Resources tagged with: Sine, cosine, tangent

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

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.

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.

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.

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.

History of Trigonometry - Part 3

Age 11 to 18

The third of three articles on the History of Trigonometry.

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.

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?

Circle Scaling

Age 14 to 16 Challenge Level:

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

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.

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.

History of Trigonometry - Part 2

Age 11 to 18

The second of three articles on the History of Trigonometry.

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

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.

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.

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.

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?

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?

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?

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?

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?

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. . . .

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. . . .

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?

Geometric Trig

Age 16 to 18 Short Challenge Level:

Trigonometry, circles and triangles combine in this short challenge.

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.

After Thought

Age 16 to 18 Challenge Level:

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

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.

Raising the Roof

Age 14 to 16 Challenge Level:

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

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 ?

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.

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.

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...

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?

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.

Three by One

Age 16 to 18 Challenge Level:

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

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.

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.

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?

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?

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.

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.

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?

Pythagoras on a Sphere

Age 16 to 18 Challenge Level:

Prove Pythagoras' Theorem for right-angled spherical triangles.

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?

Diagonals for Area

Age 16 to 18 Challenge Level:

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

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?

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?. . . .

Round and Round a Circle

Age 14 to 16 Challenge Level:

Can you explain what is happening and account for the values being displayed?

Flight Path

Age 16 to 18 Challenge Level:

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