Chapter 5.5, Problem 48E

a.

To determine

To find: The exact value of the given trigonometric function using the given figure.

Answer to Problem 48E

The value of the given expression is $\frac{\sqrt{2}}{10}$ .

Explanation of Solution

Given information:

The following figure:

  

The following expression:

  $\sin \frac{\theta }{2}$

Formula used:

The following half angle formula

  $\sin \frac{u}{2}=\pm \sqrt{\frac{1-\cos u}{2}}$

The following Pythagorean relation:

  $\cos \theta =\frac{\text{adjacent}}{\text{hypotenuse}}$

The following Pythagorean identity:

  ${\text{Hypotenuse}}^2={\text{Adjacent}}^2+{\text{Opposite}}^2$

Calculation:

From the given diagram,

  $\begin{array}{l}\text{Adjacent}=24\\ \text{Opposite}=7\\ \text{Hypotenuse}=\sqrt{{\left(24\right)}^2+{\left(7\right)}^2}\text{ [Pythagorean identity]}\\ =\sqrt{576+49}\text{ [Find square]}\\ =\sqrt{625}\text{ [Add]}\\ =25\text{ [Find square root]}\\ \text{cos}\theta \text{=}\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{24}{25}\text{ [Pythagorean relation]}\end{array}$

The value of the given expression can be calculated as:

  $\begin{array}{l}\sin \frac{\theta }{2}=\sqrt{\frac{1-\cos \theta }{2}}\text{ [Half angle formula]}\\ \text{ =}\sqrt{\frac{1-\left(\frac{24}{25}\right)}{2}}\text{ [Substitute value of cos}\theta \text{]}\\ =\sqrt{\frac{\frac{1}{25}}{2}}\text{ [Subtract fractions]}\\ =\sqrt{\frac{1}{50}}\text{ [Multiply numerator and denominator by (1/2)]}\\ =\frac{1}{5\sqrt{2}}\text{ [Find square root]}\\ =\frac{\sqrt{2}}{10}\text{ [Multiply numerator and denominator by }\sqrt{2}\text{]}\end{array}$

b.

To determine

To find: The exact value of the given trigonometric function using the given figure.

Answer to Problem 48E

The value of the given expression is $\frac{7\sqrt{2}}{10}$ .

Explanation of Solution

Given information:

The following figure:

  

The following expression:

  $\cos \frac{\theta }{2}$

Formula used:

The following half angle formula

  $\cos \frac{u}{2}=\pm \sqrt{\frac{1+\cos u}{2}}$

The following Pythagorean relation:

  $\cos \theta =\frac{\text{adjacent}}{\text{hypotenuse}}$

The following Pythagorean identity:

  ${\text{Hypotenuse}}^2={\text{Adjacent}}^2+{\text{Opposite}}^2$

Calculation:

From the given diagram,

  $\begin{array}{l}\text{Adjacent}=24\\ \text{Opposite}=7\\ \text{Hypotenuse}=\sqrt{{\left(24\right)}^2+{\left(7\right)}^2}\text{ [Pythagorean identity]}\\ =\sqrt{576+49}\text{ [Find square]}\\ =\sqrt{625}\text{ [Add]}\\ =25\text{ [Find square root]}\\ \text{cos}\theta \text{=}\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{24}{25}\text{ [Pythagorean relation]}\end{array}$

The value of the given expression can be calculated as:

  $\begin{array}{l}\cos \frac{\theta }{2}=\sqrt{\frac{1+\cos \theta }{2}}\text{ [Half angle formula]}\\ \text{ =}\sqrt{\frac{1+\left(\frac{24}{25}\right)}{2}}\text{ [Substitute value of cos}\theta \text{]}\\ =\sqrt{\frac{\frac{25+24}{25}}{2}}\text{ [Add fractions]}\\ =\sqrt{\frac{49}{50}}\text{ [Multiply numerator and denominator by }\sqrt{\text{25}}\text{]}\\ =\frac{7}{5\sqrt{2}}\text{ [Find square root]}\\ =\frac{7\sqrt{2}}{10}\text{ [Multiply numerator and denominator by }\sqrt{2}\text{]}\end{array}$

c.

To determine

To find: The exact value of the given trigonometric function using the given figure.

Answer to Problem 48E

The value of the given expression is $\frac{1}{7}$ .

Explanation of Solution

Given information:

The following figure:

  

The following expression:

  $\tan \frac{\theta }{2}$

Formula used:

The following half angle formula

  $\tan \frac{u}{2}=\frac{\sin u}{1+\cos u}$

The following Pythagorean relation:

  $\begin{array}{l}\cos \theta =\frac{\text{adjacent}}{\text{hypotenuse}}\\ \sin \theta =\frac{\text{opposite}}{\text{hypotenuse}}\end{array}$

The following Pythagorean identity:

  ${\text{Hypotenuse}}^2={\text{Adjacent}}^2+{\text{Opposite}}^2$

Calculation:

