Trigonometric Ratios Worksheet - Practice finding sine, cosine, and tangent using triangle sides.

To solve the trigonometric ratios using the given segments in the images, we need to identify the sides of the right triangles and apply the definitions of sine, cosine, tangent, secant, cosecant, and cotangent. Let's go through each problem step by step.

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Problem 1:

#### Image:
- A right triangle with labeled sides.
- Opposite side = 3, Adjacent side = 4, Hypotenuse = 5.

#### Task: Find $\sin \theta$, $\cos \theta$, $\tan \theta$.

#### Solution:
1. Sine ($\sin \theta$):
\[
\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{3}{5}
\]

2. Cosine ($\cos \theta$):
\[
\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{4}{5}
\]

3. Tangent ($\tan \theta$):
\[
\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{3}{4}
\]

#### Final Answer:
\[
\boxed{\sin \theta = \frac{3}{5}, \cos \theta = \frac{4}{5}, \tan \theta = \frac{3}{4}}
\]

---

Problem 2:

#### Image:
- A right triangle with labeled sides.
- Opposite side = 5, Adjacent side = 12, Hypotenuse = 13.

#### Task: Find $\sec \theta$, $\csc \theta$, $\cot \theta$.

#### Solution:
1. Secant ($\sec \theta$):
\[
\sec \theta = \frac{\text{Hypotenuse}}{\text{Adjacent}} = \frac{13}{12}
\]

2. Cosecant ($\csc \theta$):
\[
\csc \theta = \frac{\text{Hypotenuse}}{\text{Opposite}} = \frac{13}{5}
\]

3. Cotangent ($\cot \theta$):
\[
\cot \theta = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{12}{5}
\]

#### Final Answer:
\[
\boxed{\sec \theta = \frac{13}{12}, \csc \theta = \frac{13}{5}, \cot \theta = \frac{12}{5}}
\]

---

Problem 3:

#### Image:
- A right triangle with labeled sides.
- Opposite side = 8, Adjacent side = 15, Hypotenuse = 17.

#### Task: Find $\cos \theta$, $\tan \theta$, $\csc \theta$.

#### Solution:
1. Cosine ($\cos \theta$):
\[
\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{15}{17}
\]

2. Tangent ($\tan \theta$):
\[
\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{8}{15}
\]

3. Cosecant ($\csc \theta$):
\[
\csc \theta = \frac{\text{Hypotenuse}}{\text{Opposite}} = \frac{17}{8}
\]

#### Final Answer:
\[
\boxed{\cos \theta = \frac{15}{17}, \tan \theta = \frac{8}{15}, \csc \theta = \frac{17}{8}}
\]

---

Problem 4:

#### Image:
- A right triangle with labeled sides.
- Opposite side = 7, Adjacent side = 24, Hypotenuse = 25.

#### Task: Find $\sin \theta$, $\sec \theta$, $\cot \theta$.

#### Solution:
1. Sine ($\sin \theta$):
\[
\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{7}{25}
\]

2. Secant ($\sec \theta$):
\[
\sec \theta = \frac{\text{Hypotenuse}}{\text{Adjacent}} = \frac{25}{24}
\]

3. Cotangent ($\cot \theta$):
\[
\cot \theta = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{24}{7}
\]

#### Final Answer:
\[
\boxed{\sin \theta = \frac{7}{25}, \sec \theta = \frac{25}{24}, \cot \theta = \frac{24}{7}}
\]

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Summary of All Answers:

1. \(\boxed{\sin \theta = \frac{3}{5}, \cos \theta = \frac{4}{5}, \tan \theta = \frac{3}{4}}\)
2. \(\boxed{\sec \theta = \frac{13}{12}, \csc \theta = \frac{13}{5}, \cot \theta = \frac{12}{5}}\)
3. \(\boxed{\cos \theta = \frac{15}{17}, \tan \theta = \frac{8}{15}, \csc \theta = \frac{17}{8}}\)
4. \(\boxed{\sin \theta = \frac{7}{25}, \sec \theta = \frac{25}{24}, \cot \theta = \frac{24}{7}}\)