2 goal posts are 8m apart. A footballer is 34m from 1 post and 38m from the other, with what angle must he kick the ball if he has to score a goal.
Law of cosines
c^2 = a^2 + b^2 - 2 a b cos C
8^2 = 34^2 + 38^2 - 2* 34*38* cos C
64 = 1156 + 1444 - 2584 cos C
2584 cos C = 2536
cos C = 0.981424
C = 11.0608 degrees
To find the angle at which the footballer must kick the ball to score a goal, we can use trigonometry.
Let's refer to the point where the footballer stands as point A and the two goal posts as points B and C, where AB is 34m and AC is 38m.
First, let's find the length of the line segment BC, which represents the distance between the goal posts:
BC = AC - AB = 38m - 34m = 4m
Next, we can use the Law of Cosines to find the angle ∠BAC:
cos(∠BAC) = (AB^2 + AC^2 - BC^2) / (2 * AB * AC)
cos(∠BAC) = (34^2 + 38^2 - 4^2) / (2 * 34 * 38)
cos(∠BAC) = (1156 + 1444 - 16) / (2588)
cos(∠BAC) = (2584) / (2588)
cos(∠BAC) ≈ 0.9985
To find the angle ∠BAC, we can take the inverse cosine (arccos) of 0.9985:
∠BAC ≈ arccos(0.9985)
∠BAC ≈ 0.0774 radians
To convert the angle from radians to degrees, we multiply by 180/π:
∠BAC ≈ 0.0774 * (180/π)
∠BAC ≈ 4.44 degrees
Therefore, the footballer must kick the ball at an angle of approximately 4.44 degrees in order to score a goal.
To find the angle at which the footballer must kick the ball to score a goal, we can use trigonometry. Let's break down the problem into smaller steps.
Step 1: Draw a diagram to visualize the situation. Label the goal posts, the footballer, and the distances.
Step 2: From the diagram, we can see that the distances form a right triangle. Let's label the distances as follows:
- The distance from the first goal post to the footballer is 34m (opposite side).
- The distance from the second goal post to the footballer is 38m (adjacent side).
- The distance between the goal posts is 8m (hypotenuse).
Step 3: Use trigonometry functions to solve for the angle. In this case, we need to use the arctangent function (tan^(-1)) to find the angle. The formula is as follows:
angle = tan^(-1)(opposite/adjacent)
In this case, the angle we want to find is the angle at which the ball must be kicked to score a goal.
Let's substitute the known values:
angle = tan^(-1)(34/38)
Step 4: Use a scientific calculator to find the value of the angle:
angle ≈ tan^(-1)(0.8947) ≈ 42.3 degrees
Therefore, the footballer must kick the ball at an angle of approximately 42.3 degrees to score a goal.