A building lot in a city is shaped as a 30° -60° -90° triangle. The side opposite the 30° angle measures 41 feet.
a. Find the length of the side of the lot opposite the 60° angle.
b.Find the length of the hypotenuse of the triangular lot.
c. Find the sine, cosine, and tangent of the 30° angle in the lot. Write your answers as decimals rounded to four decimal places.
you know that for such a triangle, the sides are in the ratio
1:√3:2
So, the short side is 41
The long side is 41√3
The hypotenuse is 41*2
The functions of 30° are standard values, which it would be well to learn.
a.71 ft
To find the length of the side of the lot opposite the 60° angle, we can use the properties of a 30°-60°-90° triangle. In such a triangle, the side opposite the 30° angle is half the length of the hypotenuse, and the side opposite the 60° angle is √3 times the length of the side opposite the 30° angle.
a. The length of the side opposite the 60° angle can be found by multiplying the length of the side opposite the 30° angle by √3:
Length_of_side_opposite_60° = √3 * Length_of_side_opposite_30°
Length_of_side_opposite_60° = √3 * 41 feet
You can simplify this by using a calculator or by knowing that √3 is approximately 1.732. Multiplying this by 41, we get:
Length_of_side_opposite_60° ≈ 1.732 * 41 feet ≈ 70.852 feet
Therefore, the length of the side of the lot opposite the 60° angle is approximately 70.852 feet.
Next, let's find the length of the hypotenuse of the triangular lot. In a 30°-60°-90° triangle, the hypotenuse is twice the length of the side opposite the 30° angle.
b. Length_of_hypotenuse = 2 * Length_of_side_opposite_30°
Length_of_hypotenuse = 2 * 41 feet
Length_of_hypotenuse = 82 feet
Therefore, the length of the hypotenuse of the triangular lot is 82 feet.
Now, let's find the sine, cosine, and tangent of the 30° angle.
The sine of an angle is the ratio of the side opposite the angle to the hypotenuse.
c. Sine(30°) = Length_of_side_opposite_30° / Length_of_hypotenuse
Sine(30°) = 41 feet / 82 feet
Sine(30°) ≈ 0.500
The cosine of an angle is the ratio of the side adjacent to the angle to the hypotenuse.
c. Cosine(30°) = Length_of_side_adjacent_to_30° / Length_of_hypotenuse
In this case, the length of the side adjacent to the 30° angle is equal to the side opposite the 60° angle, which we found to be approximately 70.852 feet.
Cosine(30°) = 70.852 feet / 82 feet
Cosine(30°) ≈ 0.864
The tangent of an angle is the ratio of the sine of the angle to the cosine of the angle.
c. Tangent(30°) = Sine(30°) / Cosine(30°)
Tangent(30°) ≈ 0.500 / 0.864
Tangent(30°) ≈ 0.578
Therefore, the sine, cosine, and tangent of the 30° angle in the lot are approximately 0.500, 0.864, and 0.578 respectively.