# 7.1 The trig functions and 2-d problems  (Page 2/2)

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 ${0}^{\circ }$ ${30}^{\circ }$ ${45}^{\circ }$ ${60}^{\circ }$ ${90}^{\circ }$ ${180}^{\circ }$ $cos\theta$ 1 $\frac{\sqrt{3}}{2}$ $\frac{1}{\sqrt{2}}$ $\frac{1}{2}$ 0 $-1$ $sin\theta$ 0 $\frac{1}{2}$ $\frac{1}{\sqrt{2}}$ $\frac{\sqrt{3}}{2}$ 1 0 $tan\theta$ 0 $\frac{1}{\sqrt{3}}$ 1 $\sqrt{3}$ $-$ 0

These values are useful when asked to solve a problem involving trig functions without using a calculator.

Each of the trigonometric functions has a reciprocal that has a special name. The three reciprocals are cosecant (or cosec), secant (or sec) and cotangent (or cot). These reciprocals are given below:

$\begin{array}{ccc}cosec\theta & =& \frac{1}{sin\theta }\\ sec\theta & =& \frac{1}{cos\theta }\\ cot\theta & =& \frac{1}{tan\theta }\end{array}$

We can also define these reciprocals for any right angled triangle:

$\begin{array}{ccc}\hfill cosec\theta & =& \frac{\mathrm{hypotenuse}}{\mathrm{opposite}}\hfill \\ \hfill sec\theta & =& \frac{\mathrm{hypotenuse}}{\mathrm{adjacent}}\hfill \\ \hfill cot\theta & =& \frac{\mathrm{adjacent}}{\mathrm{opposite}}\hfill \end{array}$

Find the length of x in the following triangle.

1. In this case you have an angle ( ${50}^{\circ }$ ), the opposite side and the hypotenuse.

So you should use $sin$

$sin{50}^{\circ }=\frac{x}{100}$
2. $⇒x=100×sin{50}^{\circ }$
3. Use the sin button on your calculator

$⇒x=76.6\mathrm{m}$

Find the value of $\theta$ in the following triangle.

1. In this case you have the opposite side and the hypotenuse to the angle $\theta$ .

So you should use $tan$

$tan\theta =\frac{50}{100}$
2. $tan\theta =0.5$
3. Since you are finding the angle ,

use ${tan}^{-1}$ on your calculator

Don't forget to set your calculator to `deg' mode!

$\theta =26.{6}^{\circ }$

In the previous example we used ${\mathrm{tan}}^{-1}$ . This is simply the inverse of the tan function. Sin and cos also have inverses. All this means is that we want to find the angle that makes the expression true and so we must move the tan (or sin or cos) to the other side of the equals sign and leave the angle where it is. Sometimes the reciprocal trigonometric functions are also referred to as the 'inverse trigonometric functions'. You should note, however that ${\mathrm{tan}}^{-1}$ and $\mathrm{cot}$ are definitely NOT the same thing.

The following videos provide a summary of what you have learnt so far.

## Finding lengths

Find the length of the sides marked with letters. Give answers correct to 2 decimal places.

## Two-dimensional problems

We can use the trig functions to solve problems in two dimensions that involve right angled triangles. For example if you are given a quadrilateral and asked to find the one of the angles, you can construct a right angled triangle and use the trig functions to solve for the angle. This will become clearer after working through the following example.

Let ABCD be a trapezium with $\mathrm{AB}=4\phantom{\rule{1pt}{0ex}}\mathrm{cm}$ , $\mathrm{CD}=6\phantom{\rule{1pt}{0ex}}\mathrm{cm}$ , $\mathrm{BC}=5\phantom{\rule{1pt}{0ex}}\mathrm{cm}$ and $\mathrm{AD}=5\phantom{\rule{1pt}{0ex}}\mathrm{cm}$ . Point E on diagonal AC divides the diagonal such that $\mathrm{AE}=3\phantom{\rule{1pt}{0ex}}\mathrm{cm}$ . Find $A\stackrel{^}{B}C$ .

1. We draw a diagram and construct right angled triangles to help us visualize the problem.
2. We will use triangle ABE and triangle BEC to get the two angles, and then we will add these two angles together to find the angle we want.
3. We use sin for both triangles since we have the hypotenuse and the opposite side.
4. In triangle ABE we find:
$\begin{array}{ccc}\hfill sin\left(A\stackrel{^}{B}E\right)& =& \frac{\mathrm{opp}}{\mathrm{hyp}}\hfill \\ \hfill sin\left(A\stackrel{^}{B}E\right)& =& \frac{3}{4}\hfill \\ A\stackrel{^}{B}E& =& {sin}^{-1}\left(\frac{3}{4}\right)\hfill \\ A\stackrel{^}{B}E& =& {48,59}^{\circ }\hfill \end{array}$
We use the theorem of Pythagoras to find $\mathrm{EC}=4,4\phantom{\rule{1pt}{0ex}}\mathrm{cm}$ . In triangle BEC we find:
$\begin{array}{ccc}\hfill sin\left(C\stackrel{^}{B}E\right)& =& \frac{\mathrm{opp}}{\mathrm{hyp}}\hfill \\ \hfill sin\left(C\stackrel{^}{B}E\right)& =& \frac{4,4}{5}\hfill \\ A\stackrel{^}{B}E& =& {sin}^{-1}\left(\frac{4,4}{5}\right)\hfill \\ C\stackrel{^}{B}E& =& {61,64}^{\circ }\hfill \end{array}$
5. We add the two angles together to get: ${48,59}^{\circ }+{61,64}^{\circ }={110,23}^{\circ }$

#### Questions & Answers

what is the stm
Brian Reply
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
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LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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LITNING
scanning tunneling microscope
Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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Damian Reply
what king of growth are you checking .?
Renato
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Stoney Reply
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Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
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Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Damian Reply
absolutely yes
Daniel
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How can I make nanorobot?
Lily
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
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Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Maths grade 10 rought draft. OpenStax CNX. Sep 29, 2011 Download for free at http://cnx.org/content/col11363/1.1
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