Trigonometry:-
[The word itself literally means "Three-sided figure measurements". In fact, "trigon" is an archaic name for a triangle.]
At school level trigonometry begins with the study of right-angled triangles, and in particular the ratios (i.e. division) of pairs of its sides for a particular angle in the corner. As the corner angle varies, so will the ratios of the sides.
The actual size of the triangle doesn't matter, so to explain the basic idea of trigonometry we can think of a bicycle wheel spoke of unit length which rotates as the wheel turns anticlockwise, starting with the spoke horizontal and to the right of the axle.. If you then stop the wheel in any position, and measure the angle turned through between the spoke and the horizontal, then the vertical height of the end of the spoke will give a measure called the SINE of this angle, while the horizontal 'shadow' or projection of the spoke gives the value of the COSINE of the angle turned through.
More generally, the three sides of a right-angled triangle are named according to where they are in relation to the corner angle being considered at the time.
The side opposite the angle is called just that, and denoted by the letter O;
The side alongside, or next to, the angle is called the adjacent side, denoted by A;
The side opposite the right-angle is called the hypotenuse, denoted by H.
[You should have met the word 'hypotenuse' when doing the Theorem of Pythagoras, which stated that "the square of the hypotenuse is equal to the sum of the squares of the other two sides". Therefore, for the letters above, H² = A² + O² .]
The ratio (Length of O) ÷ (Length of H) is called the SINE of the angle;
The ratio (Length of A) ÷ (Length of H) is called the COSINE of the angle;
The ratio (Length of O) ÷ (Length of A) is called the TANGENT of the angle.
[Note here that the word 'tangent' doesn't now mean a line touching a circle or curve, which is what you've probably met before.]
If we call the angle ß, (Greek letter BETA) these ratios can be written as:-
Sin ß = O/H ; Cos ß = A/H ; Tan ß = O/A
(Don't forget that Greek letters are used a lot in maths, and also in science, and you can get the whole alphabet from your list of fonts if you scroll down to the one called 'symbol'. Some of them, such as ß and µ, can also be found in the list of alternative characters, using the ALT key.)
Now for a numerical example:-
Imagine a right-angled triangle with an angle of 60º in the bottom left-hand corner ( and therefore 30º in the other corner, since the three angles of a triangle have to add up to 180º).
[This can be thought of as an equilateral triangle cut in half down the middle.]
In this triangle, if the length of the long side is 2 units, the short side will be 1 unit (being half the base of the equilateral triangle), and by Pythagoras's Theorem the third side will be of length 'sq.root of 3', or 1.732 units (to 3 decimal places).
So for the 60º angle in the bottom left-hand corner, the adjacent side will have a length of 1 unit, the hypotenuse will be 2 units, and the opposite side will be 1.732 units.
We are now in a position to talk about the three ratios (sin, cos and tan) for the angle 60º using actual lengths.
Here are those three ratios for the angle 60º:-
sin60º = O/H = 1.732 ÷ 2 = 0.866
cos60º = A/H = 1 ÷ 2 = 0.5
tan60º = O/A = 1.732 ÷ 1 = 1.732
Using the same triangle, if we now turn our attention to the 30º angle, the side adjacent to the angle will now be the one of length 1.732, and the opposite side will have length 1.
(The hypotenuse, being the side opposite the right-angle, remains unchanged at 2 units long.)
So sin 30º , which = O/H, will now be 1 ÷ 2 , which = 0.5 , because the side opposite the 30º angle has now changed to the 1 unit length.
Similarly, the side of length 1.732, which was opposite the 60º angle, is now adjacent to the 30º angle, so in the formula for cos 30º we put A = 1.732 and H = 2.
Then we get cos 30º = A/H = 1.732 ÷ 2 = 0.866
Exercise:-
Draw a right-angled triangle with a hypotenuse of length 5cm and any size angle in the lower corner.
Now measure this lower corner angle with a protractor, and also the lengths of the other 2 sides of the triangle.
Divide the length of the side opposite the lower corner angle by 5 and compare your answer with the value of the sine of the angle as given by your sine tables or calculator.
For example, if you found the corner angle of the triangle you'd drawn was 23º, you should find that dividing the opposite side by the hypotenuse gives an answer of about 0.4 (or 0.39, to be more accurate).
You can practice this for different shapes of triangle provided they all have a right-angle in one corner, in each case measuring the lengths of the sides and dividing them as above, then comparing your answers with the values given by tables or calculator for the sine (or cosine or tangent) ratios of the angles of your triangle.
