How to Find the Limit in Trig
Trigonometric functions relate the angles of triangles to the ray lengths of the triangle. Trig limits are most easily solved when the expression is first put into terms of sine and cosine. Because the different trig functions of a triangle describe the same features in a different way they are all related to one another through special mathematical relationships.
Other People Are Reading
Instructions
-
-
1
Put the expression into terms of sine and cosine, if possible. For example, to solve (limit as x ---> 0) tan(x), use the property tan(x) = sin(x) / cos(x) and substitute the expression into the limit to put it into terms of sine and cosine. So, (limit as x ---> 0) tan(x) = (limit as x ---> 0) [sin(x) / cos(x)].
-
2
Factor the expression if it contains exponents. For example, (limit as x ---> 0) [1 - cos^3(x) / sin^2(x)] = (limit as x ---> 0) [(1 - cos(x))(1 + cos(x) + cos^2(x) / (1 + cos(x))(1 - cos(x)).
-
-
3
Simplify the expression as much as possible. For example, simplifying: (limit as x ---> 0) [(1 - cos(x))(1 + cos(x) + cos^2(x) / (1 + cos(x))(1 - cos(x)) = (limit as x ---> 0) [1 + cos(x) + cos^2(x) / 1 + cos(x)].
-
4
Take the limit. For example, solving the limit of the expression (limit as x ---> 0) [1 + cos(x) + cos^2(x) / 1 + cos(x)] finds: (1 + cos(0) + cos^2(0) / 1 + cos(0)) = 1 + 1 + 1 / 1 + 1 = (3 / 2).
-
1
Tips & Warnings
The most commonly used of these mathematical relationships are: tan(x) = (sin(x) / cos(x)), cot(x) = (cos(x) / sin(x)), csc(x) = (1 / sin(x)), sec(x) = (1 / cos(x)), sin(2x) = 2sin(x)cos(x), cos(2x) = cos²(x) - sin²(x), sin²(x) = ( - cos(2x) / 2), cos²(x) = (1 + cos(2x) / 2), tan²(x) = (1 - cos(2x) / 1 + cos(2x)), sin²(x) + cos²(x) = 1, 1 + tan²(x) = sec²(x) and 1 + cot²(x) = csc²(x).
