Question

How do you find the exact value of $\sin {85^ \circ }$ using the sum and difference, double angle or half angle formulas?

Answer 1

Hint: For solving this particular question , you have to simplify the given expression by reordering the equation , applying trigonometric identity , taking square root. We cannot find the exact value of $sin85$ using the sum and difference , double angle, or half angle formulas. While solving for this we get quadratic expressions with complex roots. We have an approximate value that is \[0.99619469\].

Complete step by step solution:
We know that ,
$
  \cos (6x) = 32{\cos ^6}(x) - 48{\cos ^4}(x) + 18{\cos ^2}(x) - 1 \\
   \Rightarrow \cos (6 \cdot 85) = \cos (510) = \cos (150) = - \dfrac{{\sqrt 3 }}{2} \\
   = 32{\cos ^6}(85) - 48{\cos ^4}(85) + 18{\cos ^2}(85) - 1 \\
 $
Put $\cos (85) = y$ , we will get the following ,
$32{y^6} - 48{y^4} + 18{y^2} + \dfrac{{\sqrt 3 }}{2} - 1 = 0$
For more simplification, put ${y^2} = z$ ,
$32{z^3} - 48{z^2} + 18z + \dfrac{{\sqrt 3 }}{2} - 1 = 0$
Now, if you are able to find $z$ , then you can easily get $y$and then finally you will get the value of $\sin 85$ by using the identity that is ${\sin ^2}x + {\cos ^2}x = 1$ .

Additional Information:
In arithmetic, pure mathematics identities are equalities that involve pure mathematics functions and are true for every worth of the occurring variables that every aspect of the equality is outlined. Geometrically, these are identities involving sure functions of one or additional angles. We have a trigonometry formula which says $\sin \theta = \dfrac{p}{h}$ , where $p$ represent length of perpendicular side and $h$represent length of hypotenuse side. We have another trigonometry formula which says $\cos \theta = \dfrac{b}{h}$ , where $b$ represents length of base and $h$ represent length of hypotenuse side.

Note: Identities are helpful whenever expressions involving pure mathematics functions should be simplified. An important application is that the combination of non-trigonometric functions: a typical technique involves initial mistreatment the substitution rule with a mathematical relation, then simplifying the ensuing integral with a pure mathematics identity.