Answer key for a trigonometry worksheet on six trigonometric ratios.

Problem: Find the exact values of six trigonometric ratios for the mentioned angles.

The six trigonometric ratios are:
1. Sine (sin): Opposite / Hypotenuse
2. Cosine (cos): Adjacent / Hypotenuse
3. Tangent (tan): Opposite / Adjacent
4. Cosecant (csc): Hypotenuse / Opposite
5. Secant (sec): Hypotenuse / Adjacent
6. Cotangent (cot): Adjacent / Opposite

We will solve each part step by step.

---

#### 1) ∠X

Given:
- Triangle XYZ with a right angle at Z.
- Side lengths: \( XY = 10 \) (hypotenuse), \( XZ = 8 \) (adjacent to ∠X), and \( YZ = 6 \) (opposite to ∠X).

Step 1: Verify the side lengths using the Pythagorean theorem.
\[
XY^2 = XZ^2 + YZ^2
\]
\[
10^2 = 8^2 + 6^2
\]
\[
100 = 64 + 36
\]
\[
100 = 100 \quad \text{(True)}
\]

Step 2: Calculate the trigonometric ratios.

- sin X: Opposite / Hypotenuse
\[
\sin X = \frac{YZ}{XY} = \frac{6}{10} = \frac{3}{5}
\]

- cos X: Adjacent / Hypotenuse
\[
\cos X = \frac{XZ}{XY} = \frac{8}{10} = \frac{4}{5}
\]

- tan X: Opposite / Adjacent
\[
\tan X = \frac{YZ}{XZ} = \frac{6}{8} = \frac{3}{4}
\]

- csc X: Hypotenuse / Opposite
\[
\csc X = \frac{XY}{YZ} = \frac{10}{6} = \frac{5}{3}
\]

- sec X: Hypotenuse / Adjacent
\[
\sec X = \frac{XY}{XZ} = \frac{10}{8} = \frac{5}{4}
\]

- cot X: Adjacent / Opposite
\[
\cot X = \frac{XZ}{YZ} = \frac{8}{6} = \frac{4}{3}
\]

Final Answer for ∠X:
\[
\boxed{
\begin{aligned}
\sin X &= \frac{3}{5}, & \csc X &= \frac{5}{3}, \\
\cos X &= \frac{4}{5}, & \sec X &= \frac{5}{4}, \\
\tan X &= \frac{3}{4}, & \cot X &= \frac{4}{3}
\end{aligned}
}
\]

---

#### 2) ∠Q

Given:
- Triangle PQR with a right angle at P.
- Side lengths: \( QR = 24 \) (hypotenuse), \( PR = 16\sqrt{2} \) (adjacent to ∠Q), and \( PQ = 8\sqrt{2} \) (opposite to ∠Q).

Step 1: Verify the side lengths using the Pythagorean theorem.
\[
QR^2 = PQ^2 + PR^2
\]
\[
24^2 = (8\sqrt{2})^2 + (16\sqrt{2})^2
\]
\[
576 = 64 \cdot 2 + 256 \cdot 2
\]
\[
576 = 128 + 512
\]
\[
576 = 576 \quad \text{(True)}
\]

Step 2: Calculate the trigonometric ratios.

- sin Q: Opposite / Hypotenuse
\[
\sin Q = \frac{PQ}{QR} = \frac{8\sqrt{2}}{24} = \frac{\sqrt{2}}{3}
\]

- cos Q: Adjacent / Hypotenuse
\[
\cos Q = \frac{PR}{QR} = \frac{16\sqrt{2}}{24} = \frac{2\sqrt{2}}{3}
\]

- tan Q: Opposite / Adjacent
\[
\tan Q = \frac{PQ}{PR} = \frac{8\sqrt{2}}{16\sqrt{2}} = \frac{1}{2}
\]

