We will learn about the trigonometric ratios of angle \(\frac{A}{2}\) in terms of angle A.

How to express sin A, cos A and tan A in terms of \(\frac{A}{2}\)?

(i) For all values of the angle A we know that, sin 2A = 2 sin A cos A

Now replacing A by \(\frac{A}{2}\) in the above relation then we obtain the relation as,

**sin ****A = 2 ****sin ****\(\frac{A}{2}\) cos**** ****\(\frac{A}{2}\)**

(ii) For all values of the angle A we know that, cos 2A = cos\(^{2}\) A – sin\(^{2}\) A

Now replacing A by \(\frac{A}{2}\) in the above relation then we obtain the relation as,

**cos ****A = cos****\(^{2}\)**** ****\(\frac{A}{2}\) – sin****\(^{2}\)**** ****\(\frac{A}{2}\)**

(iii) For all values of the angle A we know that, cos 2A = 2 cos\(^{2}\) A - 1 or 1 + cos 2A = 2 cos\(^{2}\) A

Now replacing A by \(\frac{A}{2}\) in the above relation then we obtain the relation as,

**cos A = 2 cos****\(^{2}\) \(\frac{A}{2}\) - 1 or 1 + cos A = 2 cos****\(^{2}\) \(\frac{A}{2}\)**

(iv) For all values of the angle A we know that, cos 2A = 1 - 2 sin\(^{2}\) A or 1 - cos 2A = 2 sin\(^{2}\) A

Now replacing A by \(\frac{A}{2}\) in the above relation then we obtain the relation as,

**cos A = 1 - 2 sin****\(^{2}\) \(\frac{A}{2}\) or 1 - cos A = 2 sin****\(^{2}\) \(\frac{A}{2}\)**

(v) For all values of the angle A we know that, tan 2A = 2 tan A/1 – tan^2 A

Now replacing A by A/2 in the above relation then we obtain the relation as,

**tan A =**** \(\frac{2 tan
\frac{A}{2}}{1 - tan^{2} \frac{A}{2}}\)**

(vi) For all values of the angle A we know that, sin 2A = 2 tan A/1 + tan^2 A

Now replacing A by A/2 in the above relation then we obtain the relation as,

**sin A = \(\frac{2 tan
\frac{A}{2}}{1 + tan^{2} \frac{A}{2}}\)**

(vii) For all values of the angle A we know that, cos 2A = 1 - tan^2 A /1 + tan^2 A

Now replacing A by A/2 in the above relation then we obtain the relation as,

**cos A = \(\frac{1 -
tan^{2} \frac{A}{2}}{1 + tan^{2} \frac{A}{2}}\)**

**Note:** Formulas of trigonometric ratios of angle A in
terms of angle \(\frac{A}{2}\) is also known as sub-multiple angle.

**Trigonometric Ratios of Angle A2A2****Trigonometric Ratios of Angle****A3A3****Trigonometric Ratios of Angle A2A2 in Terms of cos A****tan A2A2 in Terms of tan A****Exact value of sin 7½°****Exact value of cos 7½°****Exact value of tan 7½°****Exact Value of cot 7½°****Exact Value of tan 11¼°****Exact Value of sin 15°****Exact Value of cos 15°****Exact Value of tan 15°****Exact Value of sin 18°****Exact Value of cos 18°****Exact Value of sin 22½°****Exact Value of cos 22½°****Exact Value of tan 22½°****Exact Value of sin 27°****Exact Value of cos 27°****Exact Value of tan 27°****Exact Value of sin 36°****Exact Value of cos 36°****Exact Value of sin 54°****Exact Value of cos 54°****Exact Value of tan 54°****Exact Value of sin 72°****Exact Value of cos 72°****Exact Value of tan 72°****Exact Value of tan 142½°****Submultiple Angle Formulae****Problems on Submultiple Angles**

**11 and 12 Grade Math**

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