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 negative.

Limit of Trigonometric Ratios

We get from the definitions of trigonometrical ratios that,

Sin θ = PM/OP and Cos θ = OM/OP …….. (A)

From the above picture, OP is the hypotenuse of the triangle POM; hence, PMOP and OMOP.

Therefore, from (A) we get the values of sin θ and cos θ cannot be greater than 1.

Again, csc θ = OP/PM and sec θ = OP/OM

Therefore, it is clearly seen that the values of csc θ and sec θ can never be less than 1.

Finally, tan θ = PM/OM and cot θ = OM/PM

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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