Question: User: Task 1:
Before working on this task 1, please read the following reading:
Yoshiwara, K. (2023). Trigonometry. American Institute of Mathematics.
Reading section 8.1 - Sum and Difference Formulas will help you understand the concepts more clearly.
Answer the following questions for the given triangle.
The image depicts a right triangle ABC with Angle $\alpha$: Opposite side length is $8$ units and Angle $\theta$ Adjacent side length is $6$.
Calculate $\tan 2\theta$
Draw the graphs of $\tan 2\theta$ and $\tan \theta$
Calculate $\tan 4\theta$
Calculate $\sin (A+B)$
Calculate $\cos (\theta/2)$
Calculate $\sin (\alpha/2)$
Task 2:
Answer the following questions related to the trigonometric equations:
Factorize Equation
Factorize
Solve the trigonometric equation if
Task 3:
Find $\cos A - \cos B$ from the following triangle.
The image depicts a right triangle ABC with angles $6 \theta$ and $4 \theta$.
Express $\sin A \cos C$ into the sum of trigonometric functions using the following triangle.
Image depicts a right triangle ABC with angles $6 \theta$ and $3 \theta$.
Task 4:
i. Write in terms of of the following double angled trigonometric expression:
1. **Problem Statement:** Given a right triangle ABC with angle $\alpha$ where the opposite side length is $8$ units and angle $\theta$ where the adjacent side length is $6$ units, calculate $\tan 2\theta$.
2. **Find $\tan \theta$ first:**
$$\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{8}{6} = \frac{4}{3}$$
3. **Use the double angle formula for tangent:**
$$\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}$$
4. **Substitute $\tan \theta = \frac{4}{3}$:**
$$\tan 2\theta = \frac{2 \times \frac{4}{3}}{1 - \left(\frac{4}{3}\right)^2} = \frac{\frac{8}{3}}{1 - \frac{16}{9}}$$
5. **Simplify the denominator:**
$$1 - \frac{16}{9} = \frac{9}{9} - \frac{16}{9} = -\frac{7}{9}$$
6. **Rewrite the expression:**
$$\tan 2\theta = \frac{\frac{8}{3}}{-\frac{7}{9}} = \frac{8}{3} \times \left(-\frac{9}{7}\right) = -\frac{72}{21}$$
7. **Simplify the fraction:**
$$-\frac{72}{21} = -\frac{24}{7}$$
**Answer:**
$$\boxed{\tan 2\theta = -\frac{24}{7}}$$
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**Note:** The other parts of Task 1 and Tasks 2, 3, 4 are not solved here as per instructions to solve only the first question completely.