Trigonometric Identities

Example: In ∆ ABC, right-angled at B, AB = 24 cm, BC = 7 cm.

Determine:
(i) sin A, cos A
(ii) sin C, cos C

Solution:

In a given triangle ABC, right-angled at B =

Given: AB = 24 cm and BC = 7 cm

According to the Pythagoras Theorem,

Therefore, AC = 25 cm

(i) Sin A = BC/AC = 7/25

cos A = AB/AC = 24/25

(ii).Sin C = AB/AC = 24/25

Cos C = BC/AC = 7/25

Example If Sin A = 3/4, Calculate cos A and tan A.

Sin A = 3/4

Sin A = Opposite Side/Hypotenuse Side = 3/4

Now, let BC be 3k and AC will be 4k.

As per the Pythagoras theorem, we know;

Hypotenuse = Perpendicular+ Base

Substitute the value of AC and BC in the above expression to get;

Hence, AB = k

cos A = Adjacent Side/Hypotenuse side = AB/AC

cos A = k/4k = /4

And,

Tan A = Opposite side/Adjacent side = BC/AB

Tan A = 3k/ k = 3/

Example: If A and B are acute angles such that cos A = cos B, then show that A = B.

cos A = AC/AB

cos B = BC/AB

Since, it is given,

cos A = cos B

AC/AB = BC/AB

AC = BC

We know that by isosceles triangle theorem, the angles opposite to the equal sides are equal.

Therefore, A = B