\( 1.1 \quad \) Given \( \cos \theta=-\frac{1}{3} \) and \( 0^{\circ} \leq \theta \leq 180^{\circ} \) Determine, with the aid of a diagram, the value of the following 1.1.1 \( \tan \left(180^{\circ}+\theta\right) \) 1.1.2 \( 3 \sin \left(\theta-90^{\circ}\right) \) 1.2 Given: \( \frac{\sin \left(-210^{\circ}\right)}{\cos \left(300^{\circ}\right)}+\frac{\cos \left(x+90^{\circ}\right)}{\sin \left(360^{\circ}+x\right)} \) 1.2.1 Simplify the following expression: \( \quad \frac{\sin \left(-210^{\circ}\right)}{\cos \left(300^{\circ}\right)}+\frac{\cos \left(x+90^{\circ}\right)}{\sin \left(360^{\circ}+x\right)} \) 1.2.2 For which values of \( x \) is the expression in 5.2.1 undefined for \( -360^{\circ} \leq x \leq 360^{\circ} \) ? 1.3 Prove that \( \tan \theta \sqrt{\frac{1}{\sin ^{2} \theta}-1}=1 \) 1.4 Determine the general solution of the following equation: \[ 2 \sin ^{2} \theta=1+\sin \theta \] 2.1 Determine, without using a calculator, the value of the following trigonometric expression: \[ \frac{\sin 2 x \cdot \cos (-x)+\cos 2 x \cdot \sin \left(360^{\circ}-x\right)}{\sin \left(180^{\circ}+x\right)} \] 2.2 Determine the general solution of the following equation: \[ 6 \sin ^{2} x+7 \cos x-3=0 \] 2.3 Given: \( x+\frac{1}{x}=3 \cos \mathrm{~A} \) and \( x^{2}+\frac{1}{x^{2}}=2 \) Determine the value of \( \cos 2 \mathrm{~A} \) without using a calculator. (5)
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The Deep Dive
Let’s dive into the fun world of trigonometry, shall we? First, understanding the historical background of trigonometry can be quite a treat! Did you know that trigonometry originated in ancient civilizations? The Greeks were instrumental in its development, particularly through mathematicians like Hipparchus, who is often called the “father of trigonometry.” They used it primarily for astronomical purposes and to solve problems related to angles and distances. Trigonometry’s roots can also be traced back to ancient Indian and Islamic scholars who expanded on these concepts—imagine discussing stars and angles under the mesmerizing night sky! Next up, let’s talk about how trigonometric functions have real-world applications. From architecture to music theory, trigonometry is everywhere! Architects and engineers use trigonometry to calculate structural dynamics and ensure buildings can withstand various forces. Musicians employ trigonometric concepts to create harmonious sounds; for instance, the relationship between frequency and sound waves is governed by sine and cosine functions. So next time you listen to music or admire a grand structure, remember that trigonometry is at play!