Mathematics CAPS-Grade 11 Question 3 3.1 If $$ \sin \beta=\frac{2 \sqrt{6}}{5} $$ and $$ \beta \in\left(90^{\circ} ; 270^{\circ}\right) $$ Calculate, without the use of a calculator and with the aid of a diagram, the value of: $$ 3.1 .1 \tan \beta $$ $$ 3.1 .2 \sin \left(90^{\circ}+\beta\right) $$ 3.2 Simplify to a single trigonometric function of $$ \theta $$ : $$ \sqrt{1+\tan ^{2} \theta \cdot \cos (-\theta) \cdot \cos \left(180^{\circ}-\theta\right)} \text { if } \theta \in\left(0 ; 90^{\circ}\right) $$ QUESTION 4

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### 3.1 Given $$ \sin \beta = \frac{2 \sqrt{6}}{5} $$ and $$ \beta \in (90^{\circ}, 270^{\circ}) $$

#### 3.1.1 Calculate $$ \tan \beta $$

1. **Identify the quadrant**: Since $$ \beta $$ is in the interval $$ (90^{\circ}, 270^{\circ}) $$, it is in the second or third quadrant. In these quadrants, sine is positive and cosine is negative (second quadrant) or both sine and cosine are negative (third quadrant).

2. **Find $$ \cos \beta $$**: We can use the Pythagorean identity:
   $$
   \sin^2 \beta + \cos^2 \beta = 1
   $$
   Substituting the value of $$ \sin \beta $$:
   $$
   \left(\frac{2 \sqrt{6}}{5}\right)^2 + \cos^2 \beta = 1
   $$
   $$
   \frac{24}{25} + \cos^2 \beta = 1
   $$
   $$
   \cos^2 \beta = 1 - \frac{24}{25} = \frac{1}{25}
   $$
   Therefore, $$ \cos \beta = -\frac{1}{5} $$ (negative because $$ \beta $$ is in the second quadrant).

3. **Calculate $$ \tan \beta $$**:
   $$
   \tan \beta = \frac{\sin \beta}{\cos \beta} = \frac{\frac{2 \sqrt{6}}{5}}{-\frac{1}{5}} = -2 \sqrt{6}
   $$

#### 3.1.2 Calculate $$ \sin(90^{\circ} + \beta) $$

Using the sine addition formula:
$$
\sin(90^{\circ} + \beta) = \cos \beta
$$
From our previous calculation, we found:
$$
\cos \beta = -\frac{1}{5}
$$
Thus,
$$
\sin(90^{\circ} + \beta) = -\frac{1}{5}
$$

### 3.2 Simplify the expression

We need to simplify:
$$
\sqrt{1 + \tan^2 \theta \cdot \cos(-\theta) \cdot \cos(180^{\circ} - \theta)}
$$

1. **Use trigonometric identities**:
   - $$ \cos(-\theta) = \cos \theta $$
   - $$ \cos(180^{\circ} - \theta) = -\cos \theta $$

2. **Substituting these identities**:
   $$
   \sqrt{1 + \tan^2 \theta \cdot \cos \theta \cdot (-\cos \theta)} = \sqrt{1 - \tan^2 \theta \cdot \cos^2 \theta}
   $$

3. **Using the identity $$ \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} $$**:
   $$
   \sqrt{1 - \frac{\sin^2 \theta}{\cos^2 \theta} \cdot \cos^2 \theta} = \sqrt{1 - \sin^2 \theta}
   $$

4. **Using the Pythagorean identity**:
   $$
   \sqrt{\cos^2 \theta} = |\cos \theta|
   $$
   Since $$ \theta \in (0, 90^{\circ}) $$, $$ \cos \theta $$ is positive:
   $$
   |\cos \theta| = \cos \theta
   $$

Thus, the simplified expression is:
$$
\cos \theta
$$

### Summary of Answers
- $$ 3.1.1 \tan \beta = -2 \sqrt{6} $$
- $$ 3.1.2 \sin(90^{\circ} + \beta) = -\frac{1}{5} $$
- $$ 3.2 $$ simplifies to $$ \cos \theta $$
- $$ \tan \beta = -2 \sqrt{6} $$ - $$ \sin(90^{\circ} + \beta) = -\frac{1}{5} $$ - The expression simplifies to $$ \cos \theta $$.

Mathematics
CAPS-Grade 11

Question 3
3.1 If and

Calculate, without the use of a calculator and with the aid of a diagram, the value of:

3.2 Simplify to a single trigonometric function of :

QUESTION 4

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Mathematics CAPS-Grade 11 Question 3 3.1 If and Calculate, without the use of a calculator and with the aid of a diagram, the value of: 3.2 Simplify to a single trigonometric function of : QUESTION 4