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Sine and cosine formulas
SLIDE 1
Revision – trigonometry and Pythagoras
Find
? The angle
? The perimeter of the quadrilateral ABCD
SLIDE 2
?
?
SLIDE 3
Find
? x and y in terms of r and
? r and in terms of x and y
SLIDE 4
?
?
SLIDE 4B
?
?
SLIDE 5
The expression means
In words, “sine squared x is the same as sine x all squared”.
Evaluate
? ?
? ?
? ?
SLIDE 6
?
?
?
?
?
?
SLIDE 7
Evaluate
? ?
? ?
SLIDE 8
?
?
?
?
SLIDE 9
Prove
?
?
SLIDE 10
Prove
? To prove
? To prove
SLIDE 11
In the figure , and the altitude of the triangle ABC is h. Find h, a and c, giving your answers to 3 significant figures.
SLIDE 12
SLIDE13
Find h, x, y, a and the angle .
SLIDE 14
To find h, x, y, a and the angle .
Solution
SLIDE 14B
To find h, x, y, a and the angle .
Solution
SLIDE 15
For a triangle ABC it is customary to label angles and sides in the manner shown in the figure.
The side opposite the point A is labelled a.
The angle at A is named after the point A and labelled A.
SLIDE 16
Add labels to the angles and sides of the following.
?
?
SLIDE 17
?
?
SLIDE 18
It is customary not to mark the angles into the diagram, to avoid a cluttered diagram.
If we write then we mean .
SLIDE 19
The sine rule
The sine rule is
In any given triangle, the ratio of the sine of the angle to the length of the side opposite that angle is constant.
Observations
? It is usual to use the sine rule involving just one pair of angles and sides at a time. For example,
? By cross multiplication, we can flip and rearrange the rule.
SLIDE 20
Use the sine rule to find the lengths of a and c.
SLIDE 21
The angle A is
SLIDE 22
Find the length PQ and the angles Q and R.
SLIDE 23
SLIDE 24
? Find and
? Find and
? What do you observe in both cases?
SLIDE 25
?
?
? and
SLIDE 26
? What is the definition of in the above right-angled triangle?
? An obtuse angle x is such that . Why can there be no obtuse angle in any right-angled triangle?
? If x is an obtuse angle, can there be a
SLIDE 27
Concept stretching
?
? An obtuse angle x is such that .
No right-angled triangle can have an obtuse angle. This is because the angle sum of a triangle is . Since a right-angle is the other two angles in the triangle must add to , so neither of them can be greater than (obtuse).
? If x is an obtuse angle, can there be a
By the above definition of the sine of an angle and given that there cannot be a right-angled triangle with an obtuse angle, then it appears that there cannot be a sine of an obtuse angle.
Exception. We can stretch the concept by defining the sine of the obtuse angle. The definition must be consistent with the existing definition for a right-angle triangle.
SLIDE 28
We inscribe a right-angled triangle within a circle of radius 1.
Question
What is the relationship between the sine and cosine of an angle and the height and base of a right-angle triangle?
SLIDE 29
The sine of an angle is the same as the height of a right-angled triangle with hypotenuse 1.
The cosine of an angle is the same as the base of a right-angled triangle with hypotenuse 1.
SLIDE 30
In navigation angles are measured clockwise from the direction pointing North. This kind of angle is called a bearing.
In geometry and mathematics angles are measured anti-clockwise from the direction pointing along the positive x-axis. This kind of angle is called an angle or circle measure.
SLIDE 31
The xy¬-plane (the coordinate plane) is divided into four quadrants.
SLIDE 32
The sine of an obtuse angle is the same as the “height” of the right-angled triangle with unit hypotenuse drawn in the second quadrant. In this quadrant is positive.
The sine of a reflect angle is the same as the “height” of the right-angled triangle with unit hypotenuse drawn in the third or fourth quadrants. In this quadrant as the “height” is pointing downwards below the x-axis, is negative.
SLIDE 33
Second quadrant
Third quadrant
Fourth quadrant
SLIDE 34
Repeat the analysis for the cosine of an obtuse and reflex angle, stating the sign of this angle in the second, third and fourth quadrants.
