**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, PM ≮ OP and OM ≮ OP.

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**

**Basic Trigonometric Ratios and Their Names****Restrictions of Trigonometrical Ratios****Reciprocal Relations of Trigonometric Ratios****Quotient Relations of Trigonometric Ratios****Limit of Trigonometric Ratios****Trigonometrical Identity****Problems on Trigonometric Identities****Elimination of Trigonometric Ratios****Eliminate Theta between the equations****Problems on Eliminate Theta****Trig Ratio Problems****Proving Trigonometric Ratios****Trig Ratios Proving Problems****Verify Trigonometric Identities****Trigonometrical Ratios of 0°****Trigonometrical Ratios of 30°****Trigonometrical Ratios of 45°****Trigonometrical Ratios of 60°****Trigonometrical Ratios of 90°****Trigonometrical Ratios Table****Problems on Trigonometric Ratio of Standard Angle****Trigonometrical Ratios of Complementary Angles****Rules of Trigonometric Signs****Signs of Trigonometrical Ratios****All Sin Tan Cos Rule****Trigonometrical Ratios of (- θ)****Trigonometrical Ratios of (90° + θ)****Trigonometrical Ratios of (90° - θ)****Trigonometrical Ratios of (180° + θ)****Trigonometrical Ratios of (180° - θ)****Trigonometrical Ratios of (270° + θ)****Trigonometrical Ratios of (270° - θ)****Trigonometrical Ratios of (360° + θ)****Trigonometrical Ratios of (360° - θ)****Trigonometrical Ratios of any Angle****Trigonometrical Ratios of some Particular Angles****Trigonometric Ratios of an Angle****Trigonometric Functions of any Angles****Problems on Trigonometric Ratios of an Angle****Problems on Signs of Trigonometrical Ratios**

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