To solve for the length of $b$ in this right triangle using trigonometric ratios, we can apply the SOHCAHTOA mnemonic, which helps us remember the relationships between the sides of a right triangle and the angles.

Here, we are given:

• The angle $74^\circ$
• The adjacent side to the angle $74^\circ$ (which is 10 cm)
• The side we want to find is the opposite side, labeled $b$

Since we are dealing with the opposite side and the adjacent side, we use the tangent function, according to SOHCAHTOA:

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

Substituting the values:

$\tan(74^\circ) = \frac{b}{10}$

Now, solving for $b$:

$b = 10 \times \tan(74^\circ)$

Using a calculator to find $\tan(74^\circ)$:

$\tan(74^\circ) \approx 3.487$

Now, multiply:

$b \approx 10 \times 3.487 = 34.87 \, \text{cm}$

Thus, the length of $b$ is approximately 34.87 cm.

Would you like more details on this solution, or do you have any questions?

Here are 5 related questions you could explore:

1. How do we solve for the hypotenuse using trigonometric ratios?
2. What is the sine of a right triangle, and when is it useful?
3. How would we solve this triangle if the angle given was less than $45^\circ$?
4. How does the cosine function relate to the sides of a triangle?
5. How can you solve for an angle when given two sides of a triangle?

Tip: When solving for unknown sides using trigonometry, always ensure your calculator is set to the correct mode (degrees or radians), depending on the angle units.