Finding trigonometric ratios for certain angles

78 Pages·2016·0.96 MB·English

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Finding trigonometric ratios for certain angles
(mα+hs)Smart Workshop
Semester2,2016
Geoff Coates
These slides describe some quick ways to:
find trigonometric ratios for the angles π, π and π,
6 4 3
use these to find trig ratios for related angles greater than π and
2
solve trigonometric equations.
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Contents
Trigonometric ratios for certain angles Go
Trigonometric ratios for angles > π Go
2
Go Theunitcircle
Solving trigonometric equations Go
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For example,
(cid:0) (cid:1) √
sin π = 3 exactly (rather than 0.866...).
3 2
The other key angles whose trig ratios are exact are π and π.
4 6
It’s handy to know these exact values but memorizing stuff you don’t understand is
difficult (and dull). However, being able to quickly work out stuff using knowledge you
already have is easy (and fun).
The following two right angle triangles (and some very basic knowledge about the
definition of sine, cosine and tangent) will make your life a lot easier.
Trigonometric ratios for certain angles
Thetrigonometricratios(sine,cosineandtangent)foranglesareusuallyinfinitedecimals
but some have exact values.
F((inmdαin+ghst)rSigmoanrtoWmoertkrsihcopraStieomsefsotrecre2r,ta2i0n1a6n)gles Contents Prev Next 5/11
It’s handy to know these exact values but memorizing stuff you don’t understand is
difficult (and dull). However, being able to quickly work out stuff using knowledge you
already have is easy (and fun).
The following two right angle triangles (and some very basic knowledge about the
definition of sine, cosine and tangent) will make your life a lot easier.
Trigonometric ratios for certain angles
Thetrigonometricratios(sine,cosineandtangent)foranglesareusuallyinfinitedecimals
but some have exact values. For example,
(cid:0) (cid:1) √
sin π = 3 exactly (rather than 0.866...).
3 2
The other key angles whose trig ratios are exact are π and π.
4 6
F((inmdαin+ghst)rSigmoanrtoWmoertkrsihcopraStieomsefsotrecre2r,ta2i0n1a6n)gles Contents Prev Next 5/11
However, being able to quickly work out stuff using knowledge you
already have is easy (and fun).
The following two right angle triangles (and some very basic knowledge about the
definition of sine, cosine and tangent) will make your life a lot easier.
Trigonometric ratios for certain angles
Thetrigonometricratios(sine,cosineandtangent)foranglesareusuallyinfinitedecimals
but some have exact values. For example,
(cid:0) (cid:1) √
sin π = 3 exactly (rather than 0.866...).
3 2
The other key angles whose trig ratios are exact are π and π.
4 6
It’s handy to know these exact values but memorizing stuff you don’t understand is
difficult (and dull).
F((inmdαin+ghst)rSigmoanrtoWmoertkrsihcopraStieomsefsotrecre2r,ta2i0n1a6n)gles Contents Prev Next 5/11
The following two right angle triangles (and some very basic knowledge about the
definition of sine, cosine and tangent) will make your life a lot easier.
Trigonometric ratios for certain angles
Thetrigonometricratios(sine,cosineandtangent)foranglesareusuallyinfinitedecimals
but some have exact values. For example,
(cid:0) (cid:1) √
sin π = 3 exactly (rather than 0.866...).
3 2
The other key angles whose trig ratios are exact are π and π.
4 6
It’s handy to know these exact values but memorizing stuff you don’t understand is
difficult (and dull). However, being able to quickly work out stuff using knowledge you
already have is easy (and fun).
F((inmdαin+ghst)rSigmoanrtoWmoertkrsihcopraStieomsefsotrecre2r,ta2i0n1a6n)gles Contents Prev Next 5/11
Trigonometric ratios for certain angles
Thetrigonometricratios(sine,cosineandtangent)foranglesareusuallyinfinitedecimals
but some have exact values. For example,
(cid:0) (cid:1) √
sin π = 3 exactly (rather than 0.866...).
3 2
The other key angles whose trig ratios are exact are π and π.
4 6
It’s handy to know these exact values but memorizing stuff you don’t understand is
difficult (and dull). However, being able to quickly work out stuff using knowledge you
already have is easy (and fun).
The following two right angle triangles (and some very basic knowledge about the
definition of sine, cosine and tangent) will make your life a lot easier.
F((inmdαin+ghst)rSigmoanrtoWmoertkrsihcopraStieomsefsotrecre2r,ta2i0n1a6n)gles Contents Prev Next 5/11
π
√ 6 √
h2 d3
π π
4 3
1
Cut both in half as shown to create right-angle triangles.
Work out the missing side lengths using Pythagoras’ Theorem:
h2 = 12+12 22 = 12+d2
√ √
So h = 2 So d = 3
It should be clear what the angles are in both triangles.
Trigonometric ratios for certain angles
2 2
1
1 2
Start with a square of side length 1 and an equilateral triangle of side length 2.
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