Chapter 6.2, Problem 132AYU

a.

To determine

Value of the six trigonometric function at $t$ to the nearest tenth.

Answer to Problem 132AYU

  $\begin{array}{ccc}\sin 2\approx 0.9& \cos 2\approx -0.4& \csc 2\approx 1.1\\ \sec 2\approx -2.4& \tan 2\approx -2.3& \cot 2\approx -0.5\end{array}$

Explanation of Solution

Given information:

Use the figure to approximate the value of the six trigonometric function at $t$ to the nearest tenth. Then use a calculator to approximate each of the six trigonometric function at $t=2$ .

  

Calculation:

The given diagram is a unit circle. We will define trigonometric function with reference at a point $P\left(u,v\right)$ on a unit circle. Locate $P$ using a real number $x$ as follows,

  

If $x\ge 0$ travel along the circle in anticlockwise direction a distance $x$ units to arrive at $P\left(x\right)$ and if $x<0$ travel along the clockwise direction of distance $\left|x\right|$ units.

The value of $x$ indicates the directed arc length from $\left(1,0\right)$ . As $x$ varies the point $P$ moves around the unit circle.

  $\sin x=v,\cos x=u$ Coordinate of $P\left(x\right)$

To find $\sin 2and\cos 2$ locate $P\left(2\right)$ by moving along the unit circle a distance $2$ unit anticlockwise from the point $\left(1,0\right)$

Approximate the coordinate of $P\left(2\right)\approx \left(-0.4,0.9\right)$ .

Since the $v$ -coordinate represents the value of $\sin x$ , we have

  $\boxed{\sin 2=0.9}$

Since the $u,$ -coordinate represents the value of $\cos x$ , we have

  $\boxed{\cos 2=-0.4}$

Applying,

  $\begin{array}{l}\tan x=\frac{\sin x}{\cos x},\cot x=\frac{\cos x}{\sin x}\\ \tan 2=\frac{\sin 2}{\cos 2},\cot 2=\frac{\cos 2}{\sin 2}\\ =\boxed{-2.3},=\boxed{-0.4}\end{array}$

And

  $\begin{array}{l}\csc x=\frac{1}{\sin x},\sec x=\frac{1}{\cos x}\\ \csc 2=\frac{1}{\sin 2},\sec 2=\frac{1}{\cos 2}\\ =\boxed{1.1},=\boxed{-2.5}\end{array}$

Hence, all trigonometric functions are tabulated as.

  $\begin{array}{ccc}\sin 2\approx 0.9& \cos 2\approx -0.4& \csc 2\approx 1.1\\ \sec 2\approx -2.4& \tan 2\approx -2.3& \cot 2\approx -0.5\end{array}$

b.

To determine

Value of the six trigonometric function at $t$ to the nearest tenth.

Answer to Problem 132AYU

  $\begin{array}{ccc}\sin 4\approx -0.75& \cos 4\approx -0.6& \csc 4\approx -1.33\\ \sec 4\approx -1.67& \tan 4\approx 1.1& \cot 4\approx 0.85\end{array}$

Explanation of Solution

Given information:

Use the figure to approximate the value of the six trigonometric function at $t$ to the nearest tenth. Then use a calculator to approximate each of the six trigonometric function at $t=4$ .

  

Calculation:

The given diagram is a unit circle. We will define trigonometric function with reference at a point $P\left(u,v\right)$ on a unit circle. Locate $P$ using a real number $x$ as follows,

  

If $x\ge 0$ travel along the circle in anticlockwise direction a distance $x$ units to arrive at $P\left(x\right)$ and if $x<0$ travel along the clockwise direction of distance $\left|x\right|$ units.

The value of $x$ indicates the directed arc length from $\left(1,0\right)$ . As $x$ varies the point $P$ moves around the unit circle.

  $\sin x=v,\cos x=u$ Coordinate of $P\left(x\right)$

To find $\sin 4and\cos 4$ locate $P\left(4\right)$ by moving along the unit circle a distance $4$ unit anticlockwise from the point $\left(1,0\right)$

Approximate the coordinate of $P\left(4\right)\approx \left(-0.6,-0.7\right)$ .

Since the $v$ -coordinate represents the value of $\sin x$ , we have

  $\boxed{\sin 4=-0.75}$

Since the $u,$ -coordinate represents the value of $\cos x$ , we have

  $\boxed{\cos 4=-0.6}$

Applying,

  $\begin{array}{l}\tan x=\frac{\sin x}{\cos x},\cot x=\frac{\cos x}{\sin x}\\ \tan 4=\frac{\sin 2}{\cos 2},\cot 4=\frac{\cos 2}{\sin 2}\\ =\boxed{1.1},=\boxed{0.85}\end{array}$

And

  $\begin{array}{l}\csc x=\frac{1}{\sin x},\sec x=\frac{1}{\cos x}\\ \csc 4=\frac{1}{\sin 2},\sec 4=\frac{1}{\cos 2}\\ =\boxed{-1.33},=\boxed{-1.67}\end{array}$

Hence, all trigonometric functions are tabulated as.

  $\begin{array}{ccc}\sin 4\approx -0.75& \cos 4\approx -0.6& \csc 4\approx -1.33\\ \sec 4\approx -1.67& \tan 4\approx 1.1& \cot 4\approx 0.85\end{array}$