Question

Find the angle of elevation of the Sun, when the length of the shadow of a tree is \( \frac{1}{\sqrt{3}} \) times the height of the tree.

• A. \( 30^\circ \)
• B. \( 45^\circ \)
• C. \( 60^\circ \)
• D. \( 90^\circ \)

Hint:

\textbf{Trigonometry of Right Triangles.} In problems involving angles of elevation or depression, remember the basic trigonometric ratios (sine, cosine, tangent) and their relationships with the sides of a right-angled triangle. \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).

Answer 1

The Correct Option is C

Let the height of the tree be \(h\) and the length of the shadow of the tree be \(s\). According to the problem, the length of the shadow is \( \frac{1}{\sqrt{3}} \) times the height of the tree: $$ s = \frac{1}{\sqrt{3}} h $$ Let \( \theta \) be the angle of elevation of the Sun. We can consider a right-angled triangle formed by the tree (perpendicular), its shadow (base), and the line from the top of the tree to the end of the shadow (hypotenuse). The angle of elevation \( \theta \) is the angle between the ground (shadow) and the line of sight to the top of the tree. We can use the tangent function, which relates the angle \( \theta \) to the opposite side (height of the tree) and the adjacent side (length of the shadow): $$ \tan(\theta) = \frac{\text{height of the tree}}{\text{length of the shadow}} = \frac{h}{s} $$ Substitute the given relationship between \(s\) and \(h\): $$ \tan(\theta) = \frac{h}{\frac{1}{\sqrt{3}} h} $$ $$ \tan(\theta) = \frac{h \times \sqrt{3}}{h} $$ $$ \tan(\theta) = \sqrt{3} $$ We need to find the angle \( \theta \) whose tangent is \( \sqrt{3} \). We know from trigonometric values that: $$ \tan(60^\circ) = \sqrt{3} $$ Therefore, the angle of elevation of the Sun is \( 60^\circ \).