A triangular lot sits at the corner of two streets that intersect at an angle of 63 degrees. One street side of the lot is 35m and the other is 47m. how long is the back of the lot, to the nearest metre?
Straight cosine law question
solve
x^2 = 35^2 + 47^2 - 2(35)(47)cos 63
thank you very much.
To find the length of the back of the lot, we can use the trigonometric concept of the sine function.
1. First, let's label the sides of the triangular lot:
- The side along the first street (35m) as A
- The side along the second street (47m) as B
- The side we need to find, the back of the lot, as C
2. Since we know the lengths of two sides of the triangle and the included angle between them, we can use the Law of Cosines to find the length of the back of the lot (side C).
The Law of Cosines states:
C² = A² + B² - 2AB * cos(angle)
Substituting the given values:
C² = 35² + 47² - 2 * 35 * 47 * cos(63°)
3. Now, let's calculate the value of C²:
C² = 35² + 47² - 2 * 35 * 47 * cos(63°)
Using a calculator or math software to compute the expression, we get:
C² ≈ 1356.02
4. To find the length of side C, we can take the square root of C²:
C ≈ √1356.02
Calculating, we get:
C ≈ 36.83
5. Rounding the answer to the nearest meter, the length of the back of the lot is approximately 37 meters.
To find the length of the back of the lot, we can use the law of sines. The law of sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is equal for all sides and their opposite angles.
In this case, let's label the side opposite the 63-degree angle as A, the side opposite the 35m street as B, and the side opposite the 47m street as C. We are trying to find the length of side A.
Given:
Angle C = 63 degrees
Side B = 35m
Side C = 47m
The law of sines can be written as:
a/sin(A) = b/sin(B) = c/sin(C)
We know the values of sides B and C, and angle C. We can now substitute these values into the equation:
A/sin(63) = 35/sin(B) = 47/sin(63)
To find side A, rearrange the equation:
A = sin(63) * (35 / sin(B))
Now, to find angle B, we can use the fact that the sum of the angles in a triangle is 180 degrees. Since we know angle C and angle A (which is 180 - 63 = 117 degrees), we can find angle B:
Angle B = 180 - 63 - 117
Angle B = 0 degrees
Now we have angle B and side B, so we can substitute these values into the equation:
A = sin(63) * (35 / sin(0))
A = sin(63) * (35 / 0)
However, sin(0) is undefined, which means the information given is not sufficient to solve this problem.