building lot in a city is shaped as a 30° -60° -90° triangle. The side opposite the 30° angle measures 41 feet.
a. what is the length of the side of the lot opposite the 60° angle b. what is the length of the hypotenuse of the triangular lot.
c. what are the sine, cosine, and tangent of the 30° angle in the lot. Write your answers as decimals rounded to four decimal places.
Make an effort to memorize the ratio of sides of both the 30-60-90 and the 45-45-90 right-angled triangles, those values will come up again and again in your study of trig.
angles: 30-60-90
sides : 1 - √3 - 2
angles : 45-45-90
sides : 1 - 1 - √2
a)
so set up a ratio:
1/41 = √3/x
x = 41√3 = appr 71.0141
b)
Here is the advantage of knowing the sides of the 30-60-90 triangle
Look at your sketch:
sin30° = 1/2
cos30° = √3/2
tan30° = 1/√3
notice if you know the memorize the ratio of sides in that triangle, and if you know the definitions of the trig rations, you will always have them ready.
To solve this problem, we can use the properties of a 30°-60°-90° triangle. In this type of triangle, the ratio of the lengths of the sides is determined by the angles.
a. To find the length of the side opposite the 60° angle, we can use the ratio in a 30°-60°-90° triangle. The ratio is 1:√3:2, which means that the side opposite the 60° angle is √3 times the length of the side opposite the 30° angle.
So, to find the length of the side opposite the 60° angle:
Length of the side opposite the 30° angle = 41 feet
Length of the side opposite the 60° angle = √3 * 41 feet ≈ 71.135 feet (rounded to three decimal places)
b. To find the length of the hypotenuse, we can again use the ratio in a 30°-60°-90° triangle. The ratio is 1:√3:2, which means that the hypotenuse is double the length of the side opposite the 30° angle.
So, to find the length of the hypotenuse:
Length of the side opposite the 30° angle = 41 feet
Length of the hypotenuse = 2 * 41 feet = 82 feet
c. To find the sine, cosine, and tangent of the 30° angle, we can use the trigonometric functions.
The sine of an angle is equal to the length of the side opposite the angle divided by the length of the hypotenuse. So, for the 30° angle:
Sine of 30° = Opposite side/Hypotenuse = 41/82 ≈ 0.5000 (rounded to four decimal places)
The cosine of an angle is equal to the length of the side adjacent to the angle divided by the length of the hypotenuse. In this case, the side adjacent to the 30° angle is the side opposite the 60° angle. So, for the 30° angle:
Cosine of 30° = Opposite side/Hypotenuse = 71.135/82 ≈ 0.8660 (rounded to four decimal places)
The tangent of an angle is equal to the sine of the angle divided by the cosine of the angle. So, for the 30° angle:
Tangent of 30° = Sine of 30° / Cosine of 30° = 0.5000 / 0.8660 ≈ 0.5774 (rounded to four decimal places)