8.1 Q-7 | Fundamentals Of Trigonometry | Mathematics 11

Exercise 8.1 — Question 7

Problem

Simplify the following using trigonometric identities:

$sin 13 8^{\circ} cos 4 6^{\circ} - cos 13 8^{\circ} sin 4 6^{\circ}$

Solution

Recall the sine difference formula (fundamental law of trigonometry):

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

Comparing $sin 13 8^{\circ} cos 4 6^{\circ} - cos 13 8^{\circ} sin 4 6^{\circ}$ with the right-hand side, we identify:

$α = 13 8^{\circ}, β = 4 6^{\circ}$

Therefore:

$sin 13 8^{\circ} cos 4 6^{\circ} - cos 13 8^{\circ} sin 4 6^{\circ} = sin (13 8^{\circ} - 4 6^{\circ}) = sin 9 2^{\circ}$

Since $sin 9 2^{\circ} = sin (18 0^{\circ} - 8 8^{\circ}) = sin 8 8^{\circ}$ , or we can leave the answer as:

$sin 9 2^{\circ}$

Key Identity Used

Identity	Formula
Sine Difference	$sin (α - β) = sin α cos β - cos α sin β$
Sine Sum	$sin (α + β) = sin α cos β + cos α sin β$
Cosine Difference	$cos (α - β) = cos α cos β + sin α sin β$
Cosine Sum	$cos (α + β) = cos α cos β - sin α sin β$

Tip: The pattern $sin A cos B - cos A sin B$ always collapses to $sin (A - B)$ . Recognising this pattern quickly is essential for FBISE exam questions.

Exercise 8.1 — Question 7

Problem

Simplify the following using trigonometric identities:

$sin 13 8^{\circ} cos 4 6^{\circ} - cos 13 8^{\circ} sin 4 6^{\circ}$

Solution

Recall the sine difference formula (fundamental law of trigonometry):

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

Comparing $sin 13 8^{\circ} cos 4 6^{\circ} - cos 13 8^{\circ} sin 4 6^{\circ}$ with the right-hand side, we identify:

$α = 13 8^{\circ}, β = 4 6^{\circ}$

Therefore:

$sin 13 8^{\circ} cos 4 6^{\circ} - cos 13 8^{\circ} sin 4 6^{\circ} = sin (13 8^{\circ} - 4 6^{\circ}) = sin 9 2^{\circ}$

Since $sin 9 2^{\circ} = sin (18 0^{\circ} - 8 8^{\circ}) = sin 8 8^{\circ}$ , or we can leave the answer as:

$sin 9 2^{\circ}$

Key Identity Used

Identity	Formula
Sine Difference	$sin (α - β) = sin α cos β - cos α sin β$
Sine Sum	$sin (α + β) = sin α cos β + cos α sin β$
Cosine Difference	$cos (α - β) = cos α cos β + sin α sin β$
Cosine Sum	$cos (α + β) = cos α cos β - sin α sin β$

Tip: The pattern $sin A cos B - cos A sin B$ always collapses to $sin (A - B)$ . Recognising this pattern quickly is essential for FBISE exam questions.