The problem involves calculating the missing side lengths of right-angled triangles. The triangles in the image show an angle and at least one side length. We can solve for the missing sides using trigonometric functions such as sine, cosine, or tangent, depending on the given angle and side.

Let's analyze the three cases:

a) Triangle with an angle of $17^\circ$ and a given side of 5.8 cm

• Angle: $17^\circ$
• Side: adjacent to the angle = 5.8 cm
• Missing side: $b$ (opposite to the angle)

To find $b$ (opposite side), we use the tangent function: $\tan(17^\circ) = \frac{b}{5.8}$ Solving for $b$: $b = 5.8 \cdot \tan(17^\circ)$

b) Triangle with an angle of $62^\circ$ and a given side of 3.9 cm

• Angle: $62^\circ$
• Side: adjacent to the angle = 3.9 cm
• Missing side: $r$ (hypotenuse)

To find $r$, we use the cosine function: $\cos(62^\circ) = \frac{3.9}{r}$ Solving for $r$: $r = \frac{3.9}{\cos(62^\circ)}$

c) Triangle with an angle of $60^\circ$ and a given side of 4.5 cm

• Angle: $60^\circ$
• Side: adjacent to the angle = 4.5 cm
• Missing side: $d$ (hypotenuse)

To find $d$, we use the cosine function: $\cos(60^\circ) = \frac{4.5}{d}$ Solving for $d$: $d = \frac{4.5}{\cos(60^\circ)}$

Now, I'll compute the values of the missing sides.Here are the calculated missing side lengths:

1. Triangle a: The opposite side $b$ is approximately 1.77 cm.
2. Triangle b: The hypotenuse $r$ is approximately 8.31 cm.
3. Triangle c: The hypotenuse $d$ is approximately 9.00 cm.

Let me know if you need further explanation or details.

Here are some related questions:

1. How do we determine which trigonometric function to use?
2. What are the relationships between the sides of a right triangle and its angles?
3. Can the Pythagorean theorem also be used to solve these triangles?
4. How do trigonometric functions change as the angle increases?
5. How do we calculate the other sides once we know one side and the hypotenuse?

Tip: Always use the trigonometric function that relates the known side to the unknown side based on the given angle.