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A ladder 10 m long reaches a wall at a height of 8 m. What is the sine of the angle made by the ladder with the ground?
0.6
0.8
0.75
0.5
0.8
Recognize the 3-4-5 Pythagorean triple scaled by 2, which gives sides 6, 8, 10. Since sin(╬╕)=HypotenusePerpendicularтАЛ, simply compute 108тАЛ=0.8.
Length of the ladder (hypotenuse) = 10 m, Height reached on the wall (perpendicular) = 8 m.
sin(╬╕)=HypotenusePerpendicularтАЛ
Recognize the 3-4-5 Pythagorean triple scaled by 2, which gives sides 6, 8, 10. Since sin(╬╕)=HypotenusePerpendicularтАЛ, simply compute 108тАЛ=0.8.
Many students mistakenly use cos(╬╕) or take the base (6 m) instead of the height (8 m) for the sine ratio.
Identify the Triangle Sides
In a right-angled triangle formed by the wall, the ground, and the ladder, the ladder is the hypotenuse (h=10), and the height of the wall is the perpendicular (p=8).
p=8,h=10
Apply Sine Ratio
The sine of the angle ╬╕ between the ladder and the ground is the ratio of the height (perpendicular) to the length of the ladder (hypotenuse).
sin(╬╕)=hpтАЛ=108тАЛ
Calculate Final Value
Performing the division yields the value for sin(╬╕).
sin(╬╕)=0.8
B is correct because the ratio of the height of the wall to the length of the ladder is 108тАЛ=0.8.
This concept of ratios is identical to finding the slope of a line, which is a fundamental requirement in Coordinate Geometry for calculating tan(╬╕).
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