A vertical pole 12 m high casts a shadow 5√3 m long. Find the angle of elevation of the Sun.

Step 1: Identify the given information and form a right-angled triangle

We are given the height of the vertical pole and the length of its shadow. This scenario forms a right-angled triangle where the pole is the perpendicular side, the shadow is the base, and the angle of elevation of the Sun is the angle at the base.

Step 2: Choose the appropriate trigonometric ratio

In the right-angled triangle ABC, we know the perpendicular (height of the pole) and the base (length of the shadow). The tangent function relates the perpendicular and the base to the angle of elevation.

Step 3: Substitute the values and calculate the tangent

Substitute the given values for the height of the pole (AB) and the length of the shadow (BC) into the tangent formula.

Step 4: Rationalize the denominator and simplify

To simplify the expression, we rationalize the denominator by multiplying both the numerator and the denominator by $\sqrt{3}$. Then, we simplify the fraction.

Step 5: Find the angle of elevation

To find the angle $\theta$, we take the inverse tangent of the calculated value. Using a calculator, we find the approximate value of $\theta$.

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