From the given diagram,

  $\begin{array}{l}\text{Adjacent}=24\\ \text{Opposite}=7\\ \text{Hypotenuse}=\sqrt{{\left(24\right)}^2+{\left(7\right)}^2}\text{ [Pythagorean identity]}\\ =\sqrt{576+49}\text{ [Find square]}\\ =\sqrt{625}\text{ [Add]}\\ =25\text{ [Find square root]}\\ \text{cos}\theta \text{=}\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{24}{25}\text{ [Pythagorean relation]}\\ \sin \theta \text{=}\frac{\text{opposite}}{\text{hypotenuse}}=\frac{7}{25}\text{ [Pythagorean relation]}\end{array}$

The value of the given expression can be calculated as:

  $\begin{array}{l}\tan \frac{\theta }{2}=\frac{\sin \theta }{1+\cos \theta }\text{ [Half angle formula]}\\ \text{ =}\frac{\frac{7}{25}}{1+\frac{24}{25}}\text{ [Substitute values of sin}\theta \text{ and cos}\theta \text{]}\\ =\frac{\frac{7}{25}}{\frac{49}{25}}\text{ [Add fractions]}\\ =\frac{7}{49}\text{ [Multiply numerator and denominator by }\left(\text{25/49}\right)\text{]}\\ =\frac{1}{7}\text{ [Simplify]}\end{array}$

d.

To determine

To find: The exact value of the given trigonometric function using the given figure.

Answer to Problem 48E

The value of the given expression is $7$ .

Explanation of Solution

Given information:

The following figure:

  

The following expression:

  $\cot \frac{\theta }{2}$

Formula used:

The following half angle formula:

  $\tan \frac{u}{2}=\frac{\sin u}{1+\cos u}$

The following Pythagorean relation:

  $\begin{array}{l}\cos \theta =\frac{\text{adjacent}}{\text{hypotenuse}}\\ \sin \theta =\frac{\text{opposite}}{\text{hypotenuse}}\\ \cot \theta =\frac{1}{\tan \theta }\end{array}$

The following Pythagorean identity:

  ${\text{Hypotenuse}}^2={\text{Adjacent}}^2+{\text{Opposite}}^2$

Calculation:

From the given diagram,

  $\begin{array}{l}\text{Adjacent}=24\\ \text{Opposite}=7\\ \text{Hypotenuse}=\sqrt{{\left(24\right)}^2+{\left(7\right)}^2}\text{ [Pythagorean identity]}\\ =\sqrt{576+49}\text{ [Find square]}\\ =\sqrt{625}\text{ [Add]}\\ =25\text{ [Find square root]}\\ \text{cos}\theta \text{=}\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{24}{25}\text{ [Pythagorean relation]}\\ \sin \theta \text{=}\frac{\text{opposite}}{\text{hypotenuse}}=\frac{7}{25}\text{ [Pythagorean relation]}\\ \end{array}$

The value of the given expression can be calculated as:

  $\begin{array}{l}\cot \frac{\theta }{2}=\frac{1}{\tan \frac{\theta }{2}}\text{ [Pythagorean relation] }\\ \text{ = }\frac{1}{\frac{\sin \theta }{1+\cos \theta }}\text{ [Half angle formula]}\\ \text{ =}\frac{1+\cos \theta }{\sin \theta }\text{ [Multiply numerator and denominator by 1+cos}\theta \text{]}\\ =\frac{1+\frac{24}{25}}{\frac{7}{25}}\text{ [Substitute values of sin}\theta \text{ and cos}\theta \text{]}\\ =\frac{\frac{49}{25}}{\frac{7}{25}}\text{ [Add fractions] }\\ =7\text{ [Multiply numerator and denominator by }\left(25/7\right)\text{]}\end{array}$

e.

To determine

To find: The exact value of the given trigonometric function using the given figure.

Answer to Problem 48E

The value of the given expression is $\frac{5\sqrt{2}}{7}$ .

Explanation of Solution

Given information:

The following figure:

  

The following expression:

  $\sec \frac{\theta }{2}$

Formula used:

The following half angle formula

  $\cos \frac{u}{2}=\pm \sqrt{\frac{1+\cos u}{2}}$

The following Pythagorean relation:

  $\begin{array}{l}\cos \theta =\frac{\text{adjacent}}{\text{hypotenuse}}\\ \sec \theta =\frac{1}{\cos \theta }\end{array}$

The following Pythagorean identity:

  ${\text{Hypotenuse}}^2={\text{Adjacent}}^2+{\text{Opposite}}^2$

Calculation:

From the given diagram,

  $\begin{array}{l}\text{Adjacent}=24\\ \text{Opposite}=7\\ \text{Hypotenuse}=\sqrt{{\left(24\right)}^2+{\left(7\right)}^2}\text{ [Pythagorean identity]}\\ =\sqrt{576+49}\text{ [Find square]}\\ =\sqrt{625}\text{ [Add]}\\ =25\text{ [Find square root]}\\ \text{cos}\theta \text{=}\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{24}{25}\text{ [Pythagorean relation]}\\ \end{array}$

The value of the given expression can be calculated as:

