To find the cosine value for the given angle of 8 degrees with the point (5, 12) on its terminating side, we need to recall some basic trigonometric definitions and the properties of right triangles.
Firstly, remember that the cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. In this case, we can establish a right triangle where the adjacent side corresponds to the x-coordinate and the opposite side corresponds to the y-coordinate of the point (5, 12).
1. **Identify the coordinates**: The point (5, 12) means that the adjacent side has a length of 5 and the opposite side has a length of 12.
2. **Calculate the hypotenuse**: We can use the Pythagorean theorem to determine the length of the hypotenuse (h):
h = √(adjacent² + opposite²) = √(5² + 12²) = √(25 + 144) = √169 = 13.
3. **Calculate the cosine**: Now that we have the lengths of the sides, we can calculate cosine(θ):
cos(θ) = adjacent / hypotenuse = 5 / 13.
Therefore, for the angle of 8 degrees with the point (5, 12) on its terminating side, the value of cosine is approximately:
cos(8 degrees) ≈ 0.3846 (rounded to four decimal places).
Keep in mind that this value correlates with the angle at 8 degrees, not specifically the point (5, 12). The point helps visualize the situation in the Cartesian plane, while the angle’s cosine is calculated using standard definitions.
In conclusion, to answer your question, the value of cosine for the angle of 8 degrees (considering the point on its terminal side) can be approximated as:
cos(8 degrees) ≈ 0.3846.