Some Applications of Trigonometry

45°

Step 1: Height = 1.8 m, shadow = 1.8 m, angle of elevation = ? Step 2: tan(angle) = height/shadow = 1.8/1.8 = 1. Step 3: We know tan 45° = 1. Step 4: Therefore, angle of elevation = 45°.

60√3 m

Step 1: Building height = 60 m, angle of depression = 30°, horizontal distance = ? Step 2: Angle of depression equals angle of elevation from car to top. Step 3: tan 30° = height/distance = 60/distance. Step 4: 1/√3 = 60/distance, so distance = 60√3 m.

8 + 4√3 m

Step 1: Let broken part length = l, standing part = h. Distance from base = 12 m, angle = 30°. Step 2: In right triangle: cos 30° = 12/l, so √3/2 = 12/l, giving l = 24/√3 = 8√3 m. Step 3: sin 30° = h/l, so 1/2 = h/(8√3), giving h = 4√3 m. Step 4: Original height = h + l = 4√3 + 8√3 = 12√3 m. Wait, let me recalculate. Step 2: sin 30° = h/l = 1/2, and cos 30° = 12/l = √3/2. Step 3: From cos equation: l = 12/(√3/2) = 24/√3 = 8√3 m. Step 4: From sin equation: h = l/2 = 4√3 m. Actually, let me be more careful. The standing part is h, broken part makes hypotenuse l. We have base = 12, angle = 30°. S

3000√3 m

Step 1: Height = 3000 m, angle of depression = 30°, horizontal distance = ? Step 2: tan 30° = height/horizontal distance = 3000/distance. Step 3: 1/√3 = 3000/distance. Step 4: distance = 3000√3 m.