Final answer:
The exact value for tan(A) in a right triangle with a right angle at C is found using the ratio of side Ay (opposite to angle A) to side Ax (adjacent to angle A). The tangent of angle A is calculated as Ay/Ax. If Ax and Ay are known, tan(A) can be computed directly, and the Pythagorean theorem confirms the relationship between the sides of the triangle.
Step-by-step explanation:
To find the exact value for tan(A) in a right triangle where angle C is a right angle, we can use the definition of the tangent function, which is the ratio of the opposite side to the adjacent side of the angle in question. Assuming that we know the lengths of the sides adjacent to angle A (Ax) and opposite to angle A (Ay), we can simply calculate tan(A) as Ay/Ax.
In the context of vectors and trigonometry, this concept is similar. If vector A is the hypotenuse of the right triangle, its x- and y-components (Ax and Ay) can be used to calculate the direction (angle) of vector A relative to the x-axis using the tangent function. For instance, if Ax and Ay are given as 9 and 5 units respectively, tan(A) would be calculated as tan(A) = 5/9. Using an inverse tangent function, we could also find angle A, which in this case would be tan⁻¹(5/9), approximately 29.1°.
Remember that the Pythagorean theorem, which states that for a right triangle, the square of the length of the hypotenuse (A) equals the sum of the squares of the lengths of the other two sides (Ax and Ay), will also apply, confirming the relationship between the sides: A² = Ax² + Ay².