8.3 Q-2 | Fundamentals Of Trigonometry | Mathematics 11

Exercise 8.3 — Question 2

Background: Fundamental Law and Addition Formulas

The fundamental law of trigonometry states:

$cos (α - β) = cos α cos β + sin α sin β$

This is established using the distance formula between two points on the unit circle. From this law, the following addition formulas are deduced:

Identity	Formula
$cos (α + β)$	$cos α cos β - sin α sin β$
$sin (α + β)$	$sin α cos β + cos α sin β$
$sin (α - β)$	$sin α cos β - cos α sin β$

Product-to-Sum Identities

By adding and subtracting the addition formulas, we obtain the product-to-sum identities:

$2 sin A cos B = sin (A + B) + sin (A - B)$

$2 cos A sin B = sin (A + B) - sin (A - B)$

$2 cos A cos B = cos (A - B) + cos (A + B)$

$2 sin A sin B = cos (A - B) - cos (A + B)$

Question 2 — Express as Sum or Difference

Express each of the following products as a sum or difference:

Part (i): $sin 5 x cos 3 x$

Using $2 sin A cos B = sin (A + B) + sin (A - B)$ with $A = 5 x$ , $B = 3 x$ :

$2 sin 5 x cos 3 x = sin (5 x + 3 x) + sin (5 x - 3 x) = sin 8 x + sin 2 x$

$∴ sin 5 x cos 3 x = \frac{1}{2} [sin 8 x + sin 2 x]$

Part (ii): $cos 7 x sin 3 x$

Using $2 cos A sin B = sin (A + B) - sin (A - B)$ with $A = 7 x$ , $B = 3 x$ :

$2 cos 7 x sin 3 x = sin (7 x + 3 x) - sin (7 x - 3 x) = sin 10 x - sin 4 x$

$∴ cos 7 x sin 3 x = \frac{1}{2} [sin 10 x - sin 4 x]$

Part (iii): $cos 5 x cos 3 x$

Using $2 cos A cos B = cos (A - B) + cos (A + B)$ with $A = 5 x$ , $B = 3 x$ :

$2 cos 5 x cos 3 x = cos (5 x - 3 x) + cos (5 x + 3 x) = cos 2 x + cos 8 x$

$∴ cos 5 x cos 3 x = \frac{1}{2} [cos 2 x + cos 8 x]$

Part (iv): $sin 4 x sin 2 x$

Using $2 sin A sin B = cos (A - B) - cos (A + B)$ with $A = 4 x$ , $B = 2 x$ :

$2 sin 4 x sin 2 x = cos (4 x - 2 x) - cos (4 x + 2 x) = cos 2 x - cos 6 x$

$∴ sin 4 x sin 2 x = \frac{1}{2} [cos 2 x - cos 6 x]$

Exercise 8.3 — Question 2

Background: Fundamental Law and Addition Formulas

The fundamental law of trigonometry states:

$cos (α - β) = cos α cos β + sin α sin β$

This is established using the distance formula between two points on the unit circle. From this law, the following addition formulas are deduced:

Identity	Formula
$cos (α + β)$	$cos α cos β - sin α sin β$
$sin (α + β)$	$sin α cos β + cos α sin β$
$sin (α - β)$	$sin α cos β - cos α sin β$

Product-to-Sum Identities

By adding and subtracting the addition formulas, we obtain the product-to-sum identities:

$2 sin A cos B = sin (A + B) + sin (A - B)$

$2 cos A sin B = sin (A + B) - sin (A - B)$

$2 cos A cos B = cos (A - B) + cos (A + B)$

$2 sin A sin B = cos (A - B) - cos (A + B)$

Question 2 — Express as Sum or Difference

Express each of the following products as a sum or difference:

Part (i): $sin 5 x cos 3 x$

Using $2 sin A cos B = sin (A + B) + sin (A - B)$ with $A = 5 x$ , $B = 3 x$ :

$2 sin 5 x cos 3 x = sin (5 x + 3 x) + sin (5 x - 3 x) = sin 8 x + sin 2 x$

$∴ sin 5 x cos 3 x = \frac{1}{2} [sin 8 x + sin 2 x]$

Part (ii): $cos 7 x sin 3 x$

Using $2 cos A sin B = sin (A + B) - sin (A - B)$ with $A = 7 x$ , $B = 3 x$ :

$2 cos 7 x sin 3 x = sin (7 x + 3 x) - sin (7 x - 3 x) = sin 10 x - sin 4 x$

$∴ cos 7 x sin 3 x = \frac{1}{2} [sin 10 x - sin 4 x]$

Part (iii): $cos 5 x cos 3 x$

Using $2 cos A cos B = cos (A - B) + cos (A + B)$ with $A = 5 x$ , $B = 3 x$ :

$2 cos 5 x cos 3 x = cos (5 x - 3 x) + cos (5 x + 3 x) = cos 2 x + cos 8 x$

$∴ cos 5 x cos 3 x = \frac{1}{2} [cos 2 x + cos 8 x]$

Part (iv): $sin 4 x sin 2 x$

Using $2 sin A sin B = cos (A - B) - cos (A + B)$ with $A = 4 x$ , $B = 2 x$ :

$2 sin 4 x sin 2 x = cos (4 x - 2 x) - cos (4 x + 2 x) = cos 2 x - cos 6 x$

$∴ sin 4 x sin 2 x = \frac{1}{2} [cos 2 x - cos 6 x]$