The given triangle is a right triangle, and we are tasked with finding trigonometric ratios using the provided side lengths and angles. Here’s how to proceed:

Given information:

• $AB = 8$
• $BC = AC = 4\sqrt{2}$ (as given in part a)

b) $\sin(A)$:

$\sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AB}$

• $BC = 4\sqrt{2}$
• $AB = 8$

So, $\sin(A) = \frac{4\sqrt{2}}{8} = \frac{\sqrt{2}}{2} \approx 0.707$

• Decimal: $0.707$
• Fraction: $\frac{\sqrt{2}}{2}$

c) $\cos(A)$:

$\cos(A) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{AC}{AB}$

Since $AC = 4\sqrt{2}$ and $AB = 8$, this is the same as the sine function: $\cos(A) = \frac{\sqrt{2}}{2} \approx 0.707$

• Decimal: $0.707$
• Fraction: $\frac{\sqrt{2}}{2}$

d) $\tan(B)$:

$\tan(B) = \frac{\text{opposite}}{\text{adjacent}} = \frac{AB}{AC}$

Since $AB = 8$ and $AC = 4\sqrt{2}$: $\tan(B) = \frac{8}{4\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \approx 1.414$

• Decimal: $1.414$
• Fraction: $\sqrt{2}$

Would you like any further explanation on these steps or any other details?

5 related questions:

1. How do we use the Pythagorean theorem to verify side lengths?
2. What is the process of simplifying radical expressions like $\frac{4\sqrt{2}}{8}$?
3. How can we find the tangent of angle $A$ instead of angle $B$?
4. How do we convert a trigonometric ratio to its corresponding angle?
5. What happens to the trigonometric values if the side lengths change?

Tip: Always double-check radical simplifications to ensure proper trigonometric ratios!