- csc Q: Hypotenuse / Opposite
\[
\csc Q = \frac{QR}{PQ} = \frac{24}{8\sqrt{2}} = \frac{3}{\sqrt{2}} = \frac{3\sqrt{2}}{2}
\]

- sec Q: Hypotenuse / Adjacent
\[
\sec Q = \frac{QR}{PR} = \frac{24}{16\sqrt{2}} = \frac{3}{2\sqrt{2}} = \frac{3\sqrt{2}}{4}
\]

- cot Q: Adjacent / Opposite
\[
\cot Q = \frac{PR}{PQ} = \frac{16\sqrt{2}}{8\sqrt{2}} = 2
\]

Final Answer for ∠Q:
\[
\boxed{
\begin{aligned}
\sin Q &= \frac{\sqrt{2}}{3}, & \csc Q &= \frac{3\sqrt{2}}{2}, \\
\cos Q &= \frac{2\sqrt{2}}{3}, & \sec Q &= \frac{3\sqrt{2}}{4}, \\
\tan Q &= \frac{1}{2}, & \cot Q &= 2
\end{aligned}
}
\]

---

#### 3) ∠K

Given:
- Triangle KLM with a right angle at L.
- Side lengths: \( KM = 29 \) (hypotenuse), \( KL = 20 \) (adjacent to ∠K), and \( LM = 21 \) (opposite to ∠K).

Step 1: Verify the side lengths using the Pythagorean theorem.
\[
KM^2 = KL^2 + LM^2
\]
\[
29^2 = 20^2 + 21^2
\]
\[
841 = 400 + 441
\]
\[
841 = 841 \quad \text{(True)}
\]

Step 2: Calculate the trigonometric ratios.

- sin K: Opposite / Hypotenuse
\[
\sin K = \frac{LM}{KM} = \frac{21}{29}
\]

- cos K: Adjacent / Hypotenuse
\[
\cos K = \frac{KL}{KM} = \frac{20}{29}
\]

- tan K: Opposite / Adjacent
\[
\tan K = \frac{LM}{KL} = \frac{21}{20}
\]

- csc K: Hypotenuse / Opposite
\[
\csc K = \frac{KM}{LM} = \frac{29}{21}
\]

- sec K: Hypotenuse / Adjacent
\[
\sec K = \frac{KM}{KL} = \frac{29}{20}
\]

- cot K: Adjacent / Opposite
\[
\cot K = \frac{KL}{LM} = \frac{20}{21}
\]

Final Answer for ∠K:
\[
\boxed{
\begin{aligned}
\sin K &= \frac{21}{29}, & \csc K &= \frac{29}{21}, \\
\cos K &= \frac{20}{29}, & \sec K &= \frac{29}{20}, \\
\tan K &= \frac{21}{20}, & \cot K &= \frac{20}{21}
\end{aligned}
}
\]

---

Final Answers:

1. ∠X:
\[
\boxed{
\begin{aligned}
\sin X &= \frac{3}{5}, & \csc X &= \frac{5}{3}, \\
\cos X &= \frac{4}{5}, & \sec X &= \frac{5}{4}, \\
\tan X &= \frac{3}{4}, & \cot X &= \frac{4}{3}
\end{aligned}
}
\]

2. ∠Q:
\[
\boxed{
\begin{aligned}
\sin Q &= \frac{\sqrt{2}}{3}, & \csc Q &= \frac{3\sqrt{2}}{2}, \\
\cos Q &= \frac{2\sqrt{2}}{3}, & \sec Q &= \frac{3\sqrt{2}}{4}, \\
\tan Q &= \frac{1}{2}, & \cot Q &= 2
\end{aligned}
}
\]

3. ∠K:
\[
\boxed{
\begin{aligned}
\sin K &= \frac{21}{29}, & \csc K &= \frac{29}{21}, \\
\cos K &= \frac{20}{29}, & \sec K &= \frac{29}{20}, \\
\tan K &= \frac{21}{20}, & \cot K &= \frac{20}{21}
\end{aligned}
}
\]