3. If \( \cos x=\frac{3}{5} \), \( x \) lies in the second quadrant, find the values of \( \sin x \), \( \tan x, \sec x \) 4. Prove that \( \frac{\sin 5 x+\sin 3 x}{\cos 5 x+\cos 3 x}=\tan 4 x \) 8. d) The value of \( f^{2 a} \) is b) Find the muliplicative inverse of \( 1- \) in atib form 6.i) If \( (x+1, y-2)=(3,1) \), find the values of \( x \) and \( y \) ii) Find the domain and range of the function \( \sin x \) (i) Solve the inequality \( x+\frac{x}{2}+\frac{x}{3}<10+\frac{x}{6} \) ii) Mark the solution in a number line

Let's solve each of the problems step by step. ### Problem 3 Given \( \cos x = \frac{3}{5} \) and \( x \) lies in the second quadrant, we need to find \( \sin x \), \( \tan x \), and \( \sec x \). 1. **Finding \( \sin x \)**: Using the Pythagorean identity: \[ \sin^2 x + \cos^2 x = 1 \] Substituting \( \cos x \): \[ \sin^2 x + \left(\frac{3}{5}\right)^2 = 1 \] \[ \sin^2 x + \frac{9}{25} = 1 \] \[ \sin^2 x = 1 - \frac{9}{25} = \frac{25}{25} - \frac{9}{25} = \frac{16}{25} \] \[ \sin x = \pm \sqrt{\frac{16}{25}} = \pm \frac{4}{5} \] Since \( x \) is in the second quadrant, \( \sin x \) is positive: \[ \sin x = \frac{4}{5} \] 2. **Finding \( \tan x \)**: \[ \tan x = \frac{\sin x}{\cos x} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3} \] 3. **Finding \( \sec x \)**: \[ \sec x = \frac{1}{\cos x} = \frac{1}{\frac{3}{5}} = \frac{5}{3} \] Thus, the values are: - \( \sin x = \frac{4}{5} \) - \( \tan x = \frac{4}{3} \) - \( \sec x = \frac{5}{3} \) --- ### Problem 4 We need to prove that: \[ \frac{\sin 5x + \sin 3x}{\cos 5x + \cos 3x} = \tan 4x \] Using the sum-to-product identities: \[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \] \[ \cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \] Let \( A = 5x \) and \( B = 3x \): \[ \sin 5x + \sin 3x = 2 \sin\left(4x\right) \cos\left(x\right) \] \[ \cos 5x + \cos 3x = 2 \cos\left(4x\right) \cos\left(x\right) \] Now substituting these into the left-hand side: \[ \frac{2 \sin(4x) \cos(x)}{2 \cos(4x) \cos(x)} = \frac{\sin(4x)}{\cos(4x)} = \tan(4x) \] Thus, we have proved: \[ \frac{\sin 5x + \sin 3x}{\cos 5x + \cos 3x} = \tan 4x \] --- ### Problem 6 **i)** Given \( (x+1, y-2) = (3, 1) \), we can set up the equations: \[ x + 1 = 3 \quad \Rightarrow \quad x = 2 \] \[ y - 2 = 1 \quad \Rightarrow \quad y = 3 \] Thus, \( x = 2 \) and \( y = 3 \). **ii)** The domain and range of the function \( \sin x \): - **Domain**: All real numbers, \( (-\infty, \infty) \) - **Range**: \( [-1, 1] \) --- ### Problem 8 **d)** The value of \( f^{2a} \) is not specified in the question. Please provide more context or details. **b)** To find the multiplicative inverse of \( 1 - \) in "atib" form, we need clarification on what "atib" refers to