a Let's draw a right triangle and label its non-right angles, A and B.
If A has a slope ratio of ab, it means that its adjacent and opposite sides have lengths of b and a units, respectively.
The ratio of the opposite side to the adjacent side in a right triangle is the same thing as that angle's tangent value.
tan A = a/b
Therefore, the tangent value of the complementary angle must have a ratio of ba.
tan B = b/a
b The sine of an angle is the ratio of its opposite side to the hypotenuse
sin θ =Opposite side/Hypotenuse
The cosine of an angle is the ratio of its adjacent side to the hypotenuse.
cos θ =Adjacent side/Hypotenuse
Let's draw a new right triangle and include the length of the hypotenuse, which we will call c.
Using the information in the diagram, we can write equations for the sine and cosine of each angle.
|c|c|
A & B [0.1em]
[-1em]
sin A = a/c & sin B = b/c [1.2em]
cos A = b/c & cos B = a/c [0.8em]
Using the information from the table, we can make a conjecture that the sine ratio of an angle equals the cosine ratio of its complementary angle. Similarly, the cosine ratio of an angle equals the sine ratio of its complementary angle. Let's try this with a few angles that are complementary using a calculator.