[The word itself literally means "Three-sided figure measurements". In fact, "trigon" is an archaic name for a triangle.]
At school level trigonometry begins with the study of right-angled triangles, and in particular the ratios (i.e. division) of pairs of its sides for a particular angle in the corner. As the corner angle varies, so will the ratios of the sides.
The actual size of the triangle doesn't matter, so to explain the basic idea of trigonometry we can think of a bicycle wheel spoke of unit length which rotates as the wheel turns anticlockwise, starting with the spoke horizontal and to the right of the axle.. If you then stop the wheel in any position, and measure the angle turned through between the spoke and the horizontal, then the vertical height of the end of the spoke will give a measure called the SINE of this angle, while the horizontal 'shadow' or projection of the spoke gives the value of the COSINE of the angle turned through.
More generally, the three sides of a right-angled triangle are named according to where they are in relation to the corner angle being considered at the time.
The side opposite the angle is called just that, and denoted by the letter O;
The side alongside, or next to, the angle is called the adjacent side, denoted by A;
The side opposite the right-angle is called the hypotenuse, denoted by H.
[You should have met the word 'hypotenuse' when doing the Theorem of Pythagoras, which stated that "the square of the hypotenuse is equal to the sum of the squares of the other two sides". Therefore, for the letters above, H² = A² + O² .]
The ratio (Length of O) ÷ (Length of H) is called the SINE of the angle;
The ratio (Length of A) ÷ (Length of H) is called the COSINE of the angle;
The ratio (Length of O) ÷ (Length of A) is called the TANGENT of the angle.
[Note here that the word 'tangent' doesn't now mean a line touching a circle or curve, which is what you've probably met before.]
If we call the angle ß, (Greek letter BETA) these ratios can be written as:-
Sin ß = O/H ; Cos ß = A/H ; Tan ß = O/A
(Don't forget that Greek letters are used a lot in maths, and also in science, and you can get the whole alphabet from your list of fonts if you scroll down to the one called 'symbol'. Some of them, such as ß and µ, can also be found in the list of alternative characters, using the ALT key.)
Now for a numerical example:-
Imagine a right-angled triangle with an angle of 60º in the bottom left-hand corner ( and therefore 30º in the other corner, since the three angles of a triangle have to add up to 180º).
[This can be thought of as an equilateral triangle cut in half down the middle.]
In this triangle, if the length of the long side is 2 units, the short side will be 1 unit (being half the base of the equilateral triangle), and by Pythagoras's Theorem the third side will be of length 'sq.root of 3', or 1.732 units (to 3 decimal places).
So for the 60º angle in the bottom left-hand corner, the adjacent side will have a length of 1 unit, the hypotenuse will be 2 units, and the opposite side will be 1.732 units.
We are now in a position to talk about the three ratios (sin, cos and tan) for the angle 60º using actual lengths.
Here are those three ratios for the angle 60º:-
sin60º = O/H = 1.732 ÷ 2 = 0.866
cos60º = A/H = 1 ÷ 2 = 0.5
tan60º = O/A = 1.732 ÷ 1 = 1.732
Using the same triangle, if we now turn our attention to the 30º angle, the side adjacent to the angle will now be the one of length 1.732, and the opposite side will have length 1.
(The hypotenuse, being the side opposite the right-angle, remains unchanged at 2 units long.)
So sin 30º , which = O/H, will now be 1 ÷ 2 , which = 0.5 , because the side opposite the 30º angle has now changed to the 1 unit length.
Similarly, the side of length 1.732, which was opposite the 60º angle, is now adjacent to the 30º angle, so in the formula for cos 30º we put A = 1.732 and H = 2.
Then we get cos 30º = A/H = 1.732 ÷ 2 = 0.866
Exercise:-
Draw a right-angled triangle with a hypotenuse of length 5cm and any size angle in the lower corner.
Now measure this lower corner angle with a protractor, and also the lengths of the other 2 sides of the triangle.
Divide the length of the side opposite the lower corner angle by 5 and compare your answer with the value of the sine of the angle as given by your sine tables or calculator.
For example, if you found the corner angle of the triangle you'd drawn was 23º, you should find that dividing the opposite side by the hypotenuse gives an answer of about 0.4 (or 0.39, to be more accurate).
You can practice this for different shapes of triangle provided they all have a right-angle in one corner, in each case measuring the lengths of the sides and dividing them as above, then comparing your answers with the values given by tables or calculator for the sine (or cosine or tangent) ratios of the angles of your triangle.
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