To find the angle of elevation of the sun when the length of a pole's shadow is equal to its height, we can model this situation using a right-angled triangle.

Trigonometry for Angle of Elevation

Let the height of the pole be denoted by $h$ and the length of its shadow be denoted by $s$. We are given that the length of the shadow is equal to the height of the pole, so $s = h$.

The angle of elevation of the sun ($\theta$) is the angle between the horizontal ground (the shadow) and the line of sight from the tip of the shadow to the top of the pole.

In the right-angled triangle formed:

• The side opposite to the angle $\theta$ is the height of the pole ($h$).
• The side adjacent to the angle $\theta$ is the length of the shadow ($s$).

We use the tangent trigonometric function, which is defined as the ratio of the opposite side to the adjacent side:

$ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} $

Substituting the given values:

$ \tan(\theta) = \frac{h}{s} $

Since $h = s$, the equation becomes:

$ \tan(\theta) = \frac{h}{h} = 1 $

Calculating the Angle

We need to find the angle $\theta$ whose tangent is 1. This is a standard trigonometric value:

$ \theta = \arctan(1) $

The angle for which the tangent is 1 is $45^\circ$.

Therefore, the angle of elevation of the sun when the length of the shadow of a pole is equal to its height is $45^\circ$.