Hi, I'm Dr Ian van Loosen from the Department of Maths and Stats at Curtin University.
Let's say you're a caveman and you see a Woolly Mammoth. For protection you lie on the ground,
45 meters away from the Woolly Mammoth. You look up at an angle of elevation of 4 degrees
and you see the top of the Woolly Mammoths head.
What you would like to know is what is the height of this Woolly Mammoth? Problems such
as these can be solved by using trigonometry of a right angled triangle.
Let's start by looking at the theory.
Trigonometric functions are a fundamental tool in modelling and solving many science
and engineering problems. Trigonometry can be applied to find the height of an object,
the distance between bodies, as well as the rotation of an object.
We will begin by defining the trigonometric functions as the ratios in a
right angled triangle.
When we say we want to solve a right angled triangle, we mean we want to determine any
unknown angles, as well as any unknown side lengths. The easiest triangle to solve is
a right triangle.
So lets consider a right triangle. A right triangle is any triangle which contains an
angle of exactly 90 degrees. The way we denote that is as follows.
Let's also assume that this triangle contains the angle theta. With respect to theta, we
classify the sides as being either the opposite, the adjacent or the hypotenuse.
So the side which is opposite the angle theta is denoted as the opposite side.
The side which is adjacent - that is next to our angle theta - is denoted as our adjacent
side.
And our remaining side - which is opposite our right angle - is denoted as the hypotenuse.
It's the ratio of these three lengths which makes up our three trigonometric ratios.
The three basic trigonometric ratios are the sine, cosine and tangent.
The sine of the angle theta is denoted as the opposite side divided by the hypotenuse
side.
The cosine of the angle theta is equal to our adjacent length divided by our hypotenuse
length.
And finally the tangent of that angle theta is denoted as being the opposite length divided
by the adjacent length.
Whenever you write sine, cosine, and tan you must always associate it with an angle theta.
Without an angle theta it does not make any sense.
Also when we're trying to solve an unknown triangle, to determine any unknown side lengths
it also helps sometimes to use Pythagoras Theorem. Pythagoras Theorem tells us that
the hypotenuse side squared, must equal to our adjacent side squared
plus our opposite side squared.
Also if you're trying to determine any unknown angle within the triangles, it also helps
to use the fact that, all the angles in the triangle must add up to 180 degrees.
So let's return to our initial problem of trying to determine the height of our Woolly
Mammoth.
So here we basically have a right angled triangle.
What we'd like to do is determine this side length here. That is, the height of
the Woolly Mammoth.
So let's let this angle of 4 degrees be denoted by our angle theta. We've then classified
the sides of this triangle.
To begin with this side, which is opposite to our angle theta, we'll denote as the opposite.
This side length here, which is adjacent to our angle theta, we'll denote as
our adjacent length.
So we know theta, we know the adjacent length, we're trying to determine the opposite.
So we ask our self which of the three trigonomic ratios uses opposite and adjacent?
From before we know the tangent of an angle theta, must equal to the opposite length,
divided by our adjacent length.
So we were given the angle theta, 4 degrees.
Opposite is the height of the Woolly Mammoth, that we're trying to determine.
And our adjacent length is our distance away from the Woolly Mammoth - 45 meters.
So, trying to solve for opposite. We must make it the subject of the equation. We must
get rid of this 45. To do so we just multiple both sides of the equation by 45.
Multiplying this side by 45 will cancel out our 45 from the denominator.
So to evaluate the opposite we simply need to multiple 45 by the tan of 4 degrees. When
evaluating the tan of 4 degrees make sure your calculator is in degrees mode.
So tan of 4 degrees times by 45 will tell us our opposite length of the triangle.
So we've determined this opposite side and consequently we've therefore determined the
height of our Woolly Mammoth.
So now lets look at another scenario.
Let's say now while lying on the ground, you notice there's a stone lying on the ground
directly in front of you. You crawl forward and you pick up that stone.
Now looking up at an elevation of 6 degrees, we see the top of the Woolly Mammoths head.
