Alyssa Hanson

2023-03-18

How to find the exact values of cos(5pi/12) using the half angle formula?

Krasiniecb8rw

Beginner2023-03-19Added 7 answers

Using the formula for a half-angle:

$\text{XXXX}$$\mathrm{cos}\left(\frac{\theta}{2}\right)=\pm \sqrt{\frac{1+\mathrm{cos}\left(\theta \right)}{2}}$

If $\frac{\theta}{2}=\frac{5\pi}{12}$

$\text{XXXX}$then $\theta =\frac{5\pi}{6}$

Note that $\frac{5\pi}{6}$ is a standard angle in quadrant 2 with a reference angle of $\frac{\pi}{6}$

so $\mathrm{cos}\left(\frac{5\pi}{6}\right)=-\mathrm{cos}\left(\frac{\pi}{6}\right)=-\frac{\sqrt{3}}{2}$

Therefore

$\text{XXXX}}\mathrm{cos}\left(\frac{5\pi}{12}\right)=\pm \sqrt{\frac{1-\frac{\sqrt{3}}{2}}{2}$

$\text{XXXXXXXXXXX}}=\pm \sqrt{\frac{\frac{2-\sqrt{3}}{2}}{2}$

$\text{XXXXXXXXXXX}}=\pm \sqrt{\frac{2-\sqrt{3}}{4}$

$\text{XXXXXXXXXXX}}=\pm \frac{\sqrt{2-\sqrt{3}}}{2$

Since $\frac{5\pi}{12}<\frac{\pi}{2}$

$\text{XXXX}$$\frac{5\pi}{12}$ is in quadrant 1

$\text{XXXX}$$\to \mathrm{cos}\left(\frac{5\pi}{12}\right)$ is positive

$\text{XXXX}$(the negative solution is extraneous)

$\text{XXXX}$$\mathrm{cos}\left(\frac{\theta}{2}\right)=\pm \sqrt{\frac{1+\mathrm{cos}\left(\theta \right)}{2}}$

If $\frac{\theta}{2}=\frac{5\pi}{12}$

$\text{XXXX}$then $\theta =\frac{5\pi}{6}$

Note that $\frac{5\pi}{6}$ is a standard angle in quadrant 2 with a reference angle of $\frac{\pi}{6}$

so $\mathrm{cos}\left(\frac{5\pi}{6}\right)=-\mathrm{cos}\left(\frac{\pi}{6}\right)=-\frac{\sqrt{3}}{2}$

Therefore

$\text{XXXX}}\mathrm{cos}\left(\frac{5\pi}{12}\right)=\pm \sqrt{\frac{1-\frac{\sqrt{3}}{2}}{2}$

$\text{XXXXXXXXXXX}}=\pm \sqrt{\frac{\frac{2-\sqrt{3}}{2}}{2}$

$\text{XXXXXXXXXXX}}=\pm \sqrt{\frac{2-\sqrt{3}}{4}$

$\text{XXXXXXXXXXX}}=\pm \frac{\sqrt{2-\sqrt{3}}}{2$

Since $\frac{5\pi}{12}<\frac{\pi}{2}$

$\text{XXXX}$$\frac{5\pi}{12}$ is in quadrant 1

$\text{XXXX}$$\to \mathrm{cos}\left(\frac{5\pi}{12}\right)$ is positive

$\text{XXXX}$(the negative solution is extraneous)

Find an equation of the plane. The plane through the points (2, 1, 2), (3, −8, 6), and (−2, −3, 1), help please

A consumer in a grocery store pushes a cart with a force of 35 N directed at an angle of $25}^{\circ$ below the horizontal. The force is just enough to overcome various frictional forces, so the cart moves at a steady pace. Find the work done by the shopper as she moves down a $50.0-m$ length aisle.

??What is the derivative of $\mathrm{arcsin}\left[{x}^{\frac{1}{2}}\right]$?

What is the derivative of $y=\mathrm{arcsin}\left(\frac{3x}{4}\right)$?

Determine if the graph is symmetric about the $x$-axis, the $y$-axis, or the origin.$r=4\mathrm{cos}3\theta $.

How to differentiate $1+{\mathrm{cos}}^{2}\left(x\right)$?

What is the domain and range of $\left|\mathrm{cos}x\right|$?

How to find the value of $\mathrm{csc}74$?

How to evaluate $\mathrm{sec}\left(\pi \right)$?

Using suitable identity solve (0.99)raised to the power 2.

How to find the derivative of $y=\mathrm{tan}\left(3x\right)$?

Find the point (x,y) on the unit circle that corresponds to the real number t=pi/4

How to differentiate ${\mathrm{sin}}^{3}x$?

A,B,C are three angles of triangle. If A -B=15, B-C=30. Find A , B, C.

Find the value of $\mathrm{sin}{270}^{\circ}$.