Chapter 67, Problem 12A

To determine

(a)

The value sine function when side x is equal to side r.

Answer to Problem 12A

The value of sine function of angle 1 is $\sin \angle 1=0.7071$.

Explanation of Solution

Given information:

The given figure is

  

When the side x is equal to side y, the value of sine function of angle 1 can be calculated as follows

$\sin \angle 1=\frac{y}{r}$

Now from the Pythagoras theorem,

$\begin{array}{l}{r}^2=y^2+y^2\\ r^2=y^2+y^2\\ r^2=2\times y^2\\ r=\sqrt{\left(2\times y^2\right)}\\ r=y\sqrt{2}\end{array}$

Now $\sin \angle 1=\frac{y}{r}$

$\sin \angle 1=\frac{y}{y\sqrt{2}}$

$\sin \angle 1=\frac{1}{\sqrt{2}}=0.7071$

The value of sine function of angle 1 is $\sin \angle 1=0.7071$

Conclusion:

Thus, the value of sine function of angle 1 is $\sin \angle 1=0.7071$

To determine

(b)

The value cotangent function when side x is equal to side r.

Answer to Problem 12A

The value of cotangent function of angle 1 is $\cot \angle 1=1$.

Explanation of Solution

Given information:

The given figure is

  

When the side x is equal to side y, the value of cotangent function of angle 1 can be calculated as follows

$\cot \angle 1=\frac{x}{y}$

Since, x = y

Now $\cot \angle 1=\frac{x}{y}=\frac{x}{x}=1$

$\cot \angle 1=1$

The value of cotangent function of angle 1 is $\cot \angle 1=1$

Conclusion:

Thus, the value of cotangent function of angle 1 is $\cot \angle 1=1$

To determine

(c)

The value cosine function when side x is equal to side r.

Answer to Problem 12A

The value of sine function of angle 1 is $\cos \angle 1=0.7071$.

Explanation of Solution

Given information:

The given figure is

  

When the side x is equal to side y, the value of cosine function of angle 1 can be calculated as follows

$\cos \angle 1=\frac{x}{r}$

Now from the Pythagoras theorem,

$\begin{array}{l}{r}^2=x^2+y^2\\ r^2=x^2+x^2\\ r^2=2\times x^2\\ r=\sqrt{\left(2\times x^2\right)}\\ r=x\sqrt{2}\end{array}$

Now $\cos \angle 1=\frac{x}{x\sqrt{2}}$

$\cos \angle 1=\frac{1}{\sqrt{2}}$

$\cos \angle 1=0.7071$

The value of cosine function of angle 1 is $\cos \angle 1=0.7071$

Conclusion:

Thus, the value of cosine function of angle 1 is $\cos \angle 1=0.7071$

To determine

(d)

The value cosecant function when side x is equal to side r.

Answer to Problem 12A

The value of cosecant function of angle 1 is $\csc \angle 1=1.4142$.

Explanation of Solution

Given information:

The given figure is

  

When the side x is equal to side y, the value of cosecant function of angle 1 can be calculated as follows

$\csc \angle 1=\frac{r}{y}$

Now from the Pythagoras theorem,

$\begin{array}{l}{r}^2=y^2+y^2\\ r^2=y^2+y^2\\ r^2=2\times y^2\\ r=\sqrt{\left(2\times y^2\right)}\\ r=y\sqrt{2}\end{array}$

Now $\csc \angle 1=\frac{r}{y}$

$\csc \angle 1=\frac{y\sqrt{2}}{y}$

$\csc \angle 1=\sqrt{2}$

The value of sine function of angle 1 is $\csc \angle 1=1.4142$

Conclusion:

Thus, the value of sine function of angle 1 is $\csc \angle 1=1.4142$