To find the values of the six trigonometric functions of the angle $\theta$ in this right triangle, we can use the side lengths provided. The triangle has one leg measuring $12$ and the other leg measuring $24$.

Step 1: Calculate the Hypotenuse

Using the Pythagorean theorem: $c = \sqrt{a^2 + b^2} = \sqrt{12^2 + 24^2}$ $= \sqrt{144 + 576} = \sqrt{720} = 12\sqrt{5}$

So, the hypotenuse $c$ is $12\sqrt{5}$.

Step 2: Trigonometric Functions

Now, we can find each trigonometric function based on the angle $\theta$:

1. Sine $\sin(\theta)$: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{24}{12\sqrt{5}} = \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5}$

2. Cosine $\cos(\theta)$: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12}{12\sqrt{5}} = \frac{1}{\sqrt{5}} = \frac{\sqrt{5}}{5}$

3. Tangent $\tan(\theta)$: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{24}{12} = 2$

4. Cosecant $\csc(\theta)$ (reciprocal of sine): $\csc(\theta) = \frac{\text{hypotenuse}}{\text{opposite}} = \frac{12\sqrt{5}}{24} = \frac{\sqrt{5}}{2}$

5. Secant $\sec(\theta)$ (reciprocal of cosine): $\sec(\theta) = \frac{\text{hypotenuse}}{\text{adjacent}} = \frac{12\sqrt{5}}{12} = \sqrt{5}$

6. Cotangent $\cot(\theta)$ (reciprocal of tangent): $\cot(\theta) = \frac{\text{adjacent}}{\text{opposite}} = \frac{12}{24} = \frac{1}{2}$

Summary of Trigonometric Function Values:

• $\sin(\theta) = \frac{2\sqrt{5}}{5}$
• $\cos(\theta) = \frac{\sqrt{5}}{5}$
• $\tan(\theta) = 2$
• $\csc(\theta) = \frac{\sqrt{5}}{2}$
• $\sec(\theta) = \sqrt{5}$
• $\cot(\theta) = \frac{1}{2}$

Would you like more details on any of these steps?

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1. How can we verify the trigonometric values using a calculator?
2. How do these trigonometric functions relate to real-world applications?
3. What is the significance of each trigonometric function in a right triangle?
4. How would these values change if the triangle's dimensions were scaled?
5. Can we use these trigonometric values to find angle $\theta$ directly?

Tip: Always remember that each trigonometric function has a reciprocal pair that can help in calculations when values are known for only sine, cosine, or tangent.