Q: 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.
since the sides are in the ratio 1:√3:2
a,b) the sides are 41:41√3:82
c) as we all know (or should, because the come up repeatedly):
sin 30° = 1/2
cos 30° = √3/2
tan 30° = √3
And since the co-functions are the functions of the complementary angle, we automatically get
sin 60° = √3/2
cos 60° = 1/2
tan 60° = √3
oops: tan 30° = 1/√3
To solve this problem, we need to use the properties of a 30°-60°-90° triangle. In a 30°-60°-90° triangle, the sides have a fixed ratio.
a. To find the length of the side opposite the 60° angle, we need to use the ratio of a 30°-60°-90° triangle. In this triangle, the side opposite the 30° angle (short leg) is half the length of the hypotenuse. Therefore, if the side opposite the 30° angle measures 41 feet, the length of the side opposite the 60° angle (long leg) will be 41 x √3 (approximately 70.743 feet).
b. To find the length of the hypotenuse of the triangular lot, we still use the ratio of a 30°-60°-90° triangle. The hypotenuse is twice the length of the side opposite the 30° angle. Therefore, the length of the hypotenuse will be 41 x 2 (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 defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In this case, the sine of the 30° angle is 41/82 (approximately 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, the length of the side adjacent to the 30° angle is 41 x √3/2 (approximately 35.372 feet). Therefore, the cosine of the 30° angle is (41 x √3/2) / 82 (approximately 0.8593).
The tangent of an angle is defined as the ratio of the sine of the angle to the cosine of the angle. In this case, the tangent of the 30° angle is (41/82) / ((41 x √3/2) / 82) (approximately 0.5774).
So, the answers to part c are:
Sine of 30° = 0.5000 (rounded to four decimal places)
Cosine of 30° = 0.8593 (rounded to four decimal places)
Tangent of 30° = 0.5774 (rounded to four decimal places)