The image provided involves a problem that relates to right triangles and trigonometric functions. Let's break down the problem:

Given:

• A right triangle where one leg is $a = 5$ units and the other leg is $b = 4$ units.

Task 1:

• The problem asks to calculate the length of the hypotenuse $c$ using the Pythagorean theorem.

The Pythagorean theorem states: $c = \sqrt{a^2 + b^2}$

Task 2:

• The problem asks to calculate the sine, cosine, and tangent of the angle $\theta$ where $a$ is the opposite side and $b$ is the adjacent side to $\theta$.

1. Hypotenuse Calculation:

$c = \sqrt{5^2 + 4^2} = \sqrt{25 + 16} = \sqrt{41} \approx 6.4$

2. Trigonometric Functions:

Given:

• Opposite $= a = 5$
• Adjacent $= b = 4$
• Hypotenuse $= c \approx 6.4$

Sine $\sin(\theta)$: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{5}{6.4} \approx 0.78125$

Cosine $\cos(\theta)$: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{4}{6.4} \approx 0.625$

Tangent $\tan(\theta)$: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{5}{4} = 1.25$

Summary:

1. Length of Hypotenuse $c$: $\approx 6.4$
2. Sine $\sin(\theta)$: $\approx 0.78125$
3. Cosine $\cos(\theta)$: $\approx 0.625$
4. Tangent $\tan(\theta)$: $1.25$

Would you like more details or have any questions?

Relative Questions:

1. How would the trigonometric functions change if one of the legs of the triangle was doubled?
2. What is the value of $\theta$ in degrees given the trigonometric ratios?
3. How does the Pythagorean theorem apply to non-right triangles?
4. What are the sine, cosine, and tangent of the complementary angle of $\theta$?
5. What is the relationship between the sine and cosine of $\theta$ and $90^\circ - \theta$?

Tip:

Always remember that the sine, cosine, and tangent functions are directly tied to the angle's position in the right triangle relative to the sides.