Question

find the value for this​

Answer 1

$\begin{gathered}\begin{gathered}\bf \: To\:find - \begin{cases} &\tt{sin7\dfrac{1}{2} \degree \:cos37\dfrac{1}{2}\degree}  \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\Large{\bold{\blue{\underline{Formula \: Used \::}}}} \end{gathered}$

$(1). \: \boxed{ \green{ \bf \: 2sinx \: cosy \: = sin(x + y) + sin(x - y)}}$

$(2). \: \boxed{ \green{ \bf \:sin( - x) = - \: sinx }}$

$\large\underline\purple{\bold{Solution :- }}$

$\tt \longmapsto\:sin7\dfrac{1}{2} \degree \:cos37\dfrac{1}{2}\degree$

Multiply and divide by 2, we get

$\tt \longmapsto\: = \dfrac{1}{2}(2sin7\dfrac{1}{2} \degree \:cos37\dfrac{1}{2}\degree)$

$\tt \longmapsto\: = \dfrac{1}{2} \bigg( sin(7\dfrac{1}{2} \degree + \: 37\dfrac{1}{2}\degree) \: + sin(7\dfrac{1}{2} \degree \: - 37\dfrac{1}{2}\degree)\bigg)$

$\tt \longmapsto\: = \dfrac{1}{2} \{sin45\degree + sin( - 30\degree) \}$

$\tt \longmapsto\: = \dfrac{1}{2} (sin45\degree - sin30\degree)$

$\tt \longmapsto\: = \dfrac{1}{2} (\dfrac{1}{ \sqrt{2} } - \dfrac{1}{2} )$

$\tt \longmapsto\: = \dfrac{1}{2} \bigg(\dfrac{ \sqrt{2} - 1 }{2} \bigg)$

$\tt \longmapsto\: = \: \dfrac{ \sqrt{2} - 1}{4}$

$\rm :\implies\:\boxed{ \green{ \bf \: sin7\dfrac{1}{2} \degree \:cos37\dfrac{1}{2}\degree \: = \dfrac{ \sqrt{2} - 1 }{4} }}$

Additional Information :-

Trigonometry Formulas

sin(−θ) = −sin θ

cos(−θ) = cos θ

tan(−θ) = −tan θ

cosec(−θ) = −cosecθ

sec(−θ) = sec θ

cot(−θ) = −cot θ

Product to Sum Formulas

sin x sin y = 1/2 [cos(x–y) − cos(x+y)]

cos x cos y = 1/2[cos(x–y) + cos(x+y)]

sin x cos y = 1/2[sin(x+y) + sin(x−y)]

cos x sin y = 1/2[sin(x+y) – sin(x−y)]

Sum to Product Formulas

sin x + sin y = 2 sin [(x+y)/2] cos [(x-y)/2]

sin x – sin y = 2 cos [(x+y)/2] sin [(x-y)/2]

cos x + cos y = 2 cos [(x+y)/2] cos [(x-y)/2]

cos x – cos y = -2 sin [(x+y)/2] sin [(x-y)/2]

Sum or Difference of angles

cos (A + B) = cos A cos B – sin A sin B

cos (A – B) = cos A cos B + sin A sin B

sin (A+B) = sin A cos B + cos A sin B

sin (A -B) = sin A cos B – cos A sin B

tan(A+B) = [(tan A + tan B)/(1 – tan A tan B)]

tan(A-B) = [(tan A – tan B)/(1 + tan A tan B)]

cot(A+B) = [(cot A cot B − 1)/(cot B + cot A)]

cot(A-B) = [(cot A cot B + 1)/(cot B – cot A)]

cos(A+B) cos(A–B)=cos^2A–sin^2B=cos^2B–sin^2A

sin(A+B) sin(A–B) = sin^2A–sin^2B=cos^2B–cos^2A

Multiple and Submultiple angles

sin2A = 2sinA cosA = [2tan A /(1+tan²A)]

cos2A = cos²A–sin²A = 1–2sin²A = 2cos²A–1= [(1-tan²A)/(1+tan²A)]

tan 2A = (2 tan A)/(1-tan²A)