Download Calc - Pythagoras-and

Pythagoras and trigonometry
Pythagoras
 Used for right angled triangles
 Used to find a missing side
SOHCAHTOA
 Used for right angled triangles
 Used to find an angle, or a side when given a side and an angle.
Label the sides hypotenuse, opposite and adjacent
Deicde which ratio to use: SOH CAH or TOA
Write the formula and substitute
Finding missing sides
H
12cm
O
50°
x A
Hypotenuse is the
longest side.
Adjacent is the side
next to the angle.
We are given the Hypotenuse, we want to find
the adjacent, we don’t care about the opposite.
The ratio that uses A and H is cos.
A
CxH
We’re trying to find A, so we
need to multiply.
X = cos (50) x 12
X = 7.71cm
Finding missing angles
H
x°
9m A
We are given the adjacent and the opposite, we
are trying to find the angle.
The ratio that uses O and A is tan.
O
TxA
5m
O
We’re trying to find tan x, so
we need to divide.
Tan x = 5/9
X = 𝑡𝑎𝑛−1 (5/9)
X = 29.1°
The sine rule
 Used for triangles without a right angle
 We need sides and the angle opposite
 You are given this rule in the formula sheet
Examples
a
= b
sin A
sin B
x
= 25
sin 84
sin 47
x = 25
x sin 84
sin 47
x = 34.0 cm
sin A
a
= sin B
b
sin x = sin 40
7
6
sin x = sin 40 x 7
6
sin x = 0.7499
X = 𝑠𝑖𝑛−1 (0.7499) = 48.6°
The cosine rule
 Used for triangles without a right angle
 Two sides and the angle between them, or 3 sides
 You are given this rule in the formula sheet
Remember to
multiply by cos
a² = b² + c² - 2bc cos A
x² = 6² + 10² - 2 x 6 x 10 x cos 80°
x² = 36 + 100 -120 x cos 80°
x² = 115.16
x = 10.7
If we are trying to find an angle, we need to rearrange the cosine rule.
You are not given the rearranged formula.
Cos A = b² + c² - a²
2bc
cos x = 5² + 7² - 8²
2x5x7
cos x = 25 + 49 - 64
70
cos x = 0.1428
x = 𝑐𝑜𝑠 −1 0.1428
x = 81.8°
If you can’t remember the
rearranged formula, you
will still get marks for
substituting the values
into the cosine rule from
the formula sheet. The try
to simplify and rearrange
Mixed Exam Questions
Q1.
ABCD is a trapezium.
AD = 10 cm
AB = 9 cm
DC = 3 cm
Angle ABC = angle BCD = 90°
Calculate the length of AC.
Give your answer correct to 3 significant figures.
.................................................................................................................................
(Total for Question is 5 marks)
Q3.
The diagram shows the marking on a school playing field.
Diagram NOT accurately drawn
The diagram shows a rectangle and its diagonals.
Work out the total length of the four sides of the rectangle and its diagonals.
...................... m
(Total for Question is 5 marks)
Q4.
The diagram shows a quadrilateral ABCD.
Diagram
NOT accurately drawn
AB = 16 cm.
AD = 12 cm.
Angle BCD = 40°.
Angle ADB = angle CBD = 90°.
Calculate the length of CD.
Give your answer correct to 3 significant figures.
. . . . . . . . . . . . . . . . . . . . . . cm
(Total for Question is 5 marks)
Q5.
ABC is an isosceles triangle.
Work out the area of the triangle.
Give your answer correct to 3 significant figures.
........................................................... cm2
(Total for Question is 4 marks)
Q6.
ABCD is a parallelogram.
DC = 5 cm
Angle ADB = 36°
Calculate the length of AD.
Give your answer correct to 3 significant figures.
.................................................................................................................................
(Total for Question is 4 marks)
Q7.
In the diagram,
triangles ABD and BCD are right-angled triangles
AB = 5 cm
AD = 10 cm
CD = 4 cm
Angle ADB = 30°
Work out the value of x.
Give your answer correct to 2 decimal places.
...........................................................cm
(Total for question = 4 marks)
Q8.
In the diagram,
ABC, ACD and APD are right-angled triangles.
AB = 4 cm.
BC = 3 cm.
CD = 2 cm.
Work out the length of DP.
.................................................................................................................................
(Total for Question is 5 marks)
Q9.
The diagram represents a metal frame.
The frame is made from four metal bars, AB, AC, BC and BD.
Angle ABC = angle ADB = 90°
AB = 5 m
BC = 3 m
Work out the total length of the four metal bars of the frame.
Give your answer correct to 3 significant figures.
........................................................... m
(Total for question = 5 marks)
Q10.
The diagram shows triangle LMN.
Calculate the length of LN.
Give your answer correct to 3 significant figures.
........................................................... cm
(Total for Question is 5 marks)
Q11.
ABC is a triangle.
D is a point on AC.
Angle BAD = 45°
Angle ADB = 80°
AB = 7.4 cm
DC = 5.8 cm
Work out the length of BC.
Give your answer correct to 3 significant figures.
........................................................... cm
(Total for question = 5 marks)
Q12.
ABCD is a quadrilateral.
Diagram NOT accurately drawn
Work out the length of DC.
Give your answer correct to 3 significant figures.
. . . . . . . . . . . . . . . . . . . . . . cm
(Total for Question is 6 marks)
Q13.
ABCD is a rectangle.
CDE is a straight line.
AB = 12 cm
Angle ACB = 60°
Angle EAC = 90°
Calculate the length of CE.
You must show all your working.
........................................................... cm
(Total for question = 4 marks)
Q14.
Calculate the length of PR.
Give your answer correct to 3 significant figures.
. . . . . . . . . . . . . . . . . . . . . cm
(Total for Question is 3 marks)
Examiner's Report key points
 Think carefully which rule you will need to use. If you are not sure,
attempt one rule and if it is incorrect you will not be able to complete the
calculations
 If a question involves finding a side in a quadrilateral, try splitting it into
triangles.
 Don’t forget to square root you answer when using Pythagoras or the
cosine rule for a side- check if your answer looks sensible. If it is too big
your probably forgot this step
 Some of the more difficult questions are 5 or 6 marks. Attempt as much of
the question as you can to try and gain some of the marks- find an angle or
side that you think may be needed.
 Try not to round your answer until the end. You may lose an accuracy
mark if you round too early.
 Often, examiners comment that working lacks clarity or a logical order,
making it difficult for them to give you marks. Make sure you label each
part of your working out e.g. length AB =. This will make it easier to check
if you have found the length or angle that the question is looking for
 Remember when using Pythagoras, if you are not finding the hypotenuse
you need to subtract.
 If the question states "You must show all your working" make sure that
you show all of your calculations, because you may get 0 marks even if you
get the correct answer.
Answers
Pythagoras
19.36
44.69
18.10
29.27
SOHCAHTOA sides
a) 5.14cm
d) 7cm
b) 11.82cm
e) 26.9cm
SOHCAHTOA angles
a = 46.2o
d = 43.6o
b = 19.7o
e = 53.3o
51.85
c) 5.13cm
f) 38.8cm
c = 37.7o
f = 60.4o
Sine rule
a) 3.64m
d) 46.6°
b) 8.05cm
e) 112.0°
c) 19.4cm
f) 36.2°
Cosine rule
a) 7.71m
d) 76.2°
b) 29.1cm
e) 125.1°
c) 27.4cm
f) 90°
Mixed exam questions Mark Scheme
Q1.
Q3.
Q4.
Q5.
Q6.
Q7.
Q8.
Q9.
Q10.
Q11.
Q12.
Q13.
Q14.