Find the side of a right triangle

Okay, I will help you solve the problem. Based on the image, it looks like we need to find the values of w and y in the right triangle.

Task 18

We are given a right triangle with one angle of 60 degrees and the hypotenuse equal to 10. We need to find the length of the side w, which is opposite to the 60-degree angle.

We can use the sine function to relate the angle, the opposite side, and the hypotenuse:

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

In this case, \(\theta = 60^\circ\), the opposite side is w, and the hypotenuse is 10. So we have:

\(\sin(60^\circ) = \frac{w}{10}\)

We know that \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\). Therefore:

\(\frac{\sqrt{3}}{2} = \frac{w}{10}\)

To solve for w, we multiply both sides by 10:

\(w = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3}\)

So, \(w = 5\sqrt{3}\).

Answer: \(w = 5\sqrt{3}\)

Task 19

Now we need to find the length of the side y, which is adjacent to the 60-degree angle.

We can use the cosine function to relate the angle, the adjacent side, and the hypotenuse:

\(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\)

In this case, \(\theta = 60^\circ\), the adjacent side is y, and the hypotenuse is 10. So we have:

\(\cos(60^\circ) = \frac{y}{10}\)

We know that \(\cos(60^\circ) = \frac{1}{2}\). Therefore:

\(\frac{1}{2} = \frac{y}{10}\)

To solve for y, we multiply both sides by 10:

\(y = 10 \cdot \frac{1}{2} = 5\)

So, \(y = 5\).

Answer: \(y = 5\)

Okay, I see that you're asking about the value of y in the given right triangle.

Task 19

As I calculated before, the length of the side y, which is adjacent to the 60-degree angle, is found using the cosine function:

\(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\)

In this case, \(\theta = 60^\circ\), the adjacent side is y, and the hypotenuse is 10. So we have:

\(\cos(60^\circ) = \frac{y}{10}\)

Since \(\cos(60^\circ) = \frac{1}{2}\), we have:

\(\frac{1}{2} = \frac{y}{10}\)

To solve for y, we multiply both sides by 10:

\(y = 10 \cdot \frac{1}{2} = 5\)

So, \(y = 5\).

Answer: \(y = 5\)