# Resources tagged with: Sine, cosine, tangent

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There are 59 NRICH Mathematical resources connected to Sine, cosine, tangent, you may find related items under Pythagoras and Trigonometry.

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

##### Age 14 to 16Challenge 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 16Challenge 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? ### Spokes

##### Age 16 to 18Challenge Level

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

##### Age 14 to 16Challenge 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 18Challenge 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 16Challenge 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 16Challenge 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 18Challenge 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 18Challenge 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 18Challenge 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 16Challenge 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 16Challenge 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 18Challenge 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 18Challenge 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 18Challenge 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 18Challenge Level

Can you deduce the familiar properties of the sine and cosine functions starting from these three different mathematical representations? ##### Age 14 to 18Challenge 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 ShortChallenge Level

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

##### Age 14 to 16Challenge 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 18Challenge Level

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

##### Age 14 to 16Challenge Level

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

##### Age 14 to 16Challenge 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 18Challenge 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 16Challenge 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 18Challenge Level

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

##### Age 14 to 16Challenge Level

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

##### Age 16 to 18Challenge Level

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

##### Age 11 to 14Challenge Level

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 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 16Challenge 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 16Challenge 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 16Challenge 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 18Challenge 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 16Challenge Level

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

##### Age 14 to 16Challenge 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 16Challenge 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 18Challenge 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 16Challenge 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 18Challenge 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 16Challenge 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 16Challenge 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 14Challenge 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 16Challenge Level

Prove that the shaded area of the semicircle is equal to the area of the inner circle. ### From All Corners

##### Age 14 to 16Challenge 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. ### Diagonals for Area

##### Age 16 to 18Challenge Level

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

##### Age 16 to 18Challenge 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.