8. Simplify without using a calculator. (8.1) \( \frac{\sin \left(180^{\circ}-x\right) \cdot \tan \left(360^{\circ}-x\right)}{\cos \left(80^{\circ}-x\right)} \times \frac{\cos \left(-180^{\circ}-x\right)}{\cos \left(360^{\circ}+x\right) \sin \left(360^{\circ}-x\right)} \) \( 8.2 \frac{\cos 135^{\circ} \sin 160^{\circ}}{\sin 225^{\circ} \cos 70^{\circ}} \) (8.3) \( \frac{\sin (-\theta)+\cos 120^{\circ}+\tan \left(-180^{\circ}-\theta\right)}{\sin ^{2} 225^{\circ}-\tan (-\theta)-\cos \left(90^{\circ}+\theta\right)} \) B.4 \( 4^{x} \frac{\sin 247^{\circ} \cdot \tan 23^{\circ} \cdot \cos 113^{\circ}}{\sin \left(-157^{\circ}\right)} \) (8.5) \( \frac{3 \cos 150^{\circ} \cdot \sin 270^{\circ}}{\tan \left(-45^{\circ}\right) \cdot \cos 600^{\circ}} \) 8.6) \( \frac{\tan \left(180^{\circ}-x\right) \cdot \sin \left(90^{\circ}+x\right)}{\sin (-x)}-\sin y \cdot \cos \left(90^{\circ}-y\right) \) \( 8.7 \frac{\tan 30^{\circ} \cdot \sin 60^{\circ} \cdot \cos 25^{\circ}}{\cos 135^{\circ} \cdot \sin \left(-45^{\circ}\right) \cdot \sin 65^{\circ}} \) 6.8) \( \frac{\tan \left(180^{\circ}-x\right) \cdot \sin \left(90^{\circ}-x\right)}{\cos \left(90^{\circ}+x\right)}-\frac{\cos \left(180^{\circ}-x\right)}{\sin \left(90^{\circ}+x\right)} \) \( 8.9 \frac{\sin 189^{\circ}}{\tan 549^{\circ}}-\frac{\cos ^{2}\left(-9^{\circ}\right)}{\sin 99^{\circ}} \) Solving trigonometric equations (no calculators) (1.) If \( \sin \mathrm{A}=\frac{-3}{5} \) and \( 0^{\circ}<\mathrm{A}<270^{\circ} \) determine the value of: \( 1.1 \cos A \) \( 1.2 \tan A \). (2.) If \( -5 \tan \theta-3=0 \) and \( \sin \theta<0 \), determine: \( 2.1 \sin ^{2} \theta^{\circ} \) \( 2.25 \cos \theta \) \( 2.3 \quad 1-\cos ^{2} \theta \) 3. If \( 13 \cos \theta+12=0 \) and \( 180^{\circ}<\theta<360^{\circ} \), evaluate: \( 3.2 \tan \theta \) \( 3.1 \sin \theta \cos \theta \) \( 3.3 \sin ^{2} \theta+\cos ^{2} \theta \). (4.) If \( 3 \tan \theta-2=0 \) and \( \theta \in\left[90^{\circ} ; 360^{\circ}\right] \), determine, the value of \( \sqrt{13}(\sin \theta-\cos \theta \) (5.) If \( \cos 52^{\circ}=k \) as illustrated in the diagram, determine each of the following i

Let's simplify each of the expressions step by step. ### 8.1 We need to simplify: \[ \frac{\sin(180^{\circ}-x) \cdot \tan(360^{\circ}-x)}{\cos(80^{\circ}-x)} \times \frac{\cos(-180^{\circ}-x)}{\cos(360^{\circ}+x) \sin(360^{\circ}-x)} \] **Step 1: Simplify each trigonometric function.** - \(\sin(180^{\circ}-x) = \sin x\) - \(\tan(360^{\circ}-x) = -\tan x\) - \(\cos(-180^{\circ}-x) = -\cos x\) - \(\cos(360^{\circ}+x) = \cos x\) - \(\sin(360^{\circ}-x) = -\sin x\) **Step 2: Substitute these values into the expression.** \[ \frac{\sin x \cdot (-\tan x)}{\cos(80^{\circ}-x)} \times \frac{-\cos x}{\cos x \cdot (-\sin x)} \] **Step 3: Simplify the expression.