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Trigonometry
SLIDE 1
The two triangles above are similar triangles.
? What is the relationship between angles x and y?
? Complete the following
The ratio in the first triangle is equal to …. in the second .
SLIDE 2
If the two triangles are similar then
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SLIDE 3
When triangles are similar the ratio of their sides remains the same, regardless of the size.
This holds for any triangle, hence, for right-angled triangles. If the angle is the same in a right-angled triangle, then the ratio of the sides is the same.
Ratio of sides in a triangle has no dimension. The size of the triangle, and the units of measurement, do not change a ratio.
SLIDE 4
In a right-angled triangle
The side facing an angle is called the opposite.
The side next to an angle is called the adjacent.
The longest side, which is opposite the right-angle, is called the hypotenuse.
We use any obvious abbreviation for these, such as hyp, h and H for hypotenuse, and so on.
SLIDE 5
? In the first triangle, what is (a) the length of the side opposite the angle ? (b) The ratio of the adjacent to the hypotenuse to the angle .
? In the second triangle, find the ratio of the opposite to the adjacent to the angles (a) and (b) , giving your answer to 2 sf.
SLIDE 6
? In the first triangle
(a) The length of the side opposite the angle is 6.16
(b) The ratio of the adjacent to the hypotenuse to the angle is .
? In the second triangle, the ratio of the opposite to the adjacent to the angles
(a) is
(b) is
The two ratios are reciprocals of each other.
SLIDE 7
In a right-angled triangle,
The tangent of an angle is the ratio of the lengths of the sides opposite to adjacent.
The sine of an angle is the ratio of the lengths of the sides opposite to hypotenuse.
The cosine of the angle is the ratio of the length of the sides adjacent to hypotenuse.
These names are abbreviated to tan, sin and cos respectively.
SLIDE 8
SLIDE 9
In this exercise, give your answers to 3 sf.
Use a calculator only for division – do not use the sin, cos and tan buttons.
? In the first triangle, find the sin, cos and tan of the angle .
? In the second triangle, find the sin, cos and tan of the angle .
SLIDE 10
? In the first triangle
? In the second triangle
SLIDE 11
Remembering the ratios
A mnemonic is way that helps you remember something. People have devised various mnemonics for the trigonometric ratio.
S O H C A H T O A
Once upon a time, in the old days, when there were no calculators and we were still living in caves, there was a mathematics teacher who taught his students the following rhyme.
See Old Harry
Catch A Herring
Trawling Off America
SLIDE 12
Calculator stored values
The usefulness of all of this is that we can work out the sine, cosine and tangent of an angle without measuring a single angle or length of a triangle. The mathematics involved in this is several levels above the level of this chapter, so we ask you to take this on trust.
These values are stored in your calculator, which has buttons to find and display them.
SLIDE 13
Note. It can be useful on a calculator to use brackets when making a calculation to prevent odd things from happening. Make sure that your calculator is in degree mode. (There are ways of measuring angles, other than degrees. The degree mode is usually shown by a symbol such as “D” in the display, and the mode button can be used to select “degree mode”.)
Henceforth, we drop the symbol ° to indicate degrees, where this is obvious. All angles n this chapter are measured in degrees, so means .
Use your calculator to find the following, giving your answers to 3 significant figures.
? ?
? ?
? ?
SLIDE 14
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SLIDE 15
Example
Find the length R, giving your answer to 3 sf.
Solution
S O H C A H T O A
R is opposite the angle , so we select sine
SLIDE 16
In each case find the length of the side marked with a letter, giving your answer to 3 significant figures.
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SLIDE 17
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SLIDE 18
Example
Find the length T, giving your answer to 3 sf.
Solution
SLIDE 19
In each case find the length of the side marked with a letter, giving your answer to 3 significant figures.
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SLIDE 20
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SLIDE 21
A surveyor wishes to find the height of a building which he knows to be 82.5 m away. He uses a theodolite to measure the angle between the top of the building and the horizontal plane, as shown in the diagram.
Find the height of the building, giving your answer to 3 significant figures.
SLIDE 22
A surveyor wishes to find the height of a building which he knows to be 82.5 m away. He uses a theodolite to measure the angle between the top of the building and the horizontal plane, as shown in the diagram. Find the height of the building, giving your answer to 3 significant figures.
Solution
SLIDE 21B
A surveyor wishes to find the height of a building which he knows to be 82.5 yd away. He uses a theodolite to measure the angle between the top of the building and the horizontal plane, as shown in the diagram.
Find the height of the building, giving your answer to 3 significant figures.
SLIDE 22B
A surveyor wishes to find the height of a building which he knows to be 82.5 yd away. He uses a theodolite to measure the angle between the top of the building and the horizontal plane, as shown in the diagram. Find the height of the building, giving your answer to 3 significant figures.
Solution
SLIDE 23
The captain of a ship wishes to know how far away his ship is from a lighthouse situated near some rocks. The lighthouse is known to be 25.8 m high. He uses a sextant to measure the angle between the top of the lighthouse and the horizontal and finds it to be 0.45°. Find the distance of the ship from the lighthouse, giving your answer in km to 3 significant figures.
