91Ó°ÊÓ

Emily said that, without finding the values on a calculator, she knows that \(\sin 100^{\circ}=\cos \left(-10^{\circ}\right) .\) Do you agree with Emily? Explain why or why not.

Does \(\tan 2 \theta=\frac{\sin 2 \theta}{\cos 2 \theta} ?\) Justify your answer.

a. Explain how the identities \(1+\tan ^{2} \theta=\sec ^{2} \theta\) and \(\cot ^{2} \theta+1=\csc ^{2} \theta\) can be derived from the identity \(\cos ^{2} \theta+\sin ^{2} \theta=1\) b. The identity \(\cos ^{2} \theta+\sin ^{2} \theta=1\) is true for all real numbers. Are the identities \(1+\tan ^{2} \theta=\sec ^{2} \theta\) and \(\cot ^{2} \theta+1=\csc ^{2} \theta\) also true for all real numbers? Explain your answer.

Cory said that in Example \(3,1-\sin \theta=\frac{\cos ^{2} \theta}{1+\sin \theta}\) could have been shown to be an identity by multiplying the left side by \(\frac{1+\sin \theta}{1+\sin \theta} \cdot\) Do you agree with Cory? Explain why or why not.

Explain why \(\frac{\tan A+\tan B}{1-\tan A \tan B}\) is undefined when \(A=\frac{\pi}{6}\) and \(B=\frac{\pi}{3}\)

In \(3-8,\) for each value of \(\theta,\) use half-angle formulas to find a. \(\sin \frac{1}{2} \theta\) b. \(\cos \frac{1}{2} \theta\) c. \(\tan \frac{1}{2} \theta .\) Show all work. $$ \theta=480^{\circ} $$

In \(3-17,\) find the exact value of \(\tan (A+B)\) and of \(\tan (A-B)\) for each given pair of values. $$ A=45^{\circ}, B=30^{\circ} $$

In \(3-14,\) write each expression as a single term using \(\sin \theta, \cos \theta,\) or both. $$ \tan \theta $$

\(\ln 3-17,\) find the exact value of \(\sin (A-B)\) and of \(\sin (A+B)\) for each given pair of values. \(A=180^{\circ}, B=60^{\circ}\)

In \(3-26,\) prove that each equation is an identity. $$ \sin \theta \csc \theta \cos \theta=\cos \theta $$