Limit of Trigonometric Ratios
In limit of trigonometric ratios we will learn how to find the limits to
the values of sin θ, csc θ, cos θ, sec θ, tan θ and cot θ.
According to the definitions of the trigonometrical ratios of a positive
acute angle are always positive.
Remember that the trigonometrical ratios may be positive as well as
We get from the definitions of trigonometrical ratios that,
Sin θ = PM
and Cos θ = OM
From the above picture, OP is the hypotenuse of the triangle POM; hence, PM
Therefore, from (A) we get the values of sin θ and cos θ cannot be greater than 1.
Again, csc θ = OP
and sec θ = OP
Therefore, it is clearly seen that the values of csc θ and sec θ can never be less than 1.
Finally, tan θ = PM
and cot θ = OM
In this case, the values of PM
may be greater or less or equal to the values of OM
. Thus, the values of tan θ or cot θ may have any non-negative value.
Therefore, the limit of trigonometric ratios of a positive acute angle θ is always non-negative:
(i) The values of sin θ and cos θ cannot be greater than 1;
(ii) The values of csc θ and sec θ cannot be less than 1; and
(iii) The values of tan θ and cot θ can have any value.
● Trigonometric Functions
11 and 12 Grade Math
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