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Broad Topics > Pythagoras and Trigonometry > Cosine

### 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.

### 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?

### Three by One

##### Age 16 to 18 Challenge Level:

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

### 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.

### Round and Round a Circle

##### Age 14 to 16 Challenge Level:

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

### Pythagoras on a Sphere

##### Age 16 to 18 Challenge Level:

Prove Pythagoras' Theorem for right-angled spherical triangles.

### 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.

### 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?

### After Thought

##### Age 16 to 18 Challenge Level:

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