Download (5 lines each) and combine only three triangles all in an essential

SECTION-A
2*5=10marks
1. Originally trigonometry was understood to define relations between elements of a
triangle. In a triangle, there are six basic elements: 3 sides and 3 angles. Not any three line
segments may serve as the sides of a triangle. They do iff they satisfy the triangle inequality, or
rather three triangle inequalities. Not any three angles may be the angles of a triangle. In
Euclidean geometry, the three angles of a triangle add up to a straight angle. These
requirements impose limitations on the manner in which the relations between the elements
are defined. In modern trigonometry these relations are extended to arbitrary angles. This can
be done, for example, by observing the projections of arotating radius of a circle and a tangent
at the end of the radius.
If the sides a, b, c of a triangle lie opposite angles α, β, γ then a + b > c is oneof the inequalities
that the sides obey, and α + β + γ = 180° is the identity that holds in Euclidean geometry. We
also know that, if γ is right, then the Pythagorean theorem holds: a² + b² = c². (Its converse
holds too.) Trigonometric relations involve trigonometric functions.
2. Linear algebra is the branch of mathematics concerning vector spaces, often finite
or countably infinite dimensional, as well as linear mappings between such spaces. Such an
investigation is initially motivated by a system of linear equations containing several unknowns.
Such equations are naturally represented using the formalism of matrices and vectors.
Linear algebra is central to both pure and applied mathematics. For instance,abstract
algebra arises by relaxing the axioms of a vector space, leading to a number of
generalizations. Functional analysis studies the infinite-dimensional version of the theory of
vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems
of differential equations. Techniques from linear algebra are also used in analytic
geometry, engineering, physics, natural sciences, computer science, computer animation, and
the social sciences(particularly in economics). Because linear algebra is such a well-developed
theory, nonlinear mathematical models are sometimes approximated by linear ones.
3. Where do trigonometry use beside been engineer?
Besides those, if you type in sine or cosine into our Quandaries and Queries search,
then you'll find many problems people have sent to us where we used trigonometry to
solve them. Some examples are:
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Calculating the lot size of a property ,
Determining the distances an object has moved ,
Determining the height of a tower when you can't measure it ,
Determining how much concrete to order to pack an irrigation ditch ,
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Figuring out how much water you need to fill a semi-circular aquarium ,
Making the right angled cuts for carpentry ,
Determining the safe hours to go cliff diving into the sea,
For air traffic control ,
Making special jewelry settings,
Building a wishing well,
Designing a three dimensional star ,
Figuring out the trajectory of a ball, bullet, or anything else,
There are dozens more in our archive as well.
SECTION-B
3*20=60marks
1. Let ABC be a right-angled triangle with the right angle at C.
The side AB opposite the right angle is called the hypotenuse. This is always the longest side.
Referring to angle A, the opposite side is BC, and the remaining side AC is the adjacent side.
The ratios sine, cosine and tangent are defined as ratios of the three sides.
The mnemonic SOH CAH TOA tells you that:
sin (A) is opposite over hypotenuse (BC/AB);
cos (A) is adjacent over hypotenuse (AC/AB);
tan (A) is opposite over adjacent (BC/ AC).
The ratios for angle B are obtained in the same way, but because AC is the side opposite angle B, that
means the adjacent side is BC.
sin (B) is opposite over hypotenuse (AC/AB);
cos (B) is adjacent over hypotenuse (BC/AB);
tan (B) is opposite over adjacent (AC/ BC).
If you know AC and angle B, for example, and want to find AC, then think 'What equation can I write down
to give AC?'
AC is opposite B, and you want AB which is the hypotenuse. The ratio is therefore sine. From the
definition above:
AC / AB = sin(B).
Take reciprocals of each side to get AB:
AB / AC = 1 / sin (B)
Multiply by AC:
AB = AC / sin(B).
2. Trigonometry is a methodology for finding some unknown elements of a triangle (or
other geometric shapes) provided the data includes a sufficient amount of linear and angular
measurements to define a shape uniquely. For example, two sides a and b of a triangle and the
angle they include define the triangle uniquely. The third side c can then be found from the Law
of Cosines while the angles α and β are determined from the Law of Sines. The latter can be
used to find the circumradius. The area of the triangle can be found from S = (ab sin γ)/2 and
knowing that we can determine the inradius forms = (a + b + c)r/2, and so on.
sin² α + cos² α = 1.
Then there are double argument formulas:
sin 2α = 2 sin α cos α
cos 2α = cos² α - sin² α
tan 2α = 2 tan α / (1 - tan² α)
cot 2α = (cot² α - 1) / 2cot α.