Trigonometry

cos 60° = 1/2, tan 45° = 1, sin 90° = 1, tan 30° = 1/√3

Step-by-step verification: cos 60° = 1/2 ✓, tan 45° = 1 ✓, sin 90° = 1 ✓, cos 0° = 1 (not 0) ✗, tan 30° = 1/√3 ✓. These are standard trigonometric values.

1 : √3 : 2

In a 30-60-90 triangle, the sides are in the ratio 1 : √3 : 2, where 1 is opposite to 30°, √3 is opposite to 60°, and 2 is the hypotenuse. This comes from the properties of the equilateral triangle when cut in half.

20 meters

Step 1: Angle of depression = 45° means the angle with horizontal is 45°. Step 2: In the right triangle formed, height = 20 m (opposite), distance = adjacent. Step 3: tan 45° = opposite/adjacent = 20/distance. Step 4: Since tan 45° = 1, distance = 20 meters.

sin θ increases as θ increases from 0° to 90°, cos θ decreases as θ increases from 0° to 90°, sin 0° = 0

Analyzing each: sin θ goes from 0 to 1 as θ goes from 0° to 90° ✓; cos θ goes from 1 to 0 as θ goes from 0° to 90° ✓; tan θ increases (not decreases) from 0 to ∞ ✗; sin 0° = 0 ✓; cos 90° = 0 (not 1) ✗.