A right-angled triangle has the trigonometric ratio tan𝜃 = 3/4. Sketch and label the triangle.

To sketch and label a right-angled triangle where \( \tan(\theta) = \frac{3}{4} \), follow these steps:

### Step 1: Understand the trigonometric ratio.
- The tangent of an angle \( \theta \) in a right-angled triangle is defined as the ratio of the opposite side to the adjacent side:
  \[
  \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
  \]
- In this case, \( \tan(\theta) = \frac{3}{4} \), so the opposite side is 3 units, and the adjacent side is 4 units.

### Step 2: Apply the Pythagorean theorem.
To find the hypotenuse (\( c \)), use the Pythagorean theorem:
\[
c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]

### Step 3: Label the triangle.
- Let one of the angles be \( \theta \).
- The side opposite \( \theta \) is 3 units.
- The side adjacent to \( \theta \) is 4 units.
- The hypotenuse is 5 units.

Now you can sketch a right-angled triangle where:
- The angle \( \theta \) is between the adjacent side (4 units) and the hypotenuse (5 units).
- The side opposite \( \theta \) is 3 units.

### Diagram:
- Draw a right-angled triangle.
- Label the sides: 3 (opposite), 4 (adjacent), and 5 (hypotenuse).
- Mark \( \theta \) at the angle between the adjacent side and the hypotenuse.

Let me know if you'd like me to create this triangle diagram for you, or if you want further clarification.

### Follow-up Questions:
1. How do you calculate sine and cosine from this triangle?
2. Can you find the area of the triangle?
3. What is the exact value of \( \sin(\theta) \) in this triangle?
4. How can you calculate the angle \( \theta \) using inverse tangent?
5. What would happen if you doubled the size of all sides of the triangle?

### Tip:
When solving right-angled triangle problems, always keep the Pythagorean theorem and basic trigonometric identities in mind; they are often key to solving the problem.

Learn how to sketch and label a right-angled triangle where tan(𝜃) = 3/4. This problem demonstrates the application of trigonometric ratios and the Pythagorean theorem to find the sides and hypotenuse. Ideal for students in Grades 8-10.

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