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There are **60** NRICH Mathematical resources connected to **Sine, cosine, tangent**, you may find related items under Pythagoras and trigonometry.

Problem
Primary curriculum
Secondary curriculum
### Sine and Cosine

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Where Is the Dot?

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Spokes

Draw three equal line segments in a unit circle to divide the circle into four parts of equal area.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Figure of Eight

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 ?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Octa-flower

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?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Far Horizon

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 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Inscribed in a Circle

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Belt

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Three by One

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### 8 Methods for Three by One

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? Which do you like best?

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Doesn't Add Up

In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cosines Rule

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### So Big

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 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Shape and Territory

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Logosquares

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Over the Pole

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### A Scale for the Solar System

The Earth is further from the Sun than Venus, but how much further? Twice as far? Ten times?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pythagoras on a Sphere

Prove Pythagoras' Theorem for right-angled spherical triangles.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Raising the Roof

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Flight Path

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

Age 16 to 18

Challenge Level

General
Primary curriculum
Secondary curriculum
### Making Maths: Clinometer

You can use a clinometer to measure the height of tall things that you can't possibly reach to the top of, Make a clinometer and use it to help you estimate the heights of tall objects.

Age 11 to 14

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Muggles, Logo and Gradients

Logo helps us to understand gradients of lines and why Muggles Magic is not magic but mathematics. See the problem Muggles magic.

Age 11 to 18

Problem
Primary curriculum
Secondary curriculum
### Orbiting Billiard Balls

What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Sine and Cosine for Connected Angles

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Screen Shot

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 before being reflected across to the opposite wall and so on until it hits the screen.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Bend

What is the longest stick that can be carried horizontally along a narrow corridor and around a right-angled bend?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Circle Scaling

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Circle Box

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Squ-areas

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 triangular areas are enclosed. What is the area of this convex hexagon?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### The Dodecahedron

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 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Circumnavigation

The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Small Steps

Two problems about infinite processes where smaller and smaller steps are taken and you have to discover what happens in the limit.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Dodecawhat

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.

Age 14 to 16

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Why Stop at Three by One

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.

Age 16 to 18

Problem
Primary curriculum
Secondary curriculum
### Farhan's Poor Square

From the measurements and the clue given find the area of the square that is not covered by the triangle and the circle.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Lying and Cheating

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!

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Round and Round

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### From All Corners

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Diagonals for Area

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Sine Problem

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### 30-60-90 Polypuzzle

Re-arrange the pieces of the puzzle to form a rectangle and then to form an equilateral triangle. Calculate the angles and lengths.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Strange Rectangle 2

Find the exact values of some trig. ratios from this rectangle in which a cyclic quadrilateral cuts off four right angled triangles.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Six Discs

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Making Waves

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Gold Again

Without using a calculator, computer or tables find the exact values of cos36cos72 and also cos36 - cos72.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Ball Bearings

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Coke Machine

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