# Trigonometric ratios

Complete above steps to enable this content

Since all of these triangles are similar, we know that their sides are proportional. In particular, the following ratios are the same for all of these triangles:

Let’s try to summarise this: we picked a certain value for **α**, and got lots of similar, right-angled triangles. **All of these triangles have the same ratios of sides**. Since **α** was our only variable, there must be some relationship between **α** and those ratios.

These relationships are the **Trigonometric ratios** – and there are three of them:

The three Trigonometric ratios are relationships between the angles and the ratios of sides in a right-angles triangle. They each have a name, as well as a 3-letter abbreviation:

**Sine:**sin α = Opposite Hypotenuse **Cosine:**cos α = Adjacent Hypotenuse **Tangent:**tan α = Opposite Adjacent

Easy way to remember name and sides: SOH-CAH-TOA

Let's expand on this idea. We have seen:

and given names for these ratios.

What about the inverse of these ratios?

Complete above steps to enable this content

Let's give these ratios also a name.

The inverse of the above ratios also each have a name, as well as a 3-letter abbreviation:

**Cosecant:**cosec α = Hypotenuse Opposite **Secant:**sec α = Hypotenuse Adjacent **Cotangent:**cot α = Adjacent Opposite

These are the 6 trigonometric ratios of an acute angle. The angle

Now, when we consider

Complete above steps to enable this content

Let's consider the triangle ABC. Two sides are given, AB = 3k and BC = k. From trigonometric ratios we can find the value of

Considering angle A i.e. angle α we get: sin A =

We know that ABC is a

Complete above steps to enable this content

**Solving for AC**

Complete above steps to enable this content

We now have all the sides of the triangle.

cos A =

tan A =

The first use of the idea of **sine** in the way we use it today was in the work Aryabhatiyam by Aryabhata, in A.D. 500. Aryabhata used the word ardha-jya for the half-chord, which was shortened to jya or jiva in due course. When the Aryabhatiyam was translated into Arabic, the word jiva was retained as it is. The word jiva was translated into sinus, which means curve, when the Arabic version was translated into Latin. Soon the word sinus, also used as sine, became common in mathematical texts throughout Europe. An English Professor of astronomy Edmund Gunter (1581–1626), first used the abbreviated notation **sin**.

The origin of the terms **cosine** and **tangent** was much later. The cosine function arose from the need to compute the sine of the complementary angle. Aryabhatta called it kotijya. The name cosinus originated with Edmund Gunter. In 1674, the English Mathematician Sir Jonas Moore first used the abbreviated notation **cos**.

Complete above steps to enable this content

**Remark** : Note that the **symbol sin A is used as an abbreviation for ‘the sine of the angle A’**. sin A is **not the product of ‘sin’ and A**. ‘sin’ separated from A has no meaning. Similarly, cos A is not the product of ‘cos’ and A. Similar interpretations follow for other trigonometric ratios also.

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 (see adjacent figure), how will the trigonometric ratios of ∠A in △ PAM differ from those of ∠A in △ CAB or from those of ∠A in △ QAN ?

Complete above steps to enable this content

To answer this, first look at these triangles. Is △ PAM similar to △ CAB? From Chapter 6, recall the **AA similarity criterion**. Using the criterion, you will see that the triangles PAM and CAB are similar. Therefore, by the property of similar triangles, the corresponding sides of the triangles are proportional. So, we have

From re-arranging the last two ratios, we find

This shows that the trigonometric ratios of angle A in △ PAM do not differ from those of angle A in △ CAB. In the same way, you should check that the value of sin A (and also of other trigonometric ratios) remains the same in △ QAN also.

From our observations, it is now clear that:

**The values of the trigonometric ratios of an angle do not vary with the lengths of the sides of the triangle, if the angle remains the same**.

**Note :** For the sake of convenience, we may write **cosec A = **, which will be discussed in higher classes. Similar conventions hold for the other trigonometric ratios as well. Sometimes,

**the Greek letter θ (theta) is also used to denote an angle**.

Complete above steps to enable this content

**We have defined six trigonometric ratios of an acute angle. If we know any one of the ratios, can we obtain the other ratios?** Let us see.

If in a right triangle ABC, sin A =

**Let's change the side lengths and try it again!** Say,

Therefore, AC =

So we get AC = 5k

Now,

cosA =

Similarly, you can obtain the other trigonometric ratios of the angle A.

**Remark** : Since the hypotenuse is the longest side in a right triangle, the value of sin A or cos A is always less than 1 (or, in particular, equal to 1).

Let us consider some examples.

From the figure tan A =

Apply Pythagoras theorem.

AC =

Complete above steps to enable this content

**Example 1 :** Given tan A =

**Solution :** Let us first draw a right △ ABC.

Now, we know that tan A =

Therefore, if BC = 4k, then AB = 3k, where k is a positive number.

Now, by using the Pythagoras Theorem, we have:

So, AC =

Now, we can write all the trigonometric ratios using their definitions.

sin A =

cosA =

Therefore, cotA =

cosecA =

secA =

Complete above steps to enable this content

Complete above steps to enable this content

**Example 2 :** If ∠B and ∠Q are acute angles such that sin B = sin Q, then prove that ∠ B = ∠ Q.

**Solution :** Let us consider two right triangles ABC and PQR where sin B = sin Q (see Fig. 8.9).

We have sin B =

Then

Therefore, *k* , say (1)

Now, using Pythagoras theorem:

BC =

So,

=

From (1) and (2), we have:

Then, by using Theorem 6.4, Δ ACB ~ Δ PRQ and therefore, ∠ B = ∠ Q.

undefined

**Example 3 :** Consider Δ ACB, right-angled at C, in which AB = 29 units, BC = 21 units and ∠ ABC = θ. Determine the values of :

(i)

(ii)

**Solution :** In Δ ACB, we have:

AC =

So, sin θ =

(i)

(ii)

undefined

**Example 4 :** In a right triangle ABC, right-angled at B, if tan A = 1, then verify that 2 sin A cos A = 1.

**Solution :** In ΔABC, tan A =

Let AB = BC = k, where k is a positive number.

Now, AC =

Therefore, sin A =

So, 2 sin A cos A = 2

undefined

**Example 5 :** In Δ OPQ, right-angled at P, OP = 7 cm and OQ – PQ = 1 cm. Determine the values of sin Q and cos Q.

**Solution :** In Δ OPQ, we have:

PQ =

So, sin Q =

undefined