# Half Angle Calculator

### Half Angle Identity Tutorial

**The formulas for half angle identities are as follows:**

\text{sin}(\frac{\text{u}}{2})=\sqrt{\frac{1-\text{cos(u)}}{2}}

\text{cos}(\frac{\text{u}}{2})=\sqrt{\frac{1+\text{cosu}}{2}}

\text{tan}(\frac{\text{u}}{2})=\frac{1-\text{cos(u)}}{\text{sin(u)}}

\text{csc}(\frac{\text{u}}{2})=\sqrt{\frac{2}{1-\text{cos(u)}}}

\text{sec}(\frac{\text{u}}{2})=\sqrt{\frac{2}{1+\text{cos(u)}}}

\text{cot}(\frac{\text{u}}{2})=\frac{1+\text{cos(u)}}{\text{sin(u)}}

If you are given one trigonometric function, value, and quadrant, you are capable of finding the exact values of each and every half angle identity. For example, if told to find the exact value of $\text{cos}(\frac{\text{u}}{2})$ given $\text{csc(u)}=\frac{13}{12}$, for $0\leq\text{u}\leq\frac{\pi}{2}$ (u is in quadrant 1). You may begin solving the problem by examining the equation for $\cos\frac{\text{u}}{2}$. According to the formula for $\cos\frac{\text{u}}{2}$, the only value required is cosu. The second part of the problem is finding the exact value of cosu by using the given function and its value which is cscu = $\frac{13}{12}$. We know that csc = $\frac{\text{hypotenuse}}{\text{opposite}}$, the only unknown value is the adjacent side length. This can be calculated by using the pythagrean theorem which states:

\text{adjacent}=\sqrt{\text{hypotenuse}^2-\text{opposite}^2}

\text{adjacent}=\sqrt{(13)^2-(12)^2}

\text{adjacent}=5

\text{(in simplest form)}

Now that the adjacent side length is known, we can find the value of cosu:

\text{cos}=\frac{\text{adjacent}}{\text{hypotenuse}}

\text{cos}=\frac{(5)}{(13)}

\text{cos}=\frac{5}{13}

\text{(in simplest form)}

The final part of the half angle calculation process is substituting the found cos value into the $\text{cos}\frac{\text(u)}{2}$ equation as follows:

\text{cos}\frac{u}{2}=\sqrt{\frac{1+\text{cosu}}{2}}

\text{cos}\frac{u}{2}=\sqrt{\frac{1+(\frac{5}{13})}{2}}

\text{cos}\frac{u}{2}=\frac{3\sqrt{13}}{13}

\text{(in simplest form)}

The final value of $\text{cos}\frac{u}{2}$ is $\frac{3\sqrt{13}}{13}$.

### Half Angle Calculator Tutorial

choose formulaThe first and most obvious step in using the half angle calculator is to choose which identity you would like to calculate from the dropdown list. After having chosen an identity, you may choose which function is given and its value. The final step in using the caclculator is choosing the quadrant of the central angle. It is important that the quadrant is a valid quadrant relative to the given function and its value. the value of sin must be positive in quadrants 1 and 2, and negative in quadrants 3 and 4, the value of cos must be positive in quadrants 1 and 4, and negative in quadrants 2 and 3, the values of tan must be positive in quadrants 1 and 3, and be negative in quadrants 2 and 4. To use the identity calculator without a given function and its value is a lot simpler. Again, you begin by choosing a function from the dropdown list. After this you may enter any value for U and get the decimal approximation of the trig function. For example the value of $\text{cos}\frac{5}{2}=0.043619387365336$.