Other tags that relate to Complex Rotations
Rotations. Sine, cosine, tangent. Vector algebra. Integers. Exponential and Logarithmic Functions. Matrices. Rational and irrational numbers. Mathematical reasoning & proof. Engineering. Complex numbers.There are 57 results
Broad Topics > Pythagoras and Trigonometry > Sine, cosine, tangent
Three by One
Age 16 to 18 Challenge Level:
There are many different methods to solve this geometrical problem - how many can you find?
8 Methods for Three by One
Age 14 to 18 Challenge Level:
This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different?. . . .
Screen Shot
Age 14 to 16 Challenge Level:
A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .
Pumping the Power
Age 16 to 18 Challenge Level:
What is an AC voltage? How much power does an AC power source supply?
Coke Machine
Age 14 to 16 Challenge Level:
The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design...
Doesn't Add Up
Age 14 to 16 Challenge Level:
In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?
Circle Box
Age 14 to 16 Challenge Level:
It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?
Inscribed in a Circle
Age 14 to 16 Challenge Level:
The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?
Trig Reps
Age 16 to 18 Challenge Level:
Can you deduce the familiar properties of the sine and cosine functions starting from these three different mathematical representations?
Where Is the Dot?
Age 14 to 16 Challenge Level:
A dot starts at the point (1,0) and turns anticlockwise. Can you estimate the height of the dot after it has turned through 45 degrees? Can you calculate its height?
Six Discs
Age 14 to 16 Challenge Level:
Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?
Octa-flower
Age 16 to 18 Challenge Level:
Join some regular octahedra, face touching face and one vertex of each meeting at a point. How many octahedra can you fit around this point?
Bend
Age 16 to 18 Challenge Level:
What is the longest stick that can be carried horizontally along a narrow corridor and around a right-angled bend?
Shape and Territory
Age 16 to 18 Challenge Level:
If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?
Circle Scaling
Age 14 to 16 Challenge Level:
Describe how to construct three circles which have areas in the ratio 1:2:3.
The Dodecahedron
Age 16 to 18 Challenge Level:
What are the shortest distances between the centres of opposite faces of a regular solid dodecahedron on the surface and through the middle of the dodecahedron?
Round and Round
Age 14 to 16 Challenge Level:
Prove that the shaded area of the semicircle is equal to the area of the inner circle.
Raising the Roof
Age 14 to 16 Challenge Level:
How far should the roof overhang to shade windows from the mid-day sun?
Squ-areas
Age 14 to 16 Challenge Level:
Three squares are drawn on the sides of a triangle ABC. Their areas are respectively 18 000, 20 000 and 26 000 square centimetres. If the outer vertices of the squares are joined, three more. . . .
Circumnavigation
Age 14 to 16 Challenge Level:
The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.
Why Stop at Three by One
Age 16 to 18
Beautiful mathematics. Two 18 year old students gave eight different proofs of one result then generalised it from the 3 by 1 case to the n by 1 case and proved the general result.
The History of Trigonometry- Part 1
Age 11 to 18
The first of three articles on the History of Trigonometry. This takes us from the Egyptians to early work on trigonometry in China.
Geometric Trig
Age 16 to 18 Short Challenge Level:
Trigonometry, circles and triangles combine in this short challenge.
History of Trigonometry - Part 2
Age 11 to 18
The second of three articles on the History of Trigonometry.
History of Trigonometry - Part 3
Age 11 to 18
The third of three articles on the History of Trigonometry.
Gold Again
Age 16 to 18 Challenge Level:
Without using a calculator, computer or tables find the exact values of cos36cos72 and also cos36 - cos72.
Cosines Rule
Age 14 to 16 Challenge Level:
Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.
30-60-90 Polypuzzle
Age 16 to 18 Challenge Level:
Re-arrange the pieces of the puzzle to form a rectangle and then to form an equilateral triangle. Calculate the angles and lengths.
After Thought
Age 16 to 18 Challenge Level:
Which is larger cos(sin x) or sin(cos x) ? Does this depend on x ?
Ball Bearings
Age 16 to 18 Challenge Level:
If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.
Farhan's Poor Square
Age 14 to 16 Challenge Level:
From the measurements and the clue given find the area of the square that is not covered by the triangle and the circle.
Figure of Eight
Age 14 to 16 Challenge Level:
On a nine-point pegboard a band is stretched over 4 pegs in a "figure of 8" arrangement. How many different "figure of 8" arrangements can be made ?
Orbiting Billiard Balls
Age 14 to 16 Challenge Level:
What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?
Sine Problem
Age 16 to 18 Challenge Level:
In this 'mesh' of sine graphs, one of the graphs is the graph of the sine function. Find the equations of the other graphs to reproduce the pattern.
Muggles, Logo and Gradients
Age 11 to 18
Logo helps us to understand gradients of lines and why Muggles Magic is not magic but mathematics. See the problem Muggles magic.
Sine and Cosine for Connected Angles
Age 14 to 16 Challenge Level:
The length AM can be calculated using trigonometry in two different ways. Create this pair of equivalent calculations for different peg boards, notice a general result, and account for it.
Moving Squares
Age 14 to 16 Challenge Level:
How can you represent the curvature of a cylinder on a flat piece of paper?
Strange Rectangle 2
Age 16 to 18 Challenge Level:
Find the exact values of some trig. ratios from this rectangle in which a cyclic quadrilateral cuts off four right angled triangles.
Spokes
Age 16 to 18 Challenge Level:
Draw three equal line segments in a unit circle to divide the circle into four parts of equal area.
Pythagoras on a Sphere
Age 16 to 18 Challenge Level:
Prove Pythagoras' Theorem for right-angled spherical triangles.
Degree Ceremony
Age 16 to 18 Challenge Level:
What does Pythagoras' Theorem tell you about these angles: 90°, (45+x)° and (45-x)° in a triangle?
At a Glance
Age 14 to 16 Challenge Level:
The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?
So Big
Age 16 to 18 Challenge Level:
One side of a triangle is divided into segments of length a and b by the inscribed circle, with radius r. Prove that the area is: abr(a+b)/ab-r^2
Diagonals for Area
Age 16 to 18 Challenge Level:
Can you prove this formula for finding the area of a quadrilateral from its diagonals?
Eight Ratios
Age 14 to 16 Challenge Level:
Two perpendicular lines lie across each other and the end points are joined to form a quadrilateral. Eight ratios are defined, three are given but five need to be found.
Over the Pole
Age 16 to 18 Challenge Level:
Two places are diametrically opposite each other on the same line of latitude. Compare the distances between them travelling along the line of latitude and travelling over the nearest pole.
A Scale for the Solar System
Age 14 to 16 Challenge Level:
The Earth is further from the Sun than Venus, but how much further? Twice as far? Ten times?
Flight Path
Age 16 to 18 Challenge Level:
Use simple trigonometry to calculate the distance along the flight path from London to Sydney.
Small Steps
Age 16 to 18 Challenge Level:
Two problems about infinite processes where smaller and smaller steps are taken and you have to discover what happens in the limit.
Belt
Age 16 to 18 Challenge Level:
A belt of thin wire, length L, binds together two cylindrical welding rods, whose radii are R and r, by passing all the way around them both. Find L in terms of R and r.
NRICH is part of the family of activities in the Millennium Mathematics Project.