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For most angles θ , it is very difficult to calculate the values of sin θ , cos θ and tan θ . One usually needs to use a calculator to do so. However, we saw in the above Activity that we could work these values out for some special angles. Some of these angles are listed in the table below, along with the values of the trigonometric functions at these angles. Remember that the lengths of the sides of a right angled triangle must obey Pythagoras' theorum. The square of the hypothenuse (side opposite the 90 degree angle) equals the sum of the squares of the two other sides.

0 30 45 60 90 180
cos θ 1 3 2 1 2 1 2 0 - 1
sin θ 0 1 2 1 2 3 2 1 0
tan θ 0 1 3 1 3 - 0

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

Find the length of x in the following triangle.

  1. In this case you have an angle ( 50 ), the opposite side and the hypotenuse.

    So you should use sin

    sin 50 = x 100
  2. x = 100 × sin 50
  3. Use the sin button on your calculator

    x = 76 . 6 m

Find the value of θ in the following triangle.

  1. In this case you have the opposite side and the hypotenuse to the angle θ .

    So you should use tan

    tan θ = 50 100
  2. tan θ = 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!

    θ = 26 . 6

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

Trigonometry - 1

Khan academy video on trigonometry - 2

Finding lengths

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

Simple applications of trigonometric functions

Trigonometry was probably invented in ancient civilisations to solve practical problems such as building construction and navigating by the stars. In this section we will show how trigonometry can be used to solve some other practical problems.

Height and depth

Determining the height of a building using trigonometry.

One simple task is to find the height of a building by using trigonometry. We could just use a tape measure lowered from the roof, but this is impractical (and dangerous) for tall buildings. It is much more sensible to measure a distance along the ground and use trigonometry to find the height of the building.

[link] shows a building whose height we do not know. We have walked 100 m away from the building and measured the angle from the ground up to the top of the building. This angle is found to be 38 , 7 . We call this angle the angle of elevation . As you can see from [link] , we now have a right-angled triangle. As we know the length of one side and an angle, we can calculate the height of the triangle, which is the height of the building we are trying to find.

If we examine the figure, we see that we have the opposite and the adjacent of the angle of elevation and we can write:

tan 38 , 7 = opposite adjacent = height 100 m height = 100 m × tan 38 , 7 = 80 m

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
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Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
what is the stm
Brian Reply
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industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
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what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Math 1508 (lecture) readings in precalculus. OpenStax CNX. Aug 24, 2011 Download for free at http://cnx.org/content/col11354/1.1
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