Let's break down and solve the problem step by step:

1. Finding the Missing Side of the Triangle:

We are given a right triangle where:

• $a = 20$ (one leg),
• $b = 21$ (the other leg).

We need to find the hypotenuse $c$ using the Pythagorean Theorem: $c^2 = a^2 + b^2$ Substitute the values: $c^2 = 20^2 + 21^2 = 400 + 441 = 841$ Taking the square root: $c = \sqrt{841} = 29$ Thus, the hypotenuse $c = 29$.

2. Finding the Six Trigonometric Functions:

For the right triangle, the trigonometric ratios based on angle $\theta$ are:

• sin(θ) = opposite / hypotenuse = $\frac{a}{c} = \frac{20}{29}$
• cos(θ) = adjacent / hypotenuse = $\frac{b}{c} = \frac{21}{29}$
• tan(θ) = opposite / adjacent = $\frac{a}{b} = \frac{20}{21}$

The reciprocal functions are:

• csc(θ) = 1/sin(θ) = $\frac{29}{20}$
• sec(θ) = 1/cos(θ) = $\frac{29}{21}$
• cot(θ) = 1/tan(θ) = $\frac{21}{20}$

Summary of the Trigonometric Ratios:

1. sin(θ) = $\frac{20}{29}$
2. cos(θ) = $\frac{21}{29}$
3. tan(θ) = $\frac{20}{21}$
4. csc(θ) = $\frac{29}{20}$
5. sec(θ) = $\frac{29}{21}$
6. cot(θ) = $\frac{21}{20}$

Let me know if you'd like more details on any of the steps or concepts!

Follow-up Questions:

1. How is the Pythagorean Theorem used in right triangles?
2. Why is sin(θ) calculated as opposite over hypotenuse?
3. Can you derive trigonometric ratios for other angles in this triangle?
4. How do the reciprocal trigonometric functions relate to their primary functions?
5. What would happen if the values of $a$ and $b$ were switched?

Tip: Always check that the hypotenuse is the longest side before calculating trigonometric ratios!