Final answer:
The tangent of an angle A, written as tan(A), is defined as the ratio of sin(A) to cos(A). In the context of a right triangle, this can also be expressed as the ratio of the lengths of the opposite side to the adjacent side of angle A. tan(A) = y/x
Step-by-step explanation:
To write tan(A) in terms of sine and cosine, we simply refer to the basic trigonometric identity involving these functions. The tangent of an angle A, represented as tan(A), is defined as the ratio of the sine of angle A to the cosine of angle A. Using the notation for sine as sin(A) and cosine as cos(A), we can write the identity as:
tan(A) = sin(A) / cos(A)
This formula expresses the tangent function in terms of the other two primary trigonometric functions. Considering a right-angled triangle, if we label the side opposite to angle A as y (or Ay), the adjacent side as x (or Ax), and the hypotenuse as h (or A), according to the Pythagorean theorem, we can derive the relations:
- sin(A) = opposite/hypotenuse = y/h
- cos(A) = adjacent/hypotenuse = x/h
Substituting these back into our original definition:
tan(A) = y/x
This shows that the tangent of angle A is also the ratio of the length of the side opposite to angle A to the length of the side adjacent to angle A in a right triangle.