A diagonal is drawn in a 12-in. square floor tile. Find
the sine, cosine, and tangent of the angle formed by the
diagonal and a side.
the diagonal length is irrelevant, the fact that you have a square tile, means the angle we are working with is 45°
you should know the values of sin, cos, and tan of 45°
To find the sine, cosine, and tangent of the angle formed by the diagonal and a side of the square floor tile, we can use trigonometric ratios.
First, let's consider the triangle formed by the diagonal and one side of the square tile. Since the diagonal divides the square into two congruent right triangles, we only need to focus on one of them.
Let's assume that the length of one side of the square tile is represented as "s". Since the tile is a square, all sides are equal, so each side has a length of 12 inches.
Now, let's calculate the length of the diagonal using the Pythagorean theorem. The diagonal, being the hypotenuse of the right triangle, can be found using the formula:
diagonal^2 = side^2 + side^2
diagonal^2 = s^2 + s^2
diagonal^2 = 2s^2
Since each side of the square tile has a length of 12 inches:
diagonal^2 = 2(12^2)
diagonal^2 = 2(144)
diagonal^2 = 288
Taking the square root of both sides:
diagonal = √288
diagonal ≈ 16.97 inches
Now, let's calculate the sine, cosine, and tangent of the angle using the following trigonometric ratios:
Sine (sin): Opposite / Hypotenuse
Cosine (cos): Adjacent / Hypotenuse
Tangent (tan): Opposite / Adjacent
In our case, the side length "s" of the square tile represents the adjacent side, and the diagonal length represents the hypotenuse. The angle formed by the diagonal and a side represents the reference angle in the triangle.
sin(angle) = diagonal / s
cos(angle) = s / diagonal
tan(angle) = diagonal / s
Substituting the values:
sin(angle) ≈ 16.97 / 12
cos(angle) ≈ 12 / 16.97
tan(angle) ≈ 16.97 / 12
Now, we can calculate the approximations of sine, cosine, and tangent:
sin(angle) ≈ 1.41
cos(angle) ≈ 0.71
tan(angle) ≈ 1.41
Therefore, the sine of the angle is approximately 1.41, the cosine of the angle is approximately 0.71, and the tangent of the angle is approximately 1.41.