https://file.aiquickdraw.com/mathbot/file/1728611490326ke8ln0ok.jpg

Let's work through some of these problems together. I'll start by solving a few of the questions from the image, and then you can confirm if you'd like help with the rest or specific ones.

### Question 3:
\[
\text{Jika } \tan \theta = \frac{a}{5}, \text{ maka } \frac{x}{\sqrt{a^2 + x^2}} \text{ sama dengan?}
\]

We know that:
\[
\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{5}
\]
Thus, \(\sin \theta\) is given by:
\[
\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{\sqrt{a^2 + 5^2}} = \frac{a}{\sqrt{a^2 + 25}}
\]

The expression given, \(\frac{x}{\sqrt{a^2 + x^2}}\), represents a sine function, so the answer is:
\[
\boxed{A} \text{ (sin } \theta\text{)}
\]

### Question 4:
\[
\text{Jika cotan } \theta = \frac{7}{3} \text{ dan } 0 < \theta < \frac{\pi}{2}, \text{ nilai dari } \frac{(1 + \sin \theta)(1 - \sin \theta)}{(1 + \cos \theta)(1 - \cos \theta)} \text{ adalah?}
\]

Since \(\cot \theta = \frac{7}{3}\), then \(\tan \theta = \frac{3}{7}\). We can use this to find \(\sin \theta\) and \(\cos \theta\):
\[
\sin \theta = \frac{3}{\sqrt{3^2 + 7^2}} = \frac{3}{\sqrt{58}}, \quad \cos \theta = \frac{7}{\sqrt{58}}
\]

Now calculate:
\[
(1 + \sin \theta)(1 - \sin \theta) = 1 - \sin^2 \theta = \cos^2 \theta
\]
\[
(1 + \cos \theta)(1 - \cos \theta) = 1 - \cos^2 \theta = \sin^2 \theta
\]

Thus, the expression becomes:
\[
\frac{\cos^2 \theta}{\sin^2 \theta} = \cot^2 \theta = \left( \frac{7}{3} \right)^2 = \frac{49}{9}
\]

So the answer is:
\[
\boxed{D} \, 7/3
\]

### Question 5:
\[
\text{Jika } 3 \sin \theta + 4 \cos \theta = 5, \text{ maka nilai dari } \sin \theta \text{ adalah?}
\]

This equation represents a standard form of \(R\sin(\theta + \alpha)\), where:
\[
R = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5
\]
So, the equation becomes \(5 \sin(\theta + \alpha) = 5\), which gives:
\[
\sin(\theta + \alpha) = 1
\]
Thus, \(\theta + \alpha = \frac{\pi}{2}\), and \(\sin \theta = \frac{3}{5}\).

Therefore, the answer is:
\[
\boxed{A} \, 0.5
\]

Would you like to proceed with more solutions or explanations for any other specific question?

### Follow-up Questions:
1. How would you solve a similar trigonometric identity involving \(\cot \theta\)?
2. Can you describe how to find the angle \(\theta\) from a given sine or cosine equation?
3. What are the main trigonometric identities involved in simplifying complex expressions like in Question 4?
4. How can we express a function like \(3\sin \theta + 4\cos \theta\) in a more general form?
5. How do you calculate the hypotenuse and the angles when working with a right-angled triangle in trigonometry?

#### Tip:
Always remember to check for valid angle ranges when dealing with trigonometric identities, as certain values might restrict the possible solutions.

Jawablah soal trigonometri seperti pada gambar.

Explore detailed solutions to trigonometric problems covering identities, tangent, sine, cosine, and cotangent functions. Suitable for Grade 11 students, these exercises provide practice with key formulas and theorems.

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