### Sum-Difference Calculator Exact value of

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### Sum Difference Identity Tutorial

The formulas for Sum Difference identities are shown below:

\sin \left(\text{u}\pm \text{v}\right)=\sin \left(\text{u}\right)\cos \left(\text{v}\right)\pm \cos \left(\text{u}\right)\sin \left(\text{v}\right)

\cos \left(\text{u}\pm \text{v}\right)=\cos \left(\text{u}\right)\cos \left(\text{v}\right)\pm \sin \left(\text{u}\right)\sin \left(\text{v}\right)

\tan \left(\text{u}\pm \text{v}\right)=\frac{\text{tan(u)}\pm \text{tan(v)}}{1\pm \text{tan(u)}\cdot \text{tan(v)}}

\csc \left(\text{u}\pm \text{v}\right)=\frac{1}{\sin \left(\text{u}\right)\cos \left(\text{v}\right)\pm \cos \left(\text{u}\right)\sin \left(\text{v}\right)}

\sec \left(\text{u}\pm \text{v}\right)=\frac{1}{\cos \left(\text{u}\right)\cos \left(\text{v}\right)\pm \sin \left(\text{u}\right)\sin \left(\text{v}\right)}

\cot \left(\text{u}\pm \text{v}\right)=\frac{1\pm \text{tan(u)}\cdot \text{tan(v)}}{\text{tan(u)}\pm \text{tan(v)}}

The exact values of sum difference identities with two given functions and ratios can be calculated. If told to find the exact value of sin(u-v) given sinu = $\frac{4}{5}$ and cosv = $\frac{7}{8}$, $0$ ≤ u ≤ $\frac{\pi }{2}$, $0$ ≤ v ≤ $\frac{\pi }{2}$ you can begin solving the problem by first knowing which values are needed to solve the identity. In the sin(u-v) equation, the values sinu, cosv, sinv, cosu, and sinv are needed. Since the value of sinu and cosv are already known, the only values to be calculated are cosu and sinv. Once thes values are known, the equation can be solved. In this case angles exist in quadrant one meaning all calculated ratios will be positive values. To find the value of cosu given sinu in quadrant one, the process below can be used.

\text{sinu}=\frac{4}{5}=\frac{\text{opposite}}{\text{hypotenuse}}

\text{ opposite}=4

\text{cosu}=\frac{3}{5}

To find the exact value of sinv given cosv in quadrant one, the process below can be used:

\text{opposite}=\sqrt{8^2-7^2}

\text{opposite}=\sqrt{15}

\text{sinv}=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{\sqrt{15}}{8}

\text{sinv}=\frac{\sqrt{15}}{8}

\text{(in simplest form)}

All needed values for sin(u-v) have been found and it can now be solved. The calculation process for sin(u-v) is shown below:

\text{sin(u - v)}=\text{sinu}\cdot \text{cosv}-\text{cosu}\cdot \text{sinv}

\text{sin(u-v)}=(\frac{4}{5})(\frac{7}{8})-(\frac{3}{5})(\frac{\sqrt{15}}{8})

\text{sin(u-v)}=\frac{-3\sqrt{15}+28}{40}

\text{(in simplest form)}

### Sum Difference Identity Tutorial

Without Given Value

The sum differene identity can also be used to find the exact value of normal trig functions. For example if told to find the exact value of sin75 degrees you can use the formula for sin(u+v). The sin of 75 is also the sin of (45+30). The calculation process for sin(45+30) is shown below:
\text{sin(45 + 30)}=\sin 45\cdot \cos 30+\cos 45\cdot \sin 30

In the unit circle, the exact value of sin45 is $\frac{\sqrt{2}}{2}$, the exact value of cos30 is $\frac{\sqrt{3}}{2}$, the exact value of cos45 is $\frac{\sqrt{2}}{2}$, and the exact value of sin30 is $\frac{1}{2}$ meaning the expression can be rewritten as:
\text{sin(u-v)}=\frac{-3\sqrt{15}+28}{40}

\text{(in simplest form)}

Calculating sum and difference identites with given values may be a long process and that is why the sum difference calculator was created. This calculator returns exact values given any parameters at a very fast rate.