Question
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
Ask by Mckenzie Rose. in South Africa
Feb 22,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
### Simplified Solutions:
1. **8.1**
\[
\frac{\sin(180^{\circ}-x) \cdot \tan(360^{\circ}-x)}{\cos(80^{\circ}-x)} \times \frac{\cos(-180^{\circ}-x)}{\cos(360^{\circ}+x) \sin(360^{\circ}-x)} = \frac{\tan x}{\cos(80^{\circ}-x)}
\]
2. **8.2**
\[
\frac{\cos 135^{\circ} \sin 160^{\circ}}{\sin 225^{\circ} \cos 70^{\circ}} = 1
\]
3. **8.3**
\[
\frac{\sin(-\theta) + \cos 120^{\circ} + \tan(-180^{\circ}-\theta)}{\sin^{2} 225^{\circ} - \tan(-\theta) - \cos(90^{\circ}+\theta)} = \frac{-\sin \theta - \frac{1}{2} + \tan \theta}{\frac{1}{2} + \tan \theta + \sin \theta}
\]
4. **8.4**
\[
4^{x} \frac{\sin 247^{\circ} \cdot \tan 23^{\circ} \cdot \cos 113^{\circ}}{\sin(-157^{\circ})} = 4^{x} \frac{\sin 67^{\circ} \cdot \tan 23^{\circ} \cdot \cos 67^{\circ}}{\sin 23^{\circ}}
\]
5. **8.5**
\[
\frac{3 \cos 150^{\circ} \cdot \sin 270^{\circ}}{\tan(-45^{\circ}) \cdot \cos 600^{\circ}} = 3\sqrt{3}
\]
6. **8.6**
\[
\frac{\tan(180^{\circ}-x) \cdot \sin(90^{\circ}+x)}{\sin(-x)} - \sin y \cdot \cos(90^{\circ}-y) = \frac{-\tan x \cdot \cos x}{-\sin x} - \sin y \cdot \sin y = \cot x \cdot \cos x + \sin^2 y
\]
7. **8.7**
\[
\frac{\tan 30^{\circ} \cdot \sin 60^{\circ} \cdot \cos 25^{\circ}}{\cos 135^{\circ} \cdot \sin(-45^{\circ}) \cdot \sin 65^{\circ}} = \frac{\frac{\sqrt{3}}{3} \cdot \frac{\sqrt{3}}{2} \cdot \cos 25^{\circ}}{-\frac{1}{\sqrt{2}} \cdot -\frac{1}{\sqrt{2}} \cdot \sin 65^{\circ}} = \frac{\frac{1}{2} \cdot \cos 25^{\circ}}{\frac{1}{2} \cdot \sin 65^{\circ}} = \frac{\cos 25^{\circ}}{\sin 65^{\circ}}
\]
8. **8.8**
\[
\frac{\tan(180^{\circ}-x) \cdot \sin(90^{\circ}-x)}{\cos(90^{\circ}+x)} - \frac{\cos(180^{\circ}-x)}{\sin(90^{\circ}+x)} = \frac{-\tan x \cdot \cos x}{-\sin x} - \frac{-\cos x}{\sin x} = \cot x \cdot \cos x + \cot x = \cot x (\cos x + 1)
\]
9. **8.9**
\[
\frac{\sin 189^{\circ}}{\tan 549^{\circ}} - \frac
Solution
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Beyond the Answer
To enhance your understanding of these trigonometric problems, let’s dive into some exciting insights! Trigonometry isn't just an abstract concept. Its principles are deeply embedded in real-life scenarios, such as architecture, engineering, and even game development. Think about it—when constructing tall buildings or designing video game graphics, sine, cosine, and tangent help ensure that everything fits together perfectly and looks stunning on screen! When tackling trigonometric equations, one common mistake is neglecting the signs of trigonometric functions in different quadrants. Remember, sine is positive in the first and second quadrants, while cosine is positive in the first and fourth. Don’t forget to sketch the unit circle or a quadrant diagram—this will keep you visually oriented and clear-headed as you solve your problems!
