Let's go through the two problems one by one:

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1. Right-Angled Triangle (Finding Angle $x$)

The first image shows a right-angled triangle with two equal sides, each of length 1 cm. The task is to find the size of the angle $x$.

This is a basic trigonometry problem involving a right triangle with two equal sides. The sides of the triangle form an isosceles right triangle, where the two angles other than the right angle are equal.

For an isosceles right triangle, both of the non-right angles are $45^\circ$. Therefore, the angle $x$ in this triangle is:

$x = 45^\circ$

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2. Flagpole and Wire Problem (Finding Height $w$)

The second image involves a flagpole with a wire extending from the top of the pole to the ground. The wire makes an angle of $68^\circ$ with the ground. The length of the wire is $1.3$ meters, and the height of the pole is $w$, which we need to find.

This can be solved using trigonometry. Specifically, we can use the sine function, since we have the opposite side (the height of the flagpole) and the hypotenuse (the length of the wire). The sine function is defined as:

$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

Where:

• $\theta = 68^\circ$
• The hypotenuse is $1.3 \, \text{m}$
• The opposite is $w$, the height of the flagpole.

So,

$\sin(68^\circ) = \frac{w}{1.3}$

To solve for $w$:

$w = 1.3 \times \sin(68^\circ)$

Now, calculating the value:

$w = 1.3 \times 0.9272 = 1.206 \, \text{m}$

Thus, the height of the flagpole is approximately:

$w = 1.2 \, \text{m} \quad (\text{to 1 decimal place})$

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Let me know if you'd like any further explanation or if you have additional questions! Here are some related questions to help reinforce the concepts:

1. How do you solve for angles in other types of right triangles, such as when given only one side and the hypotenuse?
2. What other trigonometric ratios can be used to solve for sides or angles in right triangles (besides sine)?
3. How do you calculate the length of the hypotenuse in a right triangle?
4. Can trigonometry be applied in non-right-angled triangles? If so, how?
5. What are the differences between using sine, cosine, and tangent in trigonometry?

Tip: For calculating trigonometric functions without a calculator, familiarize yourself with basic sine, cosine, and tangent values for common angles (e.g., 30°, 45°, 60°) to make mental math faster.