Chapter 12

Karla said that if \(\cos A\) is positive, then \(-\frac{\pi}{2} < A < \frac{\pi}{2},-\frac{\pi}{4} < \frac{1}{2} A < \frac{\pi}{4},\) and \(\cos \frac{1}{2} A\) is positive. Do you agree with Karla? Explain why or why not.

Does \(\cos 2 \theta=\sin 2\left(90^{\circ}-\theta\right) ?\) Justify your answer.

Explain why the identity \(\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B}\) is not valid when \(A\) or \(B\) is equal to \(\frac{\pi}{2}+n \pi\) for any integer \(n .\)

William said that \(\sin (A+B)+\sin (A-B)=\sin 2 A .\) Do you agree with William? Justify your answer.

Is \(\sin \theta=\sqrt{1-\cos ^{2} \theta}\) an identity? Explain why or why not.

Maggie said that \(\cos (A+B)+\cos (A-B)=\cos 2 A .\) Do you agree with Maggie? Justify your answer.

Are the equations \(\sin \theta=\cos \left(90^{\circ}-\theta\right)\) and \(\cos \theta=\sin \left(90^{\circ}-\theta\right)\) true for all real numbers or only for values of \(\theta\) in the interval \(0<\theta<90^{\circ} ?\)

If we know the value of \(\sin \theta,\) is it possible to find the other five trigonometric function values? If not, what other information is needed?

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

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

Freddy said that \(\sin (A+B)+\sin (A-B)=2 \sin A \cos B .\) Do you agree with Freddy? Justify 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.

Germaine said \(\cos (A+B)+\cos (A-B)=2 \cos A \cos B .\) Do you agree with Germaine? Justify your answer.

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.

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.

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 \(3-8,\) for each value of \(\theta,\) use double-angle formulas to find a. \(\sin 2 \theta,\) b. \(\cos 2 \theta,\) c. \(\tan 2 \theta .\) Show all work. $$ \theta=30^{\circ} $$

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

\(\ln 3-17,\) 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 \(3-17,\) find the exact value of \(\cos (A+B)\) for each given pair of values. \(A=90^{\circ}, B=60^{\circ}\)

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

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

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. $$ \theta=120^{\circ} $$

In \(3-8,\) for each value of \(\theta,\) use double-angle formulas to find a. \(\sin 2 \theta,\) b. \(\cos 2 \theta,\) c. \(\tan 2 \theta .\) Show all work. $$ \theta=225^{\circ} $$

In \(3-17,\) find the exact value of \(\tan (A+B)\) and of \(\tan (A-B)\) for each given pair of values. $$ A=45^{\circ}, B=60^{\circ} $$

In \(3-26,\) prove that each equation is an identity. $$ \tan \theta \sin \theta \cos \theta=\sin ^{2} \theta $$

\(\ln 3-17,\) find the exact value of \(\sin (A-B)\) and of \(\sin (A+B)\) for each given pair of values. \(A=180^{\circ}, B=45^{\circ}\)

In \(3-17,\) find the exact value of \(\cos (A+B)\) for each given pair of values. \(A=90^{\circ}, B=45^{\circ}\)

In \(3-17,\) find the exact value of \(\cos (A-B)\) for each given pair of values. \(A=180^{\circ}, B=45^{\circ}\)

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

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. $$ \theta=300^{\circ} $$

In \(3-8,\) for each value of \(\theta,\) use double-angle formulas to find a. \(\sin 2 \theta,\) b. \(\cos 2 \theta,\) c. \(\tan 2 \theta .\) Show all work. $$ \theta=330^{\circ} $$

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

In \(3-26,\) prove that each equation is an identity. $$ \cot \theta \sin \theta \cos \theta=\cos ^{2} \theta $$

\(\ln 3-17,\) find the exact value of \(\sin (A-B)\) and of \(\sin (A+B)\) for each given pair of values. \(A=180^{\circ}, B=30^{\circ}\)}

In \(3-17,\) find the exact value of \(\cos (A+B)\) for each given pair of values. \(A=90^{\circ}, B=30^{\circ}\)

In \(3-17,\) find the exact value of \(\cos (A-B)\) for each given pair of values. \(A=180^{\circ}, B=30^{\circ}\)

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

In \(3-8,\) for each value of \(\theta,\) use double-angle formulas to find a. \(\sin 2 \theta,\) b. \(\cos 2 \theta,\) c. \(\tan 2 \theta .\) Show all work. $$ \theta=\frac{\pi}{4} $$

In \(3-17,\) find the exact value of \(\tan (A+B)\) and of \(\tan (A-B)\) for each given pair of values. $$ A=180^{\circ}, B=30^{\circ} $$

In \(3-26,\) prove that each equation is an identity. $$ \sec \theta(\cos \theta-\cot \theta)=1-\csc \theta $$

\(\ln 3-17,\) find the exact value of \(\sin (A-B)\) and of \(\sin (A+B)\) for each given pair of values. \(A=270^{\circ}, B=60^{\circ}\)

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

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

In \(3-17,\) find the exact value of \(\cos (A-B)\) for each given pair of values. \(A=270^{\circ}, B=60^{\circ}\)

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. $$ \theta=\frac{7 \pi}{2} $$

In \(3-8,\) for each value of \(\theta,\) use double-angle formulas to find a. \(\sin 2 \theta,\) b. \(\cos 2 \theta,\) c. \(\tan 2 \theta .\) Show all work. $$ \theta=\frac{7 \pi}{6} $$

In \(3-17,\) find the exact value of \(\tan (A+B)\) and of \(\tan (A-B)\) for each given pair of values. $$ A=180^{\circ}, B=45^{\circ} $$

In \(3-26,\) prove that each equation is an identity. $$ \csc \theta(\sin \theta+\tan \theta)=1+\sec \theta $$