We now want to knows how far we want to throw that stone to hit the Woolly Mammoth in
the top of the head.
So again we have our right angled triangle. We know what this angle is. We denote that
by theta. We've got a side length in our triangles - so is that the opposite, adjacent or the
hypotenuse? So with respect to this angle theta, here, this is the opposite side.
This is the length that I'm trying to determine. The length that I need to throw the stone.
That's the side which is opposite our right angle. So that's the hypotenuse.
So trigonomic ratio which uses opposite and hypotenuse?
Sine of an angle equals the opposite length, divided by the hypotenuse length. So again,
we know what our angle is.
6 degrees.
We know the opposite length, that being the height of our Woolly Mammoth.
And I'm trying to determine this hypotenuse length.
So again, trying to determine the hypotenuse I need to make it the subject of the equation.
So, what I'll do is multiple both sides of this equation by the hypotenuse.
When I multiply this right hand side by the hypotenuse. The hypotenuse on the numerator
will cancel that one on the denominator.
Solving the hypotenuse. Again needing to make it the subject of the equation. I must get
rid of this sine of 6 degrees. Doing so, I'll divide both sides by the sine of 6 degrees.
Dividing this left hand side by sine of 6 degrees will again cancel out with the sine
of 6 degrees on the numerator.
So I'm left with the evaluator the hypotenuse which is equal to 3.15 divided by the
sine of 6 degrees. Again, making sure your calculator is in degrees mode.
Evaluating that will give us 30.14.
So therefore the caveman needs to throw the rock 30.14 meters to hit that Mammoth
on the top of the head.
Now lets look at our third and final scenario.
So in this example lets say we have a flying fox which is located 10 meters above the ground.
The total length of the flying fox, to the ground, is 50 meters. What we would like to
know is what is the total horizontal distance traversed by taking that flying fox.
So again, in this situation we have a right angled triangle.
We're trying to determine an unknown side length, but we already have two side lengths
given. So to determine this unknown side length, here we can apply Pythagoras Theorem.
As mentioned before Pythagoras Theorem tells us that the hypotenuse length squared, must
equal to the sum of the two other sides squared.
So, 50 squared, must equal to our unknown length x squared plus our other side
10 squared.
So we need to solve for x. Make it the subject of the equation. I'll start by subtracting
10 squared from both sides.
Evaluating this.
So x squared is equal to 24 hundred.
To solve for x. Take the square root of 24 hundred.
So therefore we know that the horizontal distance covered is approximately 49 meters.
Now lets say we'd like to know what the steepness of this flying fox is. So what I want to do
is to determine this angle theta.
So again, a right angled triangle, containing an unknown angle theta. I'll label my sides
as being either the opposite, adjacent or hypotenuse.
So the side opposite to theta, is our opposite side.
The side adjacent to our angle theta is our adjacent side.
And the side opposite our right angle is my hypotenuse side.
So I've given all three side lengths, I'm trying to determine this angle. Here I basically
have the choice of three trigonomic ratios to use.
Lets say for arguments sake I decide to use cosine.
We know the cosine of the angle theta must equal to our adjacent length divided by our
hypotenuse. So cosine of theta, the adjacent length.
49, divided by our hypotenuse, 50. Here we need
to determine this angle theta from our given side lengths.
Here we therefore need to use an inverse trig function. The inverse trig functions are also
sometimes referred to as the arc functions.
So here to solve for theta, I need to take the cosine inverse of 49 divided by 50.
So evaluating this on your calculator. Cosine inverse, or arc cosine, of 49 divided by 50.
Will give us 11.5
So the steepness of this flying fox is 11 and a half degrees.
So far, we've defined the trigonometric functions as the ratio of sides of right triangles.
But not all triangles contain a right angle.
These triangles are referred to as oblique triangles. So how do we solve such a triangles
as these? Oblique triangles will require the use of either the Sine rule or Cosine rule
depending on the information given.
Thanks for watching