Graphs of Sin x, Cos x; and Tan x
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Introduction
Graphs of sinx°, cosx° and tanx°Here are the sketch graphs of the trigonometric functions f(x) = sinx°, f(x) = cosx° and f(x) = tanx°. You may be asked to draw or sketch these graphs in your exam. Try to remember what they look like, and follow these tips: If you are asked to draw or plot the graph, you will need to use your calculator to generate the yvalues. For example, if you were asked to plot the graph of f(x) = sinx° for 0° x 360° , you would use you calculator to find sin0°, sin10°, sin 20°, ...., sin 360° and then plot these values on the graph paper. Plotting a trigonometric graph is timeconsuming and it is therefore more likely that you will be asked to sketch the graph. However, even if you think that you remember what the graph looks like, your calculator can be used to check. For example, sin0° = 0 and cos0° = 1, so you have the starting points of the graphs. Tan90° has no value (your calculator will display an error message), so you know that the graph cannot cross the line x = 90°.  
Transformations of graphs; y = asinbx° and y = acosbx°
Remember
Given a graph, f(x):
 The transformation af(x) causes a stretch, parallel to the yaxis with a scale factor of a.
 The transformation f(bx) causes a stretch, parallel to the xaxis with a scale factor of .
Middle
Remember
This is the circle property which is the most difficult to spot. Look out for a triangle with one of its vertices resting on the point of contact of the tangent.
The angle between a tangent and a chord is equal to the angle made by that chord in the alternate segment.
Question 1
What is the size of:
 angle x?
 angle y?
The Answer
 Did you get x = 60°? Well done!
 Did you get y = 80°? Well done! You remembered that the angles in a triangle add up to 180°.
 Make sure that you learn these circle properties. If you are asked to find the angles in a circle, you will then be able to see which of them apply to the question.
 Do not be afraid to find the sizes of other angles first. It is not always possible to find the required angle immediately!
CongruencyIf two shapes are congruent, then they are identical in shape and size. Question 1Which of the following shapes are congruent? The AnswerDid you get the following pairs?
Well done! Remember that shapes can be congruent even if one of them has been rotated (as in A and G) or reflected (as in C and H). The symbol means 'is congruent to'. Two triangles are congruent if one of the following conditions applies: The three sides of the first triangle are equal to the three sides of the second triangle.(SSS) Two sides of the first triangle are equal to two sides of the second triangle and the included angle is equal.(SAS) Two angles in the first triangle are equal to two angles in the second triangle and one (similarly located) side is equal. (AAS) In a right angled triangle, the hypotenuse and one other side in the first triangle are equal to the hypotenuse and the corresponding side in the second triangle. (RHS) Question 2For each of the following pairs of triangles, state whether they are congruent. If they are, give a reason for your answer (SSS, SAS, AAS or RHS). Pair 1 Pair 2 Pair 3 The Answer

Conclusion
Question 1
Write down the three ways of describing the vector if the arrow is pointing in the opposite direction.
The Answer
Did you get , a and ? Well done!
Teacher's Note
If not, remember that the arrow describes the direction, so in this case, the vector is from B to A. If we move 'backwards' along a vector, it becomes negative, so a becomes a.
Vector 'arithmetic'
Equal vectors
If two vectors have the same magnitude and direction, then they are equal.
Adding vectors
Vector followed by vector represents a movement from P to R.
Subtracting vectors
Vector , followed by a backwards movement along , is equivalent to a movement from X to Z.
Multiplication by a scalar
e.g.
Question 1
If x = , y = and z = , find:
 y
 x  y
 2x + 3z
The Answer
 . Did you remember to change the signs?
To travel from X to Z, it is possible to move along vector , followed by . It is also possible to go directly along .
is therefore known as the resultant of and
Geometric problems
Question 1
Write as single vectors:
 f + g
 a + b
 e  b  a
The Answer
 e
 c Did you remember the minus sign?
 d
Remember
Two vectors are equal if they have the same magnitude and direction, regardless of where they are on the page.
Question 2
Triangles ABC and XYZ are equilateral. X is the midpoint of AB, Y is the midpoint of BC and Z is the midpoint of AC. = a , = b and = c. Express each of the following in terms of a, b and c.
This student written piece of work is one of many that can be found in our GCSE Fencing Problem section.
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