Trigonometric Ratios Worksheet - Practice Finding Tangent, Sine, and Cosine for Right Triangles.

To solve these problems, we need to find the indicated trigonometric ratios for each triangle. We'll use the definitions of sine, cosine, and tangent:

- Sine (sin) = opposite / hypotenuse
- Cosine (cos) = adjacent / hypotenuse
- Tangent (tan) = opposite / adjacent

Let's go through each problem step by step.

Problem 1: tan θ

For this triangle:
- Opposite side to θ = 40
- Adjacent side to θ = 9

Using the formula for tangent:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{40}{9} \]

So, \(\tan \theta = \frac{40}{9}\).

Problem 2: sin θ

For this triangle:
- Opposite side to θ = 7.5
- Hypotenuse = 12.5

Using the formula for sine:
\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{7.5}{12.5} \]

Simplify the fraction:
\[ \sin \theta = \frac{7.5 \div 2.5}{12.5 \div 2.5} = \frac{3}{5} \]

So, \(\sin \theta = \frac{3}{5}\).

Problem 3: cos θ

For this triangle:
- Adjacent side to θ = 8
- Hypotenuse = 17

Using the formula for cosine:
\[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{8}{17} \]

So, \(\cos \theta = \frac{8}{17}\).

Problem 4: tan θ

For this triangle:
- Opposite side to θ = 24
- Adjacent side to θ = 10

Using the formula for tangent:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{24}{10} \]

Simplify the fraction:
\[ \tan \theta = \frac{24 \div 2}{10 \div 2} = \frac{12}{5} \]

So, \(\tan \theta = \frac{12}{5}\).

Problem 5: sin θ

For this triangle:
- Opposite side to θ = 6
- Hypotenuse = 10

Using the formula for sine:
\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{6}{10} \]

Simplify the fraction:
\[ \sin \theta = \frac{6 \div 2}{10 \div 2} = \frac{3}{5} \]

So, \(\sin \theta = \frac{3}{5}\).

Problem 6: cos θ

For this triangle:
- Adjacent side to θ = 16
- Hypotenuse = 20

Using the formula for cosine:
\[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{16}{20} \]

Simplify the fraction:
\[ \cos \theta = \frac{16 \div 4}{20 \div 4} = \frac{4}{5} \]

So, \(\cos \theta = \frac{4}{5}\).

Final Answer:
1. \(\tan \theta = \frac{40}{9}\)
2. \(\sin \theta = \frac{3}{5}\)
3. \(\cos \theta = \frac{8}{17}\)
4. \(\tan \theta = \frac{12}{5}\)
5. \(\sin \theta = \frac{3}{5}\)
6. \(\cos \theta = \frac{4}{5}\)