Chapter 4.3, Problem 53E

a)

To determine

To calculate:The exact value of each indicated trigonometric function by using the given function.

Answer to Problem 53E

  $\sin \theta =\frac{2\sqrt{2}}{3}$

Explanation of Solution

Given: $\cos \theta =\frac{1}{3}$

Concept Used: Using Pythagorean identity ${\sin }^2\theta +{\cos }^2\theta =1$

Calculation: we have $\cos \theta =\frac{1}{3}$

  ${\sin }^2\theta +{\cos }^2\theta =1$ (Pythagorean identity)

  ${\sin }^2\theta +{\left(\frac{1}{3}\right)}^2=1$ (Putting value $\cos \theta =\frac{1}{3}$ )

  ${\sin }^2\theta =1-\frac{1}{9}$ (Simplify)${\sin }^2\theta =\frac{8}{9}$ (Simplify)$\sin \theta =\frac{2\sqrt{2}}{3}$ (Taking square root)

Conclusion: $\sin \theta =\frac{2\sqrt{2}}{3}$

b)

To determine

To calculate:The exact value of each indicated trigonometric function by using the given function

Answer to Problem 53E

  $\tan \theta =2\sqrt{2}$

Explanation of Solution

Given: $\cos \theta =\frac{1}{3}$

Concept Used: Using Quotient identity $\tan \theta =\frac{\sin \theta }{\cos \theta }$

Calculation: We have $\cos \theta =\frac{1}{3},\sin \theta =\frac{2\sqrt{2}}{3}$

  $\tan \theta =\frac{\sin \theta }{\cos \theta }$ (Quotient identity)

  $\tan \theta =\frac{\frac{2\sqrt{2}}{3}}{\frac{1}{3}}$ (Putting values $\cos \theta =\frac{1}{3}$ and $\sin \theta =\frac{2\sqrt{2}}{3}$ )

  $\tan \theta =\frac{2\sqrt{2}}{3}.\frac{3}{1}=2\sqrt{2}$ (Simplify)

  $\tan \theta =2\sqrt{2}$

Conclusion: $\tan \theta =2\sqrt{2}$

c)

To determine

To calculate:The exact value of each indicated trigonometric function by using the given function.

Answer to Problem 53E

  $\sec \theta =3$

Explanation of Solution

Given: $\cos \theta =\frac{1}{3}$

Concept Used: Using Reciprocal identity $\sec \theta =\frac{1}{\cos \theta }$

Calculation: We have $\cos \theta =\frac{1}{3}$

  $\sec \theta =\frac{1}{\cos \theta }$ (Reciprocal identity)

  $\sec \theta =\frac{1}{\frac{1}{3}}=3$ (Putting value of $\cos \theta =\frac{1}{3}$ )

  $\sec \theta =3$

Conclusion: $\sec \theta =3$

d)

To determine

To calculate:The exact value of each indicated trigonometric function by using the given function

Answer to Problem 53E

  $\csc \left(90-\theta \right)=\frac{2\sqrt{2}}{3}$

Explanation of Solution

Given: $\cos \theta =\frac{1}{3}$

Concept Use: Using Trigonometric ratios $\csc \left(90-\theta \right)=\sin \theta$ (co functions of complimentary angles are equal).

Calculation: We have $\sin \theta =\frac{2\sqrt{2}}{3}$ (As proved above)$\csc \left(90-\theta \right)=\sin \theta =\frac{2\sqrt{2}}{3}$

  $\csc \left(90-\theta \right)=\frac{2\sqrt{2}}{3}$

Conclusion: $\csc \left(90-\theta \right)=\frac{2\sqrt{2}}{3}$