https://file.aiquickdraw.com/mathbot/file/1725899973087rreoemk4.jpg

Let's break down the problem step by step based on the triangle provided in the image.

### Given Information:

- Right triangle \( \triangle ABC \)
- Side lengths:
  - \( BC = 2\sqrt{3} \)
  - \( AC = 2 \)
  
Using the Pythagorean theorem, we can find the length of the hypotenuse \( AB \).

### Step 1: Calculate \( AB \)
By the Pythagorean theorem:
\[
AB^2 = AC^2 + BC^2
\]
\[
AB^2 = 2^2 + (2\sqrt{3})^2 = 4 + 12 = 16
\]
\[
AB = \sqrt{16} = 4
\]

Now that we know all three sides:
- \( AB = 4 \)
- \( BC = 2\sqrt{3} \)
- \( AC = 2 \)

We can use these values to find the trigonometric ratios.

### Step 2: Solve for the trigonometric ratios

#### (a) \( \cos(B) \)
\[
\cos(B) = \frac{\text{adjacent to } B}{\text{hypotenuse}} = \frac{AC}{AB} = \frac{2}{4} = \frac{1}{2}
\]

#### (b) \( \sin(A) \)
\[
\sin(A) = \frac{\text{opposite to } A}{\text{hypotenuse}} = \frac{BC}{AB} = \frac{2\sqrt{3}}{4} = \frac{\sqrt{3}}{2}
\]

#### (c) \( \sin(\text{unknown}) = \frac{1}{2} \)
The sine value of \( \frac{1}{2} \) corresponds to a standard angle of \( 30^\circ \), which means the unknown angle is \( B \), i.e., \( B = 30^\circ \).

#### (d) \( ?(A) = \frac{\sqrt{3}}{2} \)
From standard trigonometric values, \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \), so \( ?(A) \) corresponds to \( \cos(A) \).

#### (e) \( \tan(?) = \sqrt{3} \)
Since \( \tan(60^\circ) = \sqrt{3} \), this corresponds to \( A = 60^\circ \).

### Summary of Answers:

- (a) \( \cos(B) = \frac{1}{2} \)
- (b) \( \sin(A) = \frac{\sqrt{3}}{2} \)
- (c) \( \sin(B) = \frac{1}{2} \)
- (d) \( \cos(A) = \frac{\sqrt{3}}{2} \)
- (e) \( \tan(A) = \sqrt{3} \)

Would you like further details or have any additional questions?

Here are 5 related questions to explore:
1. What are the values of \( \cos(A) \) and \( \sin(B) \) for a right triangle?
2. How do you determine the hypotenuse of a right triangle using the Pythagorean theorem?
3. Why is the sine of angle \( B \) equal to \( \frac{1}{2} \)?
4. What is the significance of the angles 30° and 60° in trigonometry?
5. How do trigonometric ratios help in solving right triangles?

**Tip:** Memorizing the trigonometric ratios of special angles like 30°, 45°, and 60° will help solve problems faster.

Find the desired ratios of the given triangle and fill in the blanks with simplified fractions.

Learn how to solve a right triangle problem by finding cos(B), sin(A), and other trigonometric ratios. The problem includes using the Pythagorean theorem to find the hypotenuse and calculating key trigonometric values.

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