# Compute the Laplace transform of f(t)=\cos (Rt-7) take R as 70

Compute the Laplace transform of $$\displaystyle{f{{\left({t}\right)}}}={\cos{{\left({R}{t}-{7}\right)}}}$$ take R as 70

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Solution:
Given: $$\displaystyle{f{{\left({t}\right)}}}={\cos{{\left({R}{t}-{7}\right)}}}$$ take R as 70
$$\displaystyle\Rightarrow{f{{\left({t}\right)}}}={\cos{{\left({70}{t}-{7}\right)}}}$$
Step 2
Note that $$\displaystyle{\cos{{\left({A}-{B}\right)}}}={\cos{{A}}}\cdot{\cos{{B}}}+{\sin{{A}}}\cdot{\sin{{B}}}$$
$$\displaystyle\Rightarrow{\cos{{\left({70}{t}-{7}\right)}}}={\cos{{\left({70}{t}\right)}}}\cdot{\cos{{7}}}+{\sin{{\left({70}{t}\right)}}}\cdot{\sin{{7}}}$$
Now , $$L\left\{f(t)\right\}$$
$$=L\left\{\cos(70t-7)\right\}$$
$$=L\left\{\cos(70t) \cdot \cos 7 + \sin (70t) \cdot \sin 7 \right\}$$
$$=\cos 7 \cdot L\left\{\cos(70t)\right\}+\sin 7 \cdot L\left\{\sin(70t)\right\}$$
$$\displaystyle{\left(\because{\sin{{7}}}\ \text{ and }\ {\cos{{7}}}\ \text{ are constants}\right)}$$
$$\displaystyle={\cos{{7}}}{\left[{\frac{{{s}}}{{{s}^{{2}}+{\left({70}\right)}^{{2}}}}}\right]}+{\sin{{7}}}{\left[{\frac{{{70}}}{{{s}^{{2}}+{\left({70}\right)}^{{2}}}}}\right]}$$
$$(\because L\left\{\sin at\right\}=\frac{a}{s^2+a^2} , L\left\{\cos at\right\}=\frac{s}{s^2+a^2})$$
$$\displaystyle={\frac{{{s}\cdot{\cos{{\left({7}\right)}}}}}{{{s}^{{2}}+{4900}}}}+{\frac{{{70}\cdot{\sin{{\left({7}\right)}}}}}{{{s}^{{2}}+{4900}}}}$$
Hence, Laplace transform of $$\displaystyle{f{{\left({t}\right)}}}={\cos{{\left({70}{t}-{7}\right)}}}\ \text{ is }\ {\frac{{{s}\cdot{\cos{{\left({7}\right)}}}+{70}\cdot{\sin{{\left({7}\right)}}}}}{{{s}^{{2}}+{4900}}}}$$