Question

How do you use the half angle identity to find the exact value of $\cos 75$ degrees?

Answer 1

The given degree is$\cos 75$ degrees.
We find the exact values of $\cos 75$ degrees using the half angle formula.
Half angle identities are closely related to the double angle identities. We can use half angle identities when we have an angle that is half the size of a special angle.
We use the trigonometric ratios of an angle of the form $\cos ({180^ \circ } - {30^ \circ })$

Complete step-by-step solution:
The given degree is $\cos 75$ degrees.
We find the exact values of $\cos 75$ degrees using the half angle formula.
Let us consider $\cos {75^ \circ } = \cos t$
Multiply $2$ by $t$, hence we get
$\Rightarrow \cos 2t = \cos (2 \times 75)$
Multiply $2$ by $75$, hence we get
$\Rightarrow \cos 2t = \cos {150^ \circ }$
We rewrite $\cos {150^ \circ }$, hence we get
$\Rightarrow \cos {150^ \circ } = \cos ( - {30^ \circ } + {180^ \circ })$
We use the trigonometric ratios of an angle of the form $\cos ({180^ \circ } - {30^ \circ })$, hence we get
$\Rightarrow \cos ({180^ \circ } - {30^ \circ }) = - \cos {30^ \circ }$
$\Rightarrow \cos {30^ \circ } = - \dfrac{{\sqrt 3 }}{2}$
Use trigonometric identity: $\cos 2t = 2{\cos ^2}t - 1$
We apply $\cos 2t$ in the identity, hence we get
$\Rightarrow - \dfrac{{\sqrt 3 }}{2} = 2{\cos ^2}t - 1$
Factor the constant term, hence we get
$\Rightarrow 2{\cos ^2}t = 1 - \dfrac{{\sqrt 3 }}{2}$
Now take LCM in RHS (Right Hand Side), hence we get
$\Rightarrow 2{\cos ^2}t = \dfrac{{2 - \sqrt 3 }}{2}$
Divide by $2$ on both sides, hence we get
$\Rightarrow \dfrac{{\not{2}}}{{\not{2}}}{\cos ^2}t = \dfrac{{2 - \sqrt 3 }}{{2 \times 2}}$
Multiply the denominator, hence we get
$\Rightarrow {\cos ^2}t = \dfrac{{2 - \sqrt 3 }}{4}$
Take square root on both sides, hence we get
$\Rightarrow \cos t = \pm \dfrac{{\sqrt {2 - \sqrt 3 } }}{2}$

Hence, we get
$\cos t = \cos {75^ \circ } = \pm \dfrac{{\sqrt {2 - \sqrt 3 } }}{2}$

Note: The half angle identities are often used to replace squared trigonometric function by a non-squared trigonometric function.
Half angle identities allow us to find the value of the sine and cosine of half angle if we know the value of the cosine of the original angle.
Instead of constructing the triangle, we can also find the value of cosine using $\cos \theta = \pm \sqrt {1 - {{\sin }^2}\theta }$ formula.