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.

Note:

Remember that the trigonometrical ratios may be positive as well as negative.

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