91Ó°ÊÓ

Prove the identity \(\sin \theta+\cos \theta=\sqrt{2} \sin \left(\theta+45^{\circ}\right)\). Explain why this shows that $$ -\sqrt{2} \leq \sin \theta+\cos \theta \leq \sqrt{2} $$ for all angles \(\theta\). For which \(\theta\) between \(0^{\circ}\) and \(360^{\circ}\) would \(\sin \theta+\cos \theta\) be the largest?

Prove the given identity. $$ \frac{\tan \theta}{\cot \theta}=\tan ^{2} \theta $$

Prove the given identity. $$ \tan 3 \theta=\frac{3 \tan \theta-\tan ^{3} \theta}{1-3 \tan ^{2} \theta} $$

Prove the given identity. $$ \cos (A+B+C)=\cos A \cos B \cos C-\cos A \sin B \sin C-\sin A \cos B \sin C-\sin A \sin B \cos C $$

Prove the given identity. $$ \frac{\csc \theta}{\sin \theta}=\csc ^{2} \theta $$

Prove Mollweide's second equation: For any triangle \(\triangle A B C, \frac{a+b}{c}=\frac{\cos \frac{1}{2}(A-B)}{\sin \frac{1}{2} C}\).

Prove the given identity. $$ \frac{\cos ^{2} \psi}{\cos ^{2} \theta}=\frac{1+\cos 2 \psi}{1+\cos 2 \theta} $$

Prove the given identity. $$ \frac{\cos ^{2} \theta}{1+\sin \theta}=1-\sin \theta $$

Some trigonometry textbooks used to claim incorrectly that \(\sin \theta+\cos \theta=\sqrt{1+\sin 2 \theta}\) was an identity. Give an example of a specific angle \(\theta\) that would make that equation false. Is \(\sin \theta+\cos \theta=\pm \sqrt{1+\sin 2 \theta}\) an identity? Justify your answer.

There is a more general form for the instantaneous power \(p(t)=v(t) i(t)\) in an electrical circuit than the one in Example 3.22. The voltage \(v(t)\) and current \(i(t)\) can be given by $$ \begin{array}{l} v(t)=V_{m} \cos (\omega t+\theta) \\ i(t)=I_{m} \cos (\omega t+\phi) \end{array} $$ where \(\theta\) is called the phase angle. Show that \(p(t)\) can be written as $$ p(t)=\frac{1}{2} V_{m} I_{m} \cos (\theta-\phi)+\frac{1}{2} V_{m} I_{m} \cos (2 \omega t+\theta+\phi) $$.