Evaluate
(a) \int \cos ^2 t \mathrm{~d} t
(b) \int \sin ^2 t \mathrm{~d} t
Powers of trigonometric functions, for example \sin ^2 t, do not appear in the table of standard integrals. What we must attempt to do is rewrite the integrand to obtain a standard form.
(a) From Table 3.1
\cos ^2 t=\frac{1+\cos 2 t}{2}
and so
\begin{aligned}\int \cos ^2 t \mathrm{~d} t & =\int \frac{1+\cos 2 t}{2} \mathrm{~d} t \\& =\int \frac{1}{2} \mathrm{~d} t+\int \frac{\cos 2 t}{2} \mathrm{~d} t \\& =\frac{t}{2}+\frac{\sin 2 t}{4}+c\end{aligned}
(b)
Table 3.1 Common trigonometric identities. |
\begin{aligned}& \tan A=\frac{\sin A}{\cos A} \\& \sin (A \pm B)=\sin A \cos B \pm \sin B \cos A \\& \cos (A \pm B)=\cos A \cos B \mp \sin A \sin B \\& \tan (A \pm B)=\frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} \\& 2 \sin A \cos B=\sin (A+B)+\sin (A-B) \\& 2 \cos A \cos B=\cos (A+B)+\cos (A-B) \\& 2 \sin A \sin B=\cos (A-B)-\cos (A+B) \\& \sin ^2 A+\cos ^2 A=1 \\& 1+\cot ^2 A=\operatorname{cosec^ 2} A \\& \tan ^2 A+1=\sec ^2 A \\& \cos 2 A=1-2 \sin ^2 A=2 \cos ^2 A-1=\cos ^2 A-\sin ^2 A \\& \sin 2 A=2 \sin A \cos A \\& \sin ^2 A=\frac{1-\cos 2 A}{2} \\& \cos ^2 A=\frac{1+\cos 2 A}{2}\end{aligned} |
Note: \sin ^2 A is the notation used for (\sin A)^2. Similarly \cos ^2 A means (\cos A)^2. |