Question

trigonometric ratio of complementary angle

3 cotton 31 tan 15 cot 27 tan 75 cot 63 cot 59

​

Answer 1

Correct Question:

3cot(31) tan(15) cot(27) tan(75) cot(63) cot(59)

Your Answer:

We know that

• cot(90-A) = tan A
• tan(90-A) = cot A

So,

• cot 31 = cot ( 90 - 59 ) = tan 59
• tan 15 = tan ( 90 - 75 ) = cot 75
• cot 27 = cot ( 90 - 63 ) = tan 63

Replacing Values in the Equation

= 3 tan 59. cot 75. tan 63.  tan 75. cot 63. cot 59

Taking them in a correct order

= 3 tan 59. cot 59 . cot 75. tan 75. tan 63. cot 63

We also know that cot A. tan A = 1

= 3 (1)(1)(1)

= 3

So, Answer is 3

More to know:

• cos A = sin ( 90 - A )
• sin A = cos ( 90 - A )
• sec A = cosec ( 90 - A )
• cosec A = sec ( 90 - A )
• sin A / cos A = tan A
• cos A / sin A = cot A
• 1/cos A = sec A
• 1/sin A = cosec A

Answer 2

$\large\mathcal{\underbrace{\red{SOLUTION:-}}}$

TO FIND :-

• The value of, $\rm{3\:\cot{31}\:.\tan{15}\:.\cot{27}\:.\tan{75}\:.\cot{63}\:.\cot{59}\:}$

FORMULA :-

$\mathbb{\underline{\red{Complement\:angles\::-}}}$

✍️ Two angles are said to be complementary if their sum is ‘90°’ . Thus ‘θ’ and ‘(90 - θ)°’ are complementary angles .

☞ Complementary angles in trigonometric function are,

• $\rm{\sin(90\:-\:\theta)\:=\:\cos\theta\:}$

• $\rm{\cos(90\:-\:\theta)\:=\:\sin\theta\:}$

• $\rm{\tan(90\:-\:\theta)\:=\:\cot\theta\:}$

• $\rm{\cot(90\:-\:\theta)\:=\:\tan\theta\:}$

• $\rm{\sec(90\:-\:\theta)\:=\:\cosec\theta\:}$

• $\rm{\cosec(90\:-\:\theta)\:=\:\sec\theta\:}$

✍️ Some trigonometric identities .

$\checkmark\:\rm{\underline{\sin\theta\:.\:\cosec\theta\:=\:1\:}}$

$\checkmark\:\rm{\underline{\cos\theta\:.\:\sec\theta\:=\:1\:}}$

$\checkmark\:\rm{\underline{\tan\theta\:.\:\cot\theta\:=\:1\:}}$

CALCULATION :-

• $\rm{\cot(90\:-\:59)\:=\:\tan59\:}$

• $\rm{\cot(90\:-\:63)\:=\:\tan63\:}$

• $\rm{\tan(90\:-\:75)\:=\:\cot75\:}$

$\checkmark\:\rm{\underline{3\:\cot31\:.\tan15\:.\cot27\:.\tan75\:.\cot63\:.\cot59\:}}$

$\longrightarrow\:\rm{3\:\cot(90\:-\:59)\:.\tan(90\:-\:75)\:.\cot(90\:-\:63)\:.\tan75\:.\cot63\:.\cot59\:}$

$\longrightarrow\:\rm{3\:\tan59\:.\cot75\:.\tan63\:.\tan75\:.\cot63\:.\cot59\:}$

$\longrightarrow\:\rm{3\:(\tan59\:.\cot59)\:.(\cot75\:.\tan75)\:.(\tan63\:.\cot63)\:}$

$\longrightarrow\:\rm{3\:\times{1}\:\times{1}\:\times{1}\:}$

$\longrightarrow\:\rm{\underline{\blue{3}}}$

$\bigstar\:\bf{\underline{\green{\boxed{Required\:Answer\::\:3\:}}}}$