To find the gradient (or slope) of a straight line that is inclined at an angle of 45 degrees to the positive direction of the x-axis, we can use the following relationship between the angle of inclination and the gradient:

$\text{Gradient (m)} = \tan(\theta)$

Where:

• $\theta$ is the angle of inclination of the line with respect to the x-axis.
• $\tan(\theta)$ is the tangent of the angle.

Given that $\theta = 45^\circ$, we can substitute into the formula:

$m = \tan(45^\circ)$

Since $\tan(45^\circ) = 1$, we have:

$m = 1$

Thus, the gradient of the straight line is 1.

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Would you like to see additional examples or explanations? Let me know if you have any questions!

Here are 5 related questions to explore further:

1. How does the gradient change if the angle of inclination is increased to 60 degrees?
2. What is the equation of a straight line with gradient 1 that passes through the point (2, 3)?
3. How do you find the angle of inclination given the gradient of a line?
4. What is the relationship between the slope of a line and its perpendicular slope?
5. How can the gradient be used to find the direction of a line in geometry?

Tip: The tangent function is periodic, so the angle of inclination can be expressed in terms of any angle $\theta + 180n$ for integer $n$, but the basic principle holds for each $\theta$.