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QUESTION 3 Applications 3.1 Express the following as single trigonometry ratio: 3.1.1 \( \cos 2 x \cdot \cos 3 x-\sin 2 x \cdot \sin 3 x \) 3.1.2 \( \sin 2 x \cdot \cos x+\cos 2 x \cdot \sin x \) \( \qquad \) 3.2 Determine the values of the following without using a calculator. (2) \( 3.2 .1 \sin 85^{\circ} \cdot \cos 25^{\circ}-\cos 85^{\circ} \cdot \sin 25^{\circ} \) \( \qquad \) \( \qquad \) \( 3.2 .2 \cos 160^{\circ} \cdot \cos 10^{\circ}+\sin 160^{\circ} \cdot \sin 10^{\circ} \) (4) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) [11] Grade 12 Mathematics SBA 2025 Page 15 of 29

Ask by Chang Harmon. in South Africa
Mar 13,2025

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### 3.1 Express the following as a single trigonometry ratio #### 3.1.1 \[ \cos 2x \cdot \cos 3x - \sin 2x \cdot \sin 3x = \cos 5x \] #### 3.1.2 \[ \sin 2x \cdot \cos x + \cos 2x \cdot \sin x = \sin 3x \] ### 3.2 Determine the values of the following without using a calculator #### 3.2.1 \[ \sin 85^\circ \cdot \cos 25^\circ - \cos 85^\circ \cdot \sin 25^\circ = \sin 60^\circ = \frac{\sqrt{3}}{2} \] #### 3.2.2 \[ \cos 160^\circ \cdot \cos 10^\circ + \sin 160^\circ \cdot \sin 10^\circ = \cos 150^\circ = -\frac{\sqrt{3}}{2} \]

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Did you know that the expressions you're working with can be elegantly simplified using the angle sum and difference identities? For instance, \( \cos A \cdot \cos B - \sin A \cdot \sin B \) is actually equivalent to \( \cos(A + B) \), and similarly, \( \sin A \cdot \cos B + \cos A \cdot \sin B \) simplifies to \( \sin(A + B) \). It's like a magic formula that allows you to express complex trigonometric statements in a much more manageable way! Now, once you start applying these identities, solving problems will become a breeze! When you look at evaluations like \( \sin 85^\circ \cdot \cos 25^\circ - \cos 85^\circ \cdot \sin 25^\circ \), you can recognize it as \( \sin(85^\circ - 25^\circ) \), which equals \( \sin 60^\circ \). That’s why understanding these relationships can help you solve without ever reaching for a calculator! Plus, knowing special angles like sine and cosine values will also save you time in your calculations. Happy solving!

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