SLIDE 24
The captain of a ship wishes to know how far away his ship is from a lighthouse situated near some rocks. The lighthouse is known to be 25.8 m high. He uses a sextant to measure the angle between the top of the lighthouse and the horizontal and finds it to be 0.45°. Find the distance of the ship from the lighthouse, giving your answer in km to 3 significant figures.
Solution
SLIDE 23B
The captain of a ship wishes to know how far away his ship is from a lighthouse situated near some rocks. The lighthouse is known to be 25.8 yd high. He uses a sextant to measure the angle between the top of the lighthouse and the horizontal and finds it to be 0.45°. Find the distance of the ship from the lighthouse, giving your answer in miles to 3 significant figures.
SLIDE 24B
The captain of a ship wishes to know how far away his ship is from a lighthouse situated near some rocks. The lighthouse is known to be 25.8 yd high. He uses a sextant to measure the angle between the top of the lighthouse and the horizontal and finds it to be 0.45°. Find the distance of the ship from the lighthouse, giving your answer in miles to 3 significant figures.
Solution
SLIDE 25
The trigonometric ratios are the same for any given angle.
Trigonometric ratios define trigonometric functions.
For example,
Complete the following
ratio function graph mapping
SLIDE 26
ratio function graph mapping
SLIDE 27
Greek letters
In algebra we may use any letters to represent variables. Angles are often measured with the “favourite” letters, x, y and z, so we see , and and so on, where the letters stand for angles.
Mathematicians often like to use the Greek alphabet for letters, and they often use them when measuring angles. Thus, we see expressions like
SLIDE 28
Some Greek letters
Greek letter English equivalent Pronounced
a alpha
b beta
g gamma
d delta
th theta
p pi
r rho
SLIDE 29
Trigonometric functions
The trigonometric functions are mappings from angles to ratios.
is a mapping from an angle to the ratio in any right-angled triangle with angle .
SLIDE 30
Inverse trigonometric functions
For angles between 0 and 90° the trigonometric functions all have inverses. These inverse functions are mappings from a ratio to the angle.
The inverse of a function is denoted . This is read “f inverse”. It is not read “f to the minus one”.
The inverse of is . It is read, “inverse sine”. It is not read “sin to the minus one”.
Note. In the USA inverse sine, , is written , which is pronounced arcsine.
British English US English
SLIDE 30B
Inverse trigonometric functions
For angles between 0 and 90° the trigonometric functions all have inverses. These inverse functions are mappings from a ratio to the angle.
The inverse of a function is denoted . This is read “f inverse”. It is not read “f to the minus one”.
In the USA inverse sine is written , which is pronounced arcsine.
In Britain , and on calculators, the inverse of is . It is read, “inverse sine”. It is not read “sin to the minus one”.
US English British English
SLIDE 31
Complete the following
inverse
function
graph
mapping
function
graph
mapping
function
graph
mapping
SLIDE 31B
Complete the following
inverse
function
graph
mapping
function
graph
mapping
function
graph
mapping
SLIDE 32
inverse
function
graph
mapping
function
graph
mapping
function
graph
mapping
SLIDE 32B
inverse
function
graph
mapping
function
graph
mapping
function
graph
mapping
SLIDE 33
Calculators have buttons for finding the inverse trigonometric functions. These are usually used by pressing the shift key, with expressions for the inverses put above the buttons for sine, cosine and tangent. Give the angles to 0.1 of a degree.
Use the calculator to find
? ? ?
Note. The sine and cosine of any angle is a number between and +1. The tangent may be any number.
SLIDE 33B
Calculators have buttons for finding the inverse trigonometric functions. These are usually used by pressing the shift key, with expressions for the inverses put above the buttons for sine, cosine and tangent. Give the angles to 0.1 of a degree.
Use the calculator to find
?
?
?
Note. The sine and cosine of any angle is a number between and +1. The tangent may be any number.
SLIDE 34
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SLIDE 34B
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SLIDE 35
Example
Find the angle x, giving your answer to 0.1°.
Solution
SLIDE 35B
Example
Find the angle x, giving your answer to 0.1°.
Solution
SLIDE 36
In each case find the angle to 0.1°.
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SLIDE 37
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SLIDE 37B
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SLIDE 38
A square pyramid has base 8 cm and height 10.5 cm. Find the angle marked in the diagram.
SLIDE 38B
A square pyramid has base 8 in and height 10.5 in. Find the angle marked in the diagram.
SLIDE 39
This is a two-step problem. We can find the angle in the triangle AEF provided we know the length of x.
In the square ABCD, we have
By Pythagoras,
SLIDE 39B
This is a two-step problem. We can find the angle in the triangle AEF provided we know the length of x.
In the square ABCD, we have
By Pythagoras,
SLIDE 40
The diagram shows a cuboid. The dimensions of the cuboid are given by , and . Find the length of the space diagonal x and the angle .
SLIDE 40B
The diagram shows a cuboid. The dimensions of the cuboid are given by , and . Find the length of the space diagonal x and the angle .
SLIDE 41
SLIDE 41B
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