We will learn how to find the trigonometrical ratios of any angle using the following step-by-step procedure.

Step I: To find the trigonometrical ratios of angles (n ∙ 90° ± θ); where n is an integer and θ is a positive acute angle, we will follow the below procedure.

First we need to determine the sign of the given trigonometrical ratio. Now to determine the sign of the given trigonometrical ratio we need to find the quadrant in which the angle (n ∙ 90° + θ) or (n ∙ 90° - θ) lies.

Now, using the rule “*All, sin, tan, cos*” we will find the sign of the given trigonometrical ratio.Therefore,

(i) All trigonometrical ratios are positive if the given angle (n ∙ 90° + θ) or (n .90° + θ) lies in the I quadrant (first quadrant);

(ii) Only sin and csc
ratios is positive if the given angle (n ∙
90° + θ) or (n ∙ 90° - θ) lies in the II quadrant (second quadrant);

(iii) Only tan and cot ratios is positive if the given angle (n ∙ 90° + θ) or (n ∙ 90° - θ) lies in the III quadrant (third quadrant);

(iv) Only cos and sec ratios is positive if the given angle (n ∙ 90° + θ) or (n ∙ 90° - θ) lies in the IV quadrant (fourth quadrant).

Step II: Now determine whether n is an even or odd integer.

(i) If n is an even integer the form of the given trigonometrical ratio will remain the same i.e.,

sin (n ∙ 90° + θ) = sin θ
sin (n ∙ 90° - θ) = - sin θ; cos (n ∙ 90° + θ) = cos θ; cos (n ∙ 90° - θ) = - cos θ; tan (n ∙ 90° + θ) = tan θ; tan (n ∙ 90° - θ) = - tan θ. |
csc (n ∙ 90° + θ) = csc θ
csc (n ∙ 90° - θ) = - csc θ; sec (n ∙ 90° + θ) = sec θ; sec (n ∙ 90° - θ) = - sec θ; cot (n ∙ 90° + θ) = cot θ; cot (n ∙ 90° - θ) = - cot θ. |

(ii) If n is an odd integer then the form of the given trigonometrical ratio is altered i.e.,

sin changes to cos;
i.e., sin (n ∙ 90° + θ) = cos θ or, sin (n ∙ 90° - θ) = - cos θ |
csc changes to sec;
i.e., csc (n ∙ 90° + θ) = sec θ or, csc (n ∙ 90° - θ) = - sec θ |

cos changes to sin;
i.e., cos (n ∙ 90° + θ) = sin θ or, cos (n ∙ 90° - θ) = - sin θ |
sec changes to csc;
i.e., sec (n ∙ 90° + θ) = csc θ or, sec (n ∙ 90° - θ) = - csc θ |

tan changes to cot;
i.e., tan (n ∙ 90° + θ) = cot θ or, tan (n ∙ 90° - θ) = - cot θ |
cot changes to tan;
i.e., cot (n ∙ 90° + θ) = tan θ or, cot (n ∙ 90° - θ) = - tan θ |

**●** **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**

**11 and 12 Grade Math**

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