We are given a right triangle ABC with sides $a = 10$, $c = 20$, and the right angle at $C$. We are asked to find the unknown side $b$, and then calculate the six trigonometric functions for angle $B$.

Step 1: Use the Pythagorean Theorem to find $b$

The Pythagorean theorem states that in a right triangle:

$a^2 + b^2 = c^2$

Substitute the known values $a = 10$ and $c = 20$:

$10^2 + b^2 = 20^2$ $100 + b^2 = 400$ $b^2 = 400 - 100$ $b^2 = 300$ $b = \sqrt{300} = \sqrt{100 \times 3} = 10\sqrt{3}$

Thus, the length of side $b$ is $10\sqrt{3}$.

Step 2: Find the six trigonometric functions for angle $B$

We now compute the six trigonometric functions for angle $B$. Recall that for a right triangle:

• $\sin(B) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{c}$
• $\cos(B) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{b}{c}$
• $\tan(B) = \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{b}$
• $\csc(B) = \frac{1}{\sin(B)}$
• $\sec(B) = \frac{1}{\cos(B)}$
• $\cot(B) = \frac{1}{\tan(B)}$

Let’s calculate each of these:

1. $\sin(B)$: $\sin(B) = \frac{a}{c} = \frac{10}{20} = \frac{1}{2}$

2. $\cos(B)$: $\cos(B) = \frac{b}{c} = \frac{10\sqrt{3}}{20} = \frac{\sqrt{3}}{2}$

3. $\tan(B)$: $\tan(B) = \frac{a}{b} = \frac{10}{10\sqrt{3}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$

4. $\csc(B)$: $\csc(B) = \frac{1}{\sin(B)} = \frac{1}{\frac{1}{2}} = 2$

5. $\sec(B)$: $\sec(B) = \frac{1}{\cos(B)} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$

6. $\cot(B)$: $\cot(B) = \frac{1}{\tan(B)} = \frac{1}{\frac{\sqrt{3}}{3}} = \frac{3}{\sqrt{3}} = \sqrt{3}$

Final Answers:

• The unknown side $b = 10\sqrt{3}$
• The six trigonometric functions for angle $B$ are:
  1. $\sin(B) = \frac{1}{2}$
  2. $\cos(B) = \frac{\sqrt{3}}{2}$
  3. $\tan(B) = \frac{\sqrt{3}}{3}$
  4. $\csc(B) = 2$
  5. $\sec(B) = \frac{2\sqrt{3}}{3}$
  6. $\cot(B) = \sqrt{3}$

Would you like more details on how these functions are derived or any additional questions about this problem?

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Related Questions:

1. How would the trigonometric functions change if the triangle had different side lengths?
2. How can you verify the Pythagorean theorem with an example of your own choosing?
3. What happens if the triangle is not a right triangle—how do the trigonometric functions change?
4. What is the relationship between the angles in a right triangle and the trigonometric ratios?
5. How can we use trigonometric functions to find unknown angles in a triangle?

Tip:

To quickly remember the basic trigonometric ratios, use the acronym SOH-CAH-TOA:

• Sine = Opposite/Hypotenuse
• Cosine = Adjacent/Hypotenuse
• Tangent = Opposite/Adjacent