SLIDE 35
Second quadrant
Third quadrant
Fourth quadrant
SLIDE 36
tangent of an angle
Second quadrant
Third quadrant
Fourth quadrant
SLIDE 37
? Identify the quadrants where each of , and are positive.
? In how many quadrants are all three of , and positive?
SLIDE 38
? is positive in the first and second quadrants.
is positive in the first and fourth quadrants.
is positive in the first and third quadrants.
? , and are only all positive in the first quadrant.
SLIDE 39
Quadrants in which the given trigonometric ratios take a positive value.
The first letters of the sequence, Cosine, All, Sine, Tangent spell the word CAST.
SLIDE 40
It is given that . Find the angle x.
SLIDE 41
It is given that . Find the angle x.
Solution
SLIDE 41B
It is given that . Find the angle x.
Solution
SLIDE 42
Find the length XZ and the angles Y and Z.
SLIDE 43
As the angle is bigger than , we subtract from .
SLIDE 43B
As the angle is bigger than , we subtract from .
SLIDE 44
Proof of the sine rule
? Find an expression for h in the triangle ACX involving b and .
? Find an expression for h in the triangle BCX involving a and .
? By eliminating h from the two expressions found above, prove the sine rule.
SLIDE 45
Proof of the sine rule
In the triangle ACX, .
In the triangle BCX,
Hence,
SLIDE 46
Sine rule for the area of a triangle
The area of a triangle is
where is the angle included between two sides of the triangle.
SLIDE 47
Find the area of the above triangle, giving your answer to 3 significant figures.
SLIDE 48
SLIDE 49
Find the area of the triangle, giving your answer to 3 significant figures.
SLIDE 50
SLIDE 51
Cosine formula
? We have written , because of the small c representing a length in .
? Comparing this with Pythagoras’s theorem, , we see that the cosine formula is the generalised theorem of Pythagoras as applied to any triangle. Recall that Pythagoras’s theorem is only valid for a right-angle triangle. In the cosine formula, we extend the theorem to any triangle, by subtracting a “correction factor” of .
? The cosine formula is also known as the cosine rule.
SLIDE 52
Find the length a for the above triangle, giving your answer to 3 significant figures.
SLIDE 53
As this is the first time we are substituting into the cosine formula, we add in the standard labels.
SLIDE 54
The area of the triangle ABC is . Find the length AD giving your answer to 1 decimal place.
SLIDE 55
The task is the find the length . To find this we will use the sine rule in the triangle ADC, but first require the missing angle and the side .
Substituting into the cosine formula
SLIDE 56
The area of the triangle is 10 square units. Given that , find the value of x, giving your answer to 3 significant figures.
SLIDE 57
The area of the triangle is 10 square units. Given that , find the value of x, giving your answer to 3 significant figures.
Solution
Negative solution is not possible.
SLIDE 58
The cosine formula is
? Rearrange this equation to make the subject.
? Cyclically permute the symbols according to the rule
SLIDE 59
?
?
SLIDE 60
Find the angle x.
SLIDE 61
Find the angle x.
Method 1
By substituting directly into the cosine formula and then solve
Method 2
By substituting into
The two processes are essentially the same, and the choice between them is a matter of personal preference.
SLIDE 61B
Find the angle x.
Method 1
By substituting directly into the cosine formula and then solve
Method 2
By substituting into
The two processes are essentially the same, and the choice between them is a matter of personal preference.
SLIDE 62
The diagram shows a cuboid. .
P and Q are the midpoints of EF and EH respectively. Find
? the length AR ? the angle
SLIDE 63
. P and Q are the midpoints of EF and EH respectively. To find ? the length AR, ? the angle .
Solution
SLIDE 63B
. P and Q are the midpoints of EF and EH respectively. To find ? the length AR, ? the angle .
Solution
SLIDE 64
Proof of the cosine formula
? In the triangle ADC, use Pythagoras’s theorem to find an equation relating x, b and h.
? Find an equation relating y, a and h.
? Find an expression for in terms of x and y.
? Given eliminate y form the expression you have found in part ?.
? Use trigonometry in triangle ADC to express x in terms of b and the angle A.
? By substituting for x derive the cosine formula.
SLIDE 65
Proof of the cosine formula
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