The order of physiographic units in Brazil while going from North West
to South East.
(a) Parana River Basin - Guyana Highlands - Brazilian Highlands
(b) Guyana Highlands - Amazon River Basin - Brazilian Highlands
(c) Coastal Plains - Amazon River Basin - Brazilian Highlands
Answers
Answer:
Trigonometry, as the name might suggest, is all about triangles.
More specifically, trigonometry is about right-angled triangles, where one of the internal angles is 90°. Trigonometry is a system that helps us to work out missing or unknown side lengths or angles in a triangle.
There is more about triangles on our page on Polygons should you need to brush up on the basics before you read further here.
Right-Angled Triangles: A Reminder
A right-angled triangle has a single right angle. By definition, that means that all sides cannot be the same length. A typical right-angled triangle is shown below.
Important Terms for Right-Angled Triangles
Right-angled triangle showing the Opposite, Adjacent and Hypotenuse
The right angle is indicated by the little box in the corner.
The other angle that we (usually) know is indicated by θ (theta).
The side opposite the right angle, which is the longest side, is called the hypotenuse.
The side opposite θ is called the opposite.
The side next to θ which is not the hypotenuse is called the adjacent.
Pythagoras’ Theorem vs. Trigonometry
Pythagoras was a Greek philosopher who lived over 2500 years ago. He is credited with a number of important mathematical and scientific discoveries, arguably the most significant of which has become known as Pythagoras’ Theorem.
It is an important rule that applies only to right-angled triangles. It says that ‘the square on the hypotenuse is equal to the sum of the squares on the other two sides.’
That sounds rather complicated, but it is actually quite a simple concept when we see it in a diagram:
Pythagoras' Theorem. Demonstating that the square on the hypotenuse is equal to the sum of the squares on the other two sides of a right-angled triangle.
Pythagoras’ Theorem says :
a2 + b2 = c2
So, if we know the length of two sides of a triangle and we need to calculate the third, we can use Pythagoras’ Theorem.
However, if we know only one side length and one of the internal angles, then Pythagoras is no use to us on its own and we need to use trigonometry.
Introducing Sine, Cosine and Tangent
There are three basic functions in trigonometry, each of which is one side of a right-angled triangle divided by another.
The three functions are:
Name Abbreviation Relationship to sides of the triangle
Sine Sin Sin (θ) = Opposite/hypotenuse
Cosine Cos Cos (θ) = Adjacent/hypotenuse
Tangent Tan Tan (θ) = Opposite/adjacent
Calculating Sine, Cosine and Tangent
You may find it helpful to remember Sine, Cosine and Tangent as SOH CAH TOA.
Remembering trigonometric functions can be difficult and confusing to begin with. Even SOH CAH TOA can be tricky. You could try making up a funny mnemonic to help you remember. Just keep each group of three letters in the same order.
For example, TOA SOH CAH could be ’The Old Archaeologist Sat On His Coat And Hat’.
Top Tip!
Because of the relationships between them, Tan θ can also be calculated as:
Sin θ / Cos θ.
This means that:
Sin θ = Cos θ × Tan θ and
Cos θ = Sin θ / Tan θ.
Trigonometry in a Circle
For more about circles, or a quick refresher, see our page on Circles and Curved Shapes.
When considering triangles, we are limited to angles less than 90°. However, trigonometry is equally applicable to all angles, from 0 to 360°. To understand how the trigonometric functions work with angles greater than 90°, it is helpful to think about triangles constructed within a circle.
The Cartesian Coordinates of a circle.
Consider a circle, divided into four quadrants.
Conventionally, the centre of the circle is considered as a Cartesian coordinate of (0,0). That is, the x value is 0 and the y value is 0. For more about this, see our page on Cartesian coordinates.
Anything to the left of the centre has an x value of less than 0, or is negative, while anything to the right has a positive value.
Similarly anything below the centre point has a y value of less than 0, or is negative and any point in the top of the circle has a positive y value.
Using a circle with trigonometric functions for angles greater than 90°.
Diagram i shows what happens if we draw a radius from the centre of the circle to the right along the x axis (we say this is in a positive direction).
We then rotate the radius in an anticlockwise direction through an angle theta θ. This creates a right-angled triangle.
Sin θ = opposite (red line)
hypotenuse (blue line)
Cos θ = adjacent (green line)
hypotenuse (blue line)
In Diagram ii, we have rotated the radius further in an anti-clockwise direction, past the vertical (y axis) into the next quadrant. Here θ is an obtuse angle, between 90° and 180°. The reference angle alpha α is equal to 180° − θ, and is the acute angle within the right-angled triangle.