  $\begin{array}{l}\sec \frac{\theta }{2}=\frac{1}{\cos \frac{\theta }{2}}\text{ [Pythagorean relation] }\\ \text{ = }\frac{1}{\sqrt{\frac{1+\cos \theta }{2}}}\text{ [Half angle formula]}\\ \text{ =}\frac{\sqrt{2}}{\sqrt{1+\cos \theta }}\text{ [Multiply numerator and denominator by }\sqrt{2}\text{]}\\ =\frac{\sqrt{2}}{\sqrt{1+\frac{24}{25}}}\text{ [Substitute value of cos}\theta \text{]}\\ =\frac{\sqrt{2}}{\sqrt{\frac{49}{25}}}\text{ [Add fractions] }\\ =\frac{5\sqrt{2}}{7}\text{ [Multiply numerator and denominator by }\sqrt{25}\text{, find square root]}\end{array}$

f.

To determine

To find: The exact value of the given trigonometric function using the given figure.

Answer to Problem 48E

The value of the given expression is $5\sqrt{2}$ .

Explanation of Solution

Given information:

The following figure:

  

The following expression:

  $\csc \frac{\theta }{2}$

Formula used:

The following half angle formula

  $\sin \frac{u}{2}=\pm \sqrt{\frac{1-\cos u}{2}}$

The following Pythagorean relation:

  $\begin{array}{l}\cos \theta =\frac{\text{adjacent}}{\text{hypotenuse}}\\ \sin \theta =\frac{1}{\csc \theta }\end{array}$

The following Pythagorean identity:

  ${\text{Hypotenuse}}^2={\text{Adjacent}}^2+{\text{Opposite}}^2$

Calculation:

From the given diagram,

  $\begin{array}{l}\text{Adjacent}=24\\ \text{Opposite}=7\\ \text{Hypotenuse}=\sqrt{{\left(24\right)}^2+{\left(7\right)}^2}\text{ [Pythagorean identity]}\\ =\sqrt{576+49}\text{ [Find square]}\\ =\sqrt{625}\text{ [Add]}\\ =25\text{ [Find square root]}\\ \text{cos}\theta \text{=}\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{24}{25}\text{ [Pythagorean relation]}\\ \end{array}$

The value of the given expression can be calculated as:

  $\begin{array}{l}\csc \frac{\theta }{2}=\frac{1}{\sin \frac{\theta }{2}}\text{ [Pythagorean relation] }\\ \text{ = }\frac{1}{\sqrt{\frac{1-\cos \theta }{2}}}\text{ [Half angle formula]}\\ \text{ =}\frac{\sqrt{2}}{\sqrt{1-\cos \theta }}\text{ [Multiply numerator and denominator by }\sqrt{2}\text{]}\\ =\frac{\sqrt{2}}{\sqrt{1-\frac{24}{25}}}\text{ [Substitute value of cos}\theta \text{]}\\ =\frac{\sqrt{2}}{\sqrt{\frac{1}{25}}}\text{ [Subtract fractions] }\\ =\sqrt{\frac{50}{1}}\text{ [Multiply numerator and denominator by }\sqrt{25}\text{]}\\ \text{ =5}\sqrt{2}\text{ [Simplify]}\end{array}$

g.

To determine

To find: The exact value of the given trigonometric function using the given figure.

Answer to Problem 48E

The value of the given expression is $\frac{7}{25}$ .

Explanation of Solution

Given information:

The following figure:

  

The following expression:

  $2\sin \frac{\theta }{2}\cos \frac{\theta }{2}$

Calculation:

From subpart(a),

  $\sin \frac{\theta }{2}=\frac{\sqrt{2}}{10}$

From subpart(b),

  $\cos \frac{\theta }{2}=\frac{7\sqrt{2}}{10}$

The value of the given expression can be calculated as:

  $\begin{array}{l}2\sin \frac{\theta }{2}\cos \frac{\theta }{2}\text{=2}\cdot \frac{\sqrt{2}}{10}\cdot \frac{7\sqrt{2}}{10}\text{ [Substitute values] }\\ \text{ =}\frac{28}{100}\text{ [Multiply]}\\ \text{ =}\frac{7}{25}\text{ [Simplify]}\\ \end{array}$

h.

To determine

To find: The exact value of the given trigonometric function using the given figure.

Answer to Problem 48E

The value of the given expression is $\frac{\sqrt{2}}{5}$ .

Explanation of Solution

Given information:

The following figure:

  

The following expression:

  $2\cos \frac{\theta }{2}\tan \frac{\theta }{2}$

Calculation:

From subpart(b),

  $\cos \frac{\theta }{2}=\frac{7\sqrt{2}}{10}$

From subpart(c),

  $\tan \frac{\theta }{2}=\frac{1}{7}$

The value of the given expression can be calculated as:

  $\begin{array}{l}2\cos \frac{\theta }{2}\tan \frac{\theta }{2}\text{=2}\cdot \frac{7\sqrt{2}}{10}\cdot \frac{1}{7}\text{ [Substitute values] }\\ \text{ =}\frac{\sqrt{2}}{5}\text{ [Multiply]}\end{array}$