Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Applications of Trigonometric Formulas

Properties of Triangle:

In any triangle ABC the sides  \overline{AB},\overline{BC}, \text{ and }\overline{AC} having length  c, a \text { and } b respectively.

\to  We know, the sum of lengths of two sides of a triangle is greater than the length of the third side.

\to The sum of measures of the internal triangles is equal to    180^o \text { or }\pi

(i) Sine Formula:

In any  \triangle ABC,  \dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}

Some Important Results:

(A) In any \triangle ABC, \dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}=2R , where  R is the radius of the circumscribing circle of \triangle ABC .

(B) If  \triangle denotes the area of the \triangle ABC , the \triangle =\dfrac{abc}{4R} .

(ii) Cosine Formula:

In any  \triangle ABC,

(a)  a^2=b^2+c^2-2bc \cos A

(b) b^2=c^2+a^2-2ca \cos B

(c) c^2=a^2+b^2-2ab \cos C

(iii) Tangent Formula:

In any  \triangle ABC,

(a)  \tan \dfrac{B-C}{2}=\dfrac{b-c}{b+c}\cot \dfrac{A}{2}

(b) \tan \dfrac{C-A}{2}=\dfrac{c-a}{c+a} \cot \dfrac{B}{2}

(c) \tan \dfrac{A-B}{2}=\dfrac{a-b}{a+b} \cot \dfrac{C}{2}

It is also known as Napier’s Formula.

(iv) Projection Formula:

In any \triangle ABC ,

(a)  a= b \cos C+c \cos B

(b) b= c \cos A+a \cos C

(c)  c=a \cos B +b \cos A

(v) Area Formula:

In any \triangle ABC ,

The area of \triangle ABC is given by  \triangle =\sqrt{S(s-a)(S-b)(S-c)}, \text { where } 2S=a+b+c is the perimeter of the triangle.

(I) (i) \sin \dfrac{A}{2}=\sqrt{\dfrac{(S-b)(S-c)}{bc}},\quad \sin \dfrac{B}{2}=\sqrt{\dfrac{(S-c)(S-a)}{ca}}, \quad\sin \dfrac{C}{2}=\sqrt{\dfrac{(S-a)(S-b)}{ab}}

(ii) \cos \dfrac{A}{2}=\sqrt{\dfrac{S(S-a)}{bc}}, \quad \cos \dfrac{B}{2}=\sqrt{\dfrac{S(S-b)}{ca}}, \quad \cos \dfrac{C}{2}=\sqrt{\dfrac{S(S-c)}{ab}}

(iii) \tan \dfrac{A}{2}=\sqrt{\dfrac{(S-b)(S-c)}{S(S-a)}}, \quad \tan \dfrac{B}{2}=\sqrt{\dfrac{(S-c)(S-a)}{S(S-b)}}, \quad \tan \dfrac{C}{2}=\sqrt{\dfrac{(S-a)(S-b)}{S(S-c)}}

(II) (i)  \sin A=\dfrac{2}{bc}\sqrt{S(S-a)(S-b)(S-c)}=\dfrac{2\triangle}{bc}

(ii) \sin B=\dfrac{2}{ca}\sqrt{S(S-a)(S-b)(S-c)}=\dfrac{2\triangle}{ca}

(iii) \sin C=\dfrac{2}{ab}\sqrt{S(S-a)(S-b)(S-c)}=\dfrac{2\triangle}{ab}

(III)  (i)  \tan \dfrac{A}{2}=\dfrac{\triangle}{S(S-a)}

(ii) \tan \dfrac{B}{2}=\dfrac{\triangle}{S(S-b)}

(ii) \tan \dfrac{C}{2}=\dfrac{\triangle}{S(S-a)}

 \sin \theta \cdot \sin (60-\theta)\cdot \sin (60+\theta)=\dfrac{1}{4}\sin 3 \theta

\cos \theta \cdot \cos (60-\theta)\cdot \cos (60+\theta)=\dfrac{1}{4}\cos 3 \theta

 \tan \theta \cdot \tan (60-\theta)\cdot \tan (60+\theta)=\tan 3 \theta


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