** \[ = \frac{-\sin x \cdot (-\tan x)}{\cos(80^{\circ}-x)} \times \frac{-\cos x}{\cos x \cdot (-\sin x)} = \frac{\sin x \cdot \tan x}{\cos(80^{\circ}-x)} \times \frac{\cos x}{\cos x \cdot \sin x} \] \[ = \frac{\tan x}{\cos(80^{\circ}-x)} \] ### 8.2 We need to simplify: \[ \frac{\cos 135^{\circ} \sin 160^{\circ}}{\sin 225^{\circ} \cos 70^{\circ}} \] **Step 1: Simplify each trigonometric function.** - \(\cos 135^{\circ} = -\frac{1}{\sqrt{2}}\) - \(\sin 160^{\circ} = \sin(180^{\circ}-20^{\circ}) = \sin 20^{\circ}\) - \(\sin 225^{\circ} = -\frac{1}{\sqrt{2}}\) - \(\cos 70^{\circ} = \sin 20^{\circ}\) **Step 2: Substitute these values into the expression.** \[ \frac{-\frac{1}{\sqrt{2}} \sin 20^{\circ}}{-\frac{1}{\sqrt{2}} \sin 20^{\circ}} = 1 \] ### 8.3 We need to simplify: \[ \frac{\sin(-\theta) + \cos 120^{\circ} + \tan(-180^{\circ}-\theta)}{\sin^{2} 225^{\circ} - \tan(-\theta) - \cos(90^{\circ}+\theta)} \] **Step 1: Simplify each trigonometric function.** - \(\sin(-\theta) = -\sin \theta\) - \(\cos 120^{\circ} = -\frac{1}{2}\) - \(\tan(-180^{\circ}-\theta) = \tan(-\theta) = -\tan \theta\) - \(\sin^{2} 225^{\circ} = \left(-\frac{1}{\sqrt{2}}\right)^{2} = \frac{1}{2}\) - \(\tan(-\theta) = -\tan \theta\) - \(\cos(90^{\circ}+\theta) = -\sin \theta\) **Step 2: Substitute these values into the expression.** \[ \frac{-\sin \theta - \frac{1}{2} + \tan \theta}{\frac{1}{2} + \tan \theta + \sin \theta} \] ### 8.4 We need to simplify: \[ 4^{x} \frac{\sin 247^{\circ} \cdot \tan 23^{\circ} \cdot \cos 113^{\circ}}{\sin(-157^{\circ})} \] **Step 1: Simplify each trigonometric function.** - \(\sin 247^{\circ} = -\sin 67^{\circ}\) - \(\tan 23^{\circ} = \tan 23^{\circ}\) - \(\cos 113^{\circ} = -\cos 67^{\circ}\) - \(\sin(-157^{\circ}) = -\sin 157^{\circ} = -\sin(180^{\circ}-23^{\circ}) = -\sin 23^{\circ}\) **Step 2: Substitute these values into the expression.** \[ 4^{x} \frac{-\sin 67^{\circ} \cdot \tan 23^{\circ} \cdot (-\cos 67^{\circ})}{-\sin 23^{\circ}} = 4^{x} \frac{\sin 67^{\circ} \cdot \tan 23^{\circ} \cdot \cos 67^{\circ}}{\sin 23^{\circ}} \] ### 8.5 We need to simplify: \[ \frac{3 \cos 150^{\circ} \cdot \sin 270^{\circ}}{\tan(-45^{\circ}) \cdot \cos 600^{\circ}} \] **Step 1: Simplify each trigonometric function.** - \(\cos 150^{\circ} = -\frac{\sqrt{3}}{2}\) - \(\sin 270^{\circ} = -1\) - \(\tan(-45^{\circ}) = -1\) - \(\cos 600^{\circ} = \cos(600^{\circ} - 360^{\circ}) = \cos 240^{\circ} = -\frac{1}{2}\) **Step 2: Substitute these values into the expression.** \[ \frac{3 \cdot -\frac{\sqrt{3}}{2} \cdot -1}{-1 \cdot -\frac{1}{2}} = \frac{3\sqrt{3}}{1} = 3\sqrt{3} \] ### 8.6 We need to simplify: \[ \frac{\tan(180^{\circ}-x) \cdot \sin(90^{\circ}+x)}{\sin(-x)} - \sin y \cdot \cos(90^{\circ}-y) \] **Step 1: Simplify each