SHORT TEST 1.4 Calculate the general Question 1 If $$ \theta $$ is a reztex angle and $$ \tan \theta=-\frac{3}{4} $$ with towing with $$ \quad 2 \sin (3 x+20)=2 \cos x $$ (6) determine the agram ald of a diagram (2) $$ 1.11 \sin \theta $$ (3) $$ 11.2 \cos 2 \theta $$ (3) $$ 13 \cos \left(Q+30^{\circ}\right) $$ $$ \text { Total }=27 \text { marks }+ $$ Simplizy the following to a single trigonometric ratio. $$ =\sin \left(900^{\circ}-x\right) \cdot \cos (-x)-\sin \left(x-180^{\circ}\right) \sin \left(90^{\circ}+x\right) $$ (6) 3 given the identity $$ \frac{\sin 7 x+\sin x}{2 \cos 3 x}=\sin 4 x $$ 1.3.1 Prove the identity (4) 1.3.1 For which values is the identity above undezined. (3)

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Let's break down the problem step by step.

### Question 1

1. **Given Conditions:**
   - $$ \theta $$ is a reference angle.
   - $$ \tan \theta = -\frac{3}{4} $$.
   - The equation $$ 2 \sin(3x + 20) = 2 \cos x $$.

2. **Determine $$ \sin \theta $$ and $$ \cos 2\theta $$:**
   - Since $$ \tan \theta = \frac{\sin \theta}{\cos \theta} $$, we can use the Pythagorean identity to find $$ \sin \theta $$ and $$ \cos \theta $$.
   - Let $$ \sin \theta = -3k $$ and $$ \cos \theta = 4k $$ for some $$ k $$.
   - Then, $$ \sin^2 \theta + \cos^2 \theta = 1 $$ gives us:
     $$
     (-3k)^2 + (4k)^2 = 1 \implies 9k^2 + 16k^2 = 1 \implies 25k^2 = 1 \implies k^2 = \frac{1}{25} \implies k = \frac{1}{5}
     $$
   - Thus, $$ \sin \theta = -\frac{3}{5} $$ and $$ \cos \theta = \frac{4}{5} $$.

3. **Calculate $$ \sin \theta $$:**
   $$
   \sin \theta = -\frac{3}{5}
   $$

4. **Calculate $$ \cos 2\theta $$:**
   - Using the double angle formula $$ \cos 2\theta = 2\cos^2 \theta - 1 $$:
   $$
   \cos 2\theta = 2\left(\frac{4}{5}\right)^2 - 1 = 2 \cdot \frac{16}{25} - 1 = \frac{32}{25} - 1 = \frac{32}{25} - \frac{25}{25} = \frac{7}{25}
   $$

5. **Calculate $$ 13 \cos(Q + 30^\circ) $$:**
   - This requires more information about $$ Q $$. Assuming $$ Q = \theta $$:
   $$
   \cos(Q + 30^\circ) = \cos(\theta + 30^\circ) = \cos \theta \cos 30^\circ - \sin \theta \sin 30^\circ
   $$
   - Using $$ \cos 30^\circ = \frac{\sqrt{3}}{2} $$ and $$ \sin 30^\circ = \frac{1}{2} $$:
   $$
   \cos(\theta + 30^\circ) = \frac{4}{5} \cdot \frac{\sqrt{3}}{2} - \left(-\frac{3}{5}\right) \cdot \frac{1}{2} = \frac{4\sqrt{3}}{10} + \frac{3}{10} = \frac{4\sqrt{3} + 3}{10}
   $$
   - Therefore,
   $$
   13 \cos(Q + 30^\circ) = 13 \cdot \frac{4\sqrt{3} + 3}{10} = \frac{52\sqrt{3} + 39}{10}
   $$

### Simplify the Trigonometric Expression

Now, simplify:
$$
\sin(900^\circ - x) \cdot \cos(-x) - \sin(x - 180^\circ) \sin(90^\circ + x)
$$

1. **Using Trigonometric Identities:**
   - $$ \sin(900^\circ - x) = \sin(90^\circ + 810^\circ - x) = \cos(x) $$
   - $$ \cos(-x) = \cos(x) $$
   - $$ \sin(x - 180^\circ) = -\sin(x) $$
   - $$ \sin(90^\circ + x) = \cos(x) $$

2. **Substituting:**
   $$
   \cos(x) \cdot \cos(x) - (-\sin(x) \cdot \cos(x)) = \cos^2(x) + \sin(x) \cos(x)
   $$

3. **Final Simplification:**
   $$
   = \cos^2(x) + \sin(x) \cos(x) = \cos(x)(\cos(x) + \sin(x))
   $$

### Prove the Identity

Given:
$$
\frac{\sin 7x + \sin x}{2 \cos 3x} = \sin 4x
$$

1. **Using the Sum-to-Product Identities:**
   $$
   \sin 7x + \sin x = 2 \sin\left(\frac{7x + x}{2}\right) \cos\left(\frac{7x - x}{2}\right) = 2 \sin(4x) \cos(3x)
   $$
   - Therefore,
   $$
   \frac{2 \sin(4x) \cos(3x)}{2 \cos(3x)} = \sin(4x)
   $$
   - This proves the identity.

### Values for Which the Identity is Undefined

The identity is undefined when $$ \cos(3x) = 0 $$:
$$
3x = \frac{\pi}{2} + n\pi \implies x = \frac{\pi}{6} + \frac{n\pi}{3}, \quad n \in \mathbb{Z}
$$

### Summary of Answers
1. $$ \sin \theta = -\frac{3}{5} $$
2. $$ \cos 2\theta = \frac{7}{25} $$
3. $$ 13 \cos(Q + 30^\circ) = \frac{52\sqrt{3} + 39}{10} $$
4. Simplified expression: $$ \cos(x)(\cos(x) + \sin(x)) $$
5. Identity proved.
6. Undefined for $$ x = \frac{\pi}{6} + \frac{n\pi}{3} $$.
1. $$ \sin \theta = -\frac{3}{5} $$ 2. $$ \cos 2\theta = \frac{7}{25} $$ 3. $$ 13 \cos(Q + 30^\circ) = \frac{52\sqrt{3} + 39}{10} $$ 4. Simplified expression: $$ \cos(x)(\cos(x) + \sin(x)) $$ 5. The identity is always true. 6. The identity is undefined when $$ x = \frac{\pi}{6} + \frac{n\pi}{3} $$, where $$ n $$ is an integer.

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