9. Given the sequence: \( 1 ; \sin 45^{\circ} ; \frac{1}{2} ; \ldots \) a Write down the next two terms if the pattern continue b Find the \( n \)th term.

To solve the problem, let's analyze the given sequence step by step. ### Part a: Finding the Next Two Terms The sequence provided is: 1. \( 1 \) 2. \( \sin 45^{\circ} \) 3. \( \frac{1}{2} \) First, we need to evaluate \( \sin 45^{\circ} \): \[ \sin 45^{\circ} = \frac{\sqrt{2}}{2} \] Now, let's look for a pattern in the sequence: - The first term is \( 1 \). - The second term is \( \frac{\sqrt{2}}{2} \). - The third term is \( \frac{1}{2} \). It appears that the terms are decreasing, and we can express them in terms of trigonometric functions. The first three terms can be related to angles: 1. \( 1 = \sin 90^{\circ} \) 2. \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \) 3. \( \sin 30^{\circ} = \frac{1}{2} \) Following this pattern, the next terms could be: 4. \( \sin 0^{\circ} = 0 \) 5. \( \sin (-45^{\circ}) = -\frac{\sqrt{2}}{2} \) Thus, the next two terms are: - Fourth term: \( 0 \) - Fifth term: \( -\frac{\sqrt{2}}{2} \) ### Part b: Finding the \( n \)th Term From the analysis, we can see that the sequence can be expressed in terms of sine functions with decreasing angles. The angles appear to be: - \( 90^{\circ} \) for the first term - \( 45^{\circ} \) for the second term - \( 30^{\circ} \) for the third term - \( 0^{\circ} \) for the fourth term - \( -45^{\circ} \) for the fifth term The angles can be represented as: - \( 90^{\circ} - 45(n-1) \) for the \( n \)th term. Thus, the \( n \)th term can be expressed as: \[ a_n = \sin(90^{\circ} - 45(n-1)) \] This simplifies to: \[ a_n = \cos(45(n-1)) \] Therefore, the \( n \)th term of the sequence is: \[ a_n = \cos(45(n-1)) \] ### Summary of Results - Next two terms: \( 0 \) and \( -\frac{\sqrt{2}}{2} \) - \( n \)th term: \( a_n = \cos(45(n-1)) \)