathematics are simply values that, when multiplied together, give a product of one. In trigonometry, each ratio has a reciprocal:

• For sine (sin A), the reciprocal is cosecant (cosec A), meaning cos A is equal to 1 divided by sin A.
• For cosine (cos A), the reciprocal is secant (sec A), meaning sec A is equal to 1 divided by cos A.
• For tangent (tan A), the reciprocal is cotangent (cot A), meaning cot A is equal to 1 divided by tan A.

These relationships help in solving various problems efficiently.

Imagine if you were hiking up a hill, measuring with a virtual assistant. Your assistant gives you the height of the hill in relation to how much you need to walk (sine). Then, if you ask your assistant how steep the hill is in reverse terms, they would tell you how much to walk to reach a specific height – that’s akin to using cosecant, the 'inverse' relationship.

The trigonometric ratios of angle A in ΔPAM not differ from those of angle A in ΔCAB.

This shows that the trigonometric ratios of an angle do not vary with the lengths of the sides of the triangle, if the angle remains the same.

In similar triangles, the corresponding angles are equal, and the ratios of the lengths of corresponding sides are proportional. This means, no matter how big or small the triangle, if the angle stays the same, the trigonometric ratios remain constant.

This is a key feature of trigonometric functions – they express relationships that remain consistent across different contexts, shapes, and sizes.

If you think of two similar flags flying on different poles, the angle at which they hang remains the same irrespective of the height of the poles. The angle they create with respect to the ground creates consistent ratios, just like similar triangles!

Now, if we take a point P on the hypotenuse AC or a point Q on AC extended, of the right triangle ABC and draw PM perpendicular to AB and QN perpendicular to AB extended, how will the trigonometric ratios of ∠A in ΔPAM differ from those of ∠A in ΔCAB or from those of ∠A in ΔQAN?

We can establish that the ratios do not change by using the properties of similar triangles. When constructing points on the hypotenuse and drawing perpendiculars, we find that ΔPAM and ΔCAB are similar due to equal angles. Thus, the lengths of corresponding sides have the same ratios, confirming that trigonometric ratios do not change with the size of the triangle.

This understanding helps solidify the importance of angle measures over side lengths in defining trigonometric functions.

This is like observing shadows cast by trees at different times of day. The angles of shadows remain the same (the trigonometric ratios), even as the lengths of the shadows change due to the sun's position (the sides of different triangles).

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