# Search by Topic

#### Resources tagged with Sine, cosine, tangent similar to Weird Universes:

Filter by: Content type:
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### There are 59 results

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

### 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.

### Flight Path

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

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

### 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.

### 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.

### 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.

### 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.

### 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.

##### Age 14 to 18 Challenge Level:

How would you design the tiering of seats in a stadium so that all spectators have a good view?

### Diagonals for Area

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

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

### 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.

### Moving Squares

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

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

### Geometric Trig

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

Trigonometry, circles and triangles combine in this short challenge.

### History of Trigonometry - Part 3

##### Age 11 to 18

The third of three articles on the History of Trigonometry.

### Pumping the Power

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

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

### Bend

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

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

### 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?

### 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?

### 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?

### Trigonometric Protractor

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

An environment that simulates a protractor carrying a right- angled triangle of unit hypotenuse.

### Raising the Roof

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

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

### 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?

### 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.

##### 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.

### 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?. . . .

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

If you were to set the X weight to 2 what do you think the angle might be?

### 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?

### Round and Round a Circle

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

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

### History of Trigonometry - Part 2

##### Age 11 to 18

The second of three articles on the History of Trigonometry.

### 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.

### 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?

### 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?

### Circle Scaling

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

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

### 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?

##### 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.

### 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?

### 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. . . .

### Orbiting Billiard Balls

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

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

### 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 ?

### 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. . . .

### 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?

### 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.

### 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.

##### 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?

### 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...

### 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?

### 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.
The text passage is a collection of mathematical problems and challenges, primarily focused on trigonometry, specifically sine, cosine, and tangent functions. Each problem is presented as a separate challenge with a title and a brief description of the task.
multiple_questions

the passage contains multiple distinct mathematical questions or problems, each requiring a different application of trigonometric principles. however, none of the questions are accompanied by answers within the text.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Search by Topic

#### Resources tagged with Sine, cosine, tangent similar to Weird Universes:

Filter by: Content type:
Age range:
Challenge level:

### There are 59 results

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

### 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.

### Flight Path

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

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

### 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.

### 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.

### 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.

### 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.

### 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.

##### Age 14 to 18 Challenge Level:

How would you design the tiering of seats in a stadium so that all spectators have a good view?

### Diagonals for Area

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

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

### 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.

### Moving Squares

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

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

### Geometric Trig

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

Trigonometry, circles and triangles combine in this short challenge.

### History of Trigonometry - Part 3

##### Age 11 to 18

The third of three articles on the History of Trigonometry.

### Pumping the Power

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

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

### Bend

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

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

### 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?

### 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?

### 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?

### Trigonometric Protractor

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

An environment that simulates a protractor carrying a right- angled triangle of unit hypotenuse.

### Raising the Roof

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

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

### 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?

### 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.

##### 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.

### 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?. . . .

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

If you were to set the X weight to 2 what do you think the angle might be?

### 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?

### Round and Round a Circle

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

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

### History of Trigonometry - Part 2

##### Age 11 to 18

The second of three articles on the History of Trigonometry.

### 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.

### 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?

### 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?

### Circle Scaling

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

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

### 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?

##### 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.

### 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?

### 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. . . .

### Orbiting Billiard Balls

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

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

### 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 ?

### 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. . . .

### 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?

### 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.

### 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.

##### 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?

### 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...

### 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?

### 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.
user
The text passage is a collection of mathematical problems and challenges, primarily focused on trigonometry, specifically sine, cosine, and tangent functions. Each problem is presented as a separate challenge with a title and a brief description of the task.

TYPE: MULTIPLE_QUESTIONS

The passage contains multiple distinct mathematical questions or problems, each requiring a different application of trigonometric principles. However, none of the questions are accompanied by answers within the text.