Resources tagged with: Sine, cosine, tangent

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

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?

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.

History of Trigonometry - Part 2

Age 11 to 18

The second of three articles on the History of Trigonometry.

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.

Geometric Trig

Age 16 to 18 Short
Challenge Level

Trigonometry, circles and triangles combine in this short challenge.

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.

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?

Diagonals for Area

Age 16 to 18
Challenge Level

Can you prove this formula for finding the area of a quadrilateral from its diagonals?

History of Trigonometry - Part 3

Age 11 to 18

The third of three articles on the History of Trigonometry.

After Thought

Age 16 to 18
Challenge Level

Which is larger cos(sin x) or sin(cos x) ? Does this depend on x ?

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.

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.

Pythagoras on a Sphere

Age 16 to 18
Challenge Level

Prove Pythagoras' Theorem for right-angled spherical triangles.

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.

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?. . . .

Three by One

Age 16 to 18
Challenge Level

There are many different methods to solve this geometrical problem - how many can you find?

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.

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?

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.

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?

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

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?

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.

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?

Flight Path

Age 16 to 18
Challenge Level

Use simple trigonometry to calculate the distance along the flight path from London to Sydney.

Pumping the Power

Age 16 to 18
Challenge Level

What is an AC voltage? How much power does an AC power source supply?

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.

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.

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.

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. . . .

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 ?

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?

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.

Circle Scaling

Age 14 to 16
Challenge Level

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

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.

Moving Squares

Age 14 to 16
Challenge Level

How can you represent the curvature of a cylinder on a flat piece of paper?

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.

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. . . .

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.

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?

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?

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.

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.

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?

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?

Raising the Roof

Age 14 to 16
Challenge Level

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

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.

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?

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.

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.