The problem asks to find the missing side of the right triangle using the Pythagorean theorem, followed by calculating the six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent for angle $\theta$.

Step 1: Finding the Missing Side

You are given a right triangle with:

• Opposite side ($a$) = 16
• Adjacent side ($b$) = 30
• Hypotenuse ($c$) = ?

Using the Pythagorean theorem: $c^2 = a^2 + b^2$ Substitute the values: $c^2 = 16^2 + 30^2 = 256 + 900 = 1156$ $c = \sqrt{1156} = 34$

So, the hypotenuse ($c$) is 34.

Step 2: Trigonometric Ratios

Now, using the triangle sides, we can find the six trigonometric functions:

• Sine: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{16}{34} = \frac{8}{17}$
• Cosine: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{30}{34} = \frac{15}{17}$
• Tangent: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{16}{30} = \frac{8}{15}$
• Cosecant: $\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{34}{16} = \frac{17}{8}$
• Secant: $\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{34}{30} = \frac{17}{15}$
• Cotangent: $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{30}{16} = \frac{15}{8}$

Summary:

• $\sin(\theta) = \frac{8}{17}$
• $\cos(\theta) = \frac{15}{17}$
• $\tan(\theta) = \frac{8}{15}$
• $\csc(\theta) = \frac{17}{8}$
• $\sec(\theta) = \frac{17}{15}$
• $\cot(\theta) = \frac{15}{8}$

Let me know if you would like further details or have any questions!

Here are 5 related questions to explore:

1. How can you verify the trigonometric values using a calculator?
2. What happens to these ratios if the triangle is scaled up or down?
3. Can you apply these ratios to solve for $\theta$?
4. How are trigonometric identities related to these basic ratios?
5. How would the values change if the triangle were not a right triangle?

Tip: Always reduce trigonometric ratios to their simplest form for clarity in calculations!