This is a right triangle with a hypotenuse of $\sqrt{5}$, a leg labeled $1$ (opposite to the right angle), and another leg labeled $s$. The angle $\theta$ is shown at the acute angle opposite to side $s$.

To solve for the unknowns or apply trigonometric relationships, we can use the Pythagorean Theorem or trigonometric functions. First, the Pythagorean Theorem gives us a way to relate the sides:

$\text{Hypotenuse}^2 = \text{Leg 1}^2 + \text{Leg 2}^2$

Substituting the known values:

$(\sqrt{5})^2 = 1^2 + s^2$

$5 = 1 + s^2$

$s^2 = 4$

$s = 2$

So, the length of side $s$ is 2.

If you want to calculate the trigonometric values of angle $\theta$, here’s how we can proceed:

• $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{2}{\sqrt{5}}$
• $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{\sqrt{5}}$
• $\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{2}{1} = 2$

Would you like further details or explanations? Here are some related questions to explore:

1. How can we use the sine and cosine rules in a right triangle?
2. How do you convert trigonometric values into angles?
3. What is the significance of the Pythagorean identity in trigonometry?
4. How can we apply trigonometry to solve real-world problems involving right triangles?
5. What is the relationship between the tangent of an angle and the sides of a right triangle?

Tip: Always check if a triangle is right-angled before applying the Pythagorean theorem.