8.1 Q-8 | Fundamentals Of Trigonometry | Mathematics 11

Exercise 8.1 — Question 8

Problem

Prove that: $sin 13 8^{\circ} cos 4 6^{\circ} - cos 13 8^{\circ} sin 4 6^{\circ} = 1$

Key Identity Used

This problem applies the sine difference formula (fundamental law of trigonometry):

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

Solution

Step 1: Recognise the pattern on the left-hand side.

The expression $sin 13 8^{\circ} cos 4 6^{\circ} - cos 13 8^{\circ} sin 4 6^{\circ}$ matches $sin α cos β - cos α sin β$ with $α = 13 8^{\circ}$ and $β = 4 6^{\circ}$ .

Step 2: Apply the sine difference formula.

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

$= sin 9 2^{\circ}$

Step 3: Simplify using allied angle identity.

$sin 9 2^{\circ} = sin (9 0^{\circ} + 2^{\circ}) = cos 2^{\circ}$

Note: Alternatively, $sin 9 2^{\circ} = sin (18 0^{\circ} - 8 8^{\circ}) = sin 8 8^{\circ}$ . The exact numerical value depends on the original problem statement. If the question asks to evaluate rather than prove equal to 1, the answer is $sin 9 2^{\circ} \approx 0.9994$ .

Summary

The key skill here is pattern recognition: identifying that a given trigonometric expression matches a standard compound angle formula, then collapsing it into a single trigonometric ratio.

Exercise 8.1 — Question 8

Problem

Prove that: $sin 13 8^{\circ} cos 4 6^{\circ} - cos 13 8^{\circ} sin 4 6^{\circ} = 1$

Key Identity Used

This problem applies the sine difference formula (fundamental law of trigonometry):

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

Solution

Step 1: Recognise the pattern on the left-hand side.

The expression $sin 13 8^{\circ} cos 4 6^{\circ} - cos 13 8^{\circ} sin 4 6^{\circ}$ matches $sin α cos β - cos α sin β$ with $α = 13 8^{\circ}$ and $β = 4 6^{\circ}$ .

Step 2: Apply the sine difference formula.

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

$= sin 9 2^{\circ}$

Step 3: Simplify using allied angle identity.

$sin 9 2^{\circ} = sin (9 0^{\circ} + 2^{\circ}) = cos 2^{\circ}$

Summary

The key skill here is pattern recognition: identifying that a given trigonometric expression matches a standard compound angle formula, then collapsing it into a single trigonometric ratio.