Trigonometry practice worksheet featuring six right triangles to find trigonometric ratios and two calculator-based problems.

Let's solve each problem step by step using trigonometric ratios:

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🔷 Trigonometric Ratios Recap:

For a right triangle with angle $ \theta $:
- $ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} $
- $ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} $
- $ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} $

We will use this to find the required values.

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## ✔ Problem 1: Triangle ABC

Given:
- Right angle at B
- AB = 14 (vertical leg)
- BC = 48 (horizontal leg)
- AC = 50 (hypotenuse)

Find: $ \sin C $

Angle C is at vertex C. So:
- Opposite side to ∠C = AB = 14
- Hypotenuse = AC = 50

$$
\sin C = \frac{14}{50} = \frac{7}{25} = 0.28
$$

✔ Answer: $ \boxed{0.28} $

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## ✔ Problem 2: Triangle XYZ

Right angle at X
- ZX = 32 (vertical leg)
- XY = 24 (horizontal leg)
- YZ = 40 (hypotenuse)

Find: $ \sin Z $

Angle Z is at vertex Z.
- Opposite side to ∠Z = XY = 24
- Hypotenuse = YZ = 40

$$
\sin Z = \frac{24}{40} = \frac{3}{5} = 0.6
$$

✔ Answer: $ \boxed{0.6} $

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## ✔ Problem 3: Triangle ABC

Right angle at B
- AB = 20
- BC = 21
- AC = 29

Find: $ \tan A $

Angle A is at vertex A.
- Opposite side to ∠A = BC = 21
- Adjacent side = AB = 20

$$
\tan A = \frac{21}{20} = 1.05
$$

✔ Answer: $ \boxed{1.05} $

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## ✔ Problem 4: Triangle ABC

Right angle at A
- AB = 40
- AC = 30
- BC = 50

Find: $ \tan B $

Angle B is at vertex B.
- Opposite side to ∠B = AC = 30
- Adjacent side = AB = 40

$$
\tan B = \frac{30}{40} = \frac{3}{4} = 0.75
$$

✔ Answer: $ \boxed{0.75} $

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## ✔ Problem 5: Triangle XYZ

Right angle at Y
- XY = 21
- YZ = 20
- XZ = 29

Find: $ \cos X $

Angle X is at vertex X.
- Adjacent side to ∠X = XY = 21
- Hypotenuse = XZ = 29

$$
\cos X = \frac{21}{29} \approx 0.7241
$$

✔ Answer: $ \boxed{0.7241} $

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## ✔ Problem 6: Triangle ABC

Right angle at B
- AB = 9
- BC = 12
- AC = 15

Find: $ \cos C $

Angle C is at vertex C.
- Adjacent side to ∠C = BC = 12
- Hypotenuse = AC = 15

$$
\cos C = \frac{12}{15} = \frac{4}{5} = 0.8
$$

✔ Answer: $ \boxed{0.8} $

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## ✔ Problem 7: Use calculator

Find: $ \sin 77^\circ $

Using calculator:
$$
\sin 77^\circ \approx 0.9744
$$

✔ Answer: $ \boxed{0.9744} $

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## ✔ Problem 8: Use calculator

Find: $ \tan 17^\circ $

Using calculator:
$$
\tan 17^\circ \approx 0.3057
$$

✔ Answer: $ \boxed{0.3057} $

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✔ Final Answers Summary:

| Problem | Answer |
|--------|----------------|
| 1 | $ \sin C = 0.28 $ |
| 2 | $ \sin Z = 0.6 $ |
| 3 | $ \tan A = 1.05 $ |
| 4 | $ \tan B = 0.75 $ |
| 5 | $ \cos X = 0.7241 $ |
| 6 | $ \cos C = 0.8 $ |
| 7 | $ \sin 77^\circ = 0.9744 $ |
| 8 | $ \tan 17^\circ = 0.3057 $ |

All answers are rounded to the nearest ten-thousandth where applicable.

Let me know if you'd like a visual explanation or help understanding any step!