Introduction to Trigonometry

Suppose a boy is standing in front of a lamp post at a certain distance. The height of the boy is 170 cm and the length of his shadow is 150 cm.

You can see from the above figure that the boy and his shadow form a right-angled triangle as shown in the figure below.

The ratio of the height of the boy to his shadow is 170:150 i.e., 17:15.

**Is this ratio related to either of the angles of ΔABC?**

Let us find that out by going through the given video.

We can also conclude the following:

**cos A , tan A **

Also, note that

**tan A = **** and cot A = **

Let us now solve some more examples based on trigonometric ratios.

**Example 1:**

**In a triangle ABC, right-angled at B, side AB = 40 cm and BC = 9 cm. Find the value of sin A, cos A, and tan A.**

**Solution:**

It is given that AB = 40 cm and BC = 9 cm

Using Pythagoras theorem in ΔABC, we obtain

(AC)^{2} = (AB)^{2} + (BC)^{2}

(AC)^{2} = (40)^{2} + (9)^{2}

(AC)^{2 }= 1600 + 81

(AC)^{2 }= 1681

(AC)^{2 }= (41)^{2}

AC = 41 cm

Now, sin A

cos A

tan A

**Example 2:**

**From the given figure, find the values of cosec C and cot C, if AC = BC + 1.**

**Solution:**

Now, it is given that AB = 5 cm and

AC = BC + 1 … (1)

By Pythagoras theorem, we obtain

(AB)^{2} + (BC)^{2} = (AC)^{2}

⇒ (AC)^{2} − (BC)^{2} = (AB)^{2}

⇒ (BC + 1)^{2} – (BC)^{2} = (5)^{2} [Using (1)]

⇒ (BC)^{2} + 1 + 2BC – (BC)^{2 }= 25

⇒ 2BC = 25 – 1

⇒ 2BC = 24

⇒ BC = 12 cm

∴ AC = 12 + 1 = 13 cm

Thus, cosec C =

=

And, cot C =

=

**E...**

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