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Chapter  2:  Pythagoras  Theorem  and  Trigonometry  (Revision)  

Paper  1  &  2B   2A  

3.1.3 Triangles ·  Understand  a  proof  of  Pythagoras’  Theorem.  ·  Understand  the  converse  of  Pythagoras’  Theorem.  ·  Use  Pythagoras’ Trigonometry 3.5.1 Trigonometric ratios ·  Understand, recall and use the trigonometric relationships in right-angled triangles, namely, sine, cosine and tangent. ·  Use the trigonometric ratios to solve problems in simple practical situations (e.g. in problems involving angles of elevation and depression).

 

3.1.3 Triangles  ·  Use Pythagoras’ Theorem in 3-D situations (e.g. to determine lengths inside a cuboid). Trigonometry 3.5.1 Trigonometric ratios ·  Extend the use of the sine and cosine functions to angles between 90°  and 180°. ·  Solve simple trigonometric problems in 3- D. (e.g. find the angle between a line and a plane and the angle between two planes). 3.6.2 Sine and cosine rules ·  Use the sine and cosine rules to solve any triangle.  

 

 

 

 

 

 

 

 

 

 

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Section  2.1  Pythagoras  Theorem  

If the triangle had a right angle (90°) ...

... and you made a square on each of the three sides, then ...

... the biggest square had the exact same area as the other two squares put together!

It is called "Pythagoras' Theorem" and can be written in one short equation:

a2 + b2 = c2

Note:

• c is the longest side of the triangle • a and b are the other two sides

The  longest  side  of  a  triangle  is  always  called  the  HYPOTENUSE.    

REMEMBER:  Pythagoras  Theorem  can  only  be  used  in  a  right-­‐angled  triangle.  

 

 

 

 

 

 

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Example  1  

 

 

Example  2  

 

 

 

 

 

 

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Example  3  

 

Example  4  

 

 

Support  Exercise  Edexcel  Pg  297  Ex  19A  No  1  –  5    

  Pg  300  Ex  19B  No  1  –  10    

  Handout  Pythagoras  Theorem    

 

 

 

 

 

 

 

 

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Section  2.2    Trigonometric  Ratios  

 

Trigonometry  uses  three  important  ratios  to  calculate  sides  and  angles:  sine,  cosine  and  tangent.  These  ratios  are  defined  in  terms  of  the  sides  of  a  right-­‐angled  triangle  and  an  angle.  The  angle  is  often  written  as  θ.  

 

In  a  right-­‐angled  triangle:  

• the  side  opposite  the  right  angle  is  called  the  hypotenuse  and  is  the  longest  side  

• the  side  opposite  the  angle  θ  is  called  the  opposite  side  

• the  other  side  next  to  both  the  right  angle  and  the  angle  θ  is  called  the  adjacent  side.  

 

The  sine,  cosine  and  tangent  ratios  for  θ  are  defined  as:  

 

In  order  to  remember  these  we  use  the  word  

SOH             CAH           TOA  

Sin  =  Opp  /  Hyp         Cos  =  Adj  /  Hyp         Tan  =  Opp  /  Adj  

 

 

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Example  1  

 

 

Example  2  

 

Example  3  

 

 

 

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Example  4  

A  ladder  5  m  long,  leaning  against  a  vertical  wall  makes  an  angle  of  65˚  with  the  ground.  

a)  How  high  on  the  wall  does  the  ladder  reach?  

b)  How  far  is  the  foot  of  the  ladder  from  the  wall?  

c)  What  angle  does  the  ladder  make  with  the  wall?  

Solution:  

 

a)  The  height  that  the  ladder  reaches  is  PQ  

 

PQ  =  sin  65˚  ×  5  =  4.53  m  

b)  The  distance  of  the  foot  of  the  ladder  from  the  wall  is  RQ.  

 

RQ  =  cos  65˚  ×  5  =  2.11  m  

c)  The  angle  that  the  ladder  makes  with  the  wall  is  angle  P  

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Support  Exercise  Edexcel  Pg  307  Ex  19D  Nos  1  –  5    

  Handout  

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Section 2.3 Working with Trigonometric Ratios to find Angles

 

 

Support  Exercise  Pg  309  Ex  19E  Nos  1  –  4    

    Handout  

 

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Section 2.4 Mixed Examples

Example  1  

 

ABCD  is  a  quadrilateral.    Angle  BDA  =  90˚,  angle  BCD  =  90˚,  angle  BAD  =  40˚.  

BC  =  6  cm,  BD  =  8cm.  

a) Calculate  the  length  of  DC.    Give  your  answer  correct  to  3  significant  figures.    [Hint:  use  Pythagoras’  theorem!]  

 

 

 

 

b) Calculate  the  size  of  angle  DBC.    Give  your  answer  correct  to  3  significant  figures.      

 

 

 

 

c) Calculate  the  length  of  AB.    Give  your  answer  correct  to  3  significant  figures.  

 

 

 

 

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Example  2  

 

AB  =  19.5  cm,  AC  =  19.5  cm  and  BC  =  16.4  cm.  

Angle  ADB  =  90  degrees.  

BDC  is  a  straight  line.  

Calculate  the  size  of  angle  ABC.    Give  your  answer  correct  to  1  decimal  place.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Example  3  

 

 

 

 

 

 

Find  the  value  of  l  giving  your  answer  to  three  significant  figures.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Support  Exercise  Handout  

P  

Q   R  

S  

5  cm  

13  cm  l  

l