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Broad Topics > Pythagoras and Trigonometry > Sine, cosine, tangent Sine and Cosine

Age 14 to 16 Challenge Level:

The sine of an angle is equal to the cosine of its complement. Can you explain why and does this rule extend beyond angles of 90 degrees? 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? Trigonometric Protractor

Age 14 to 16 Challenge Level:

An environment that simulates a protractor carrying a right- angled triangle of unit hypotenuse. 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. Age 14 to 16 Challenge Level:

If you were to set the X weight to 2 what do you think the angle might be? 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 ? 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? Far Horizon

Age 14 to 16 Challenge Level:

An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see? 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? 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. 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?. . . . 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? 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. 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 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? Logosquares

Age 16 to 18 Challenge Level:

Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares. 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? Age 14 to 18 Challenge Level:

How would you design the tiering of seats in a stadium so that all spectators have a good view? Geometric Trig

Age 16 to 18 Short Challenge Level:

Trigonometry, circles and triangles combine in this short challenge. Moving Squares

Age 14 to 16 Challenge Level:

How can you represent the curvature of a cylinder on a flat piece of paper? History of Trigonometry - Part 3

Age 11 to 18

The third of three articles on the History of Trigonometry. History of Trigonometry - Part 2

Age 11 to 18

The second of three articles on the History of Trigonometry. 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. Pumping the Power

Age 16 to 18 Challenge Level:

What is an AC voltage? How much power does an AC power source supply? Round and Round a Circle

Age 14 to 16 Challenge Level:

Can you explain what is happening and account for the values being displayed? 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? Pythagoras on a Sphere

Age 16 to 18 Challenge Level:

Prove Pythagoras' Theorem for right-angled spherical triangles. Raising the Roof

Age 14 to 16 Challenge Level:

How far should the roof overhang to shade windows from the mid-day sun? Flight Path

Age 16 to 18 Challenge Level:

Use simple trigonometry to calculate the distance along the flight path from London to Sydney. Making Maths: Clinometer

Age 11 to 14 Challenge Level:

Make a clinometer and use it to help you estimate the heights of tall objects. 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. 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 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. 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. . . . 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? Circle Scaling

Age 14 to 16 Challenge Level:

Describe how to construct three circles which have areas in the ratio 1:2:3. 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? 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. . . . 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? 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. 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. Dodecawhat

Age 14 to 16 Challenge Level:

Follow instructions to fold sheets of A4 paper into pentagons and assemble them to form a dodecahedron. Calculate the error in the angle of the not perfectly regular pentagons you make. 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. 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. Lying and Cheating

Age 11 to 14 Challenge Level:

Follow the instructions and you can take a rectangle, cut it into 4 pieces, discard two small triangles, put together the remaining two pieces and end up with a rectangle the same size. Try it! 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. From All Corners

Age 14 to 16 Challenge Level:

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.