To understand how the given phrases are used differently, you should consider the trigonometric ratios.
See solution.
Practice makes perfect
To understand how the phrases tangent ratio and tangent of a circle are used differently, we can begin by recalling the definitions of them.
Tangent Ratio: The tangent ratio is the ratio of the length of the leg opposite an angle to the length of the leg adjacent the same angle in a right triangle.
Tangent of a Circle: The tangent of a circle is a line in the plane of the circle that intersects the circle in exactly one point.
To compare the phrases, let's try to visualize both of them.
Tangent Ratio
Let △ ABC be a right triangle with angles α and β.
Considering the definition, we can write two different tangent ratios as tangent equals "opposite over adjacent."
tan α=AB/BC
and
tan β=BC/AB
Tangent of a Circle
Let's consider a circle centered at point O and a line that intersects the circle only in point P.
We can conclude that, by the definition, line l is the tangent line of the circle.
Comparison
When we contrast the phrases, we can see that they are completely different. The tangent ratio is a numerical value in a right triangle while the tangent of a circle is a geometric figure.
