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.
To solve these problems, we'll make use of the properties of a 30-60-90 triangle. In a 30-60-90 triangle, the side opposite the 30° angle is half the length of the hypotenuse, and the side opposite the 60° angle is sqrt(3) times the length of the side opposite the 30° angle.
a. To find the length of the side opposite the 60° angle, we'll use the property mentioned above. The side opposite the 60° angle is sqrt(3) times the length of the side opposite the 30° angle. Since the side opposite the 30° angle is given as 41 feet, we can calculate the length of the side opposite the 60° angle as follows:
Length of side opposite the 60° angle = sqrt(3) * Length of side opposite the 30° angle = sqrt(3) * 41 feet. Now we can calculate this value:
Length of side opposite the 60° angle = sqrt(3) * 41 ≈ 71.073 feet.
b. To find the length of the hypotenuse of the triangular lot, we'll use the property mentioned above. The side opposite the 30° angle is half the length of the hypotenuse. So, the length of the hypotenuse will be twice the length of the side opposite the 30° angle. Since the side opposite the 30° angle is given as 41 feet, we can calculate the length of the hypotenuse as follows:
Length of hypotenuse = 2 * Length of side opposite the 30° angle = 2 * 41 feet. Now we can calculate this value:
Length of hypotenuse = 2 * 41 = 82 feet.
c. To find the sine, cosine, and tangent of the 30° angle, we'll use the definitions of these trigonometric functions.
The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In this case, the side opposite the 30° angle is 41 feet, and the hypotenuse is 82 feet. So, the sine of the 30° angle can be calculated as follows:
Sine of 30° = (Length of side opposite 30° angle) / (Length of hypotenuse) = 41 / 82 ≈ 0.5000.
The cosine of an angle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. In this case, there is no side directly adjacent to the 30° angle. However, we can use the fact that the sum of the angles in a triangle is 180° to find the side adjacent to the 30° angle. The remaining angle in the triangle is 90°, so the side adjacent to the 30° angle is the side opposite the 90° angle, which is the hypotenuse. So, the cosine of the 30° angle can be calculated as follows:
Cosine of 30° = (Length of hypotenuse) / (Length of hypotenuse) = 82 / 82 = 1.0000.
The tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. In this case, the side opposite the 30° angle is 41 feet, and there is no side directly adjacent to the 30° angle. However, we can use the fact that the sum of the angles in a triangle is 180° to find the side adjacent to the 30° angle. The remaining angle in the triangle is 90°, so the side adjacent to the 30° angle is the side opposite the 90° angle, which is the hypotenuse. So, the tangent of the 30° angle can be calculated as follows:
Tangent of 30° = (Length of side opposite 30° angle) / (Length of hypotenuse) = 41 / 82 ≈ 0.5000.
Therefore,
Sine of 30° ≈ 0.5000,
Cosine of 30° = 1.0000, and
Tangent of 30° ≈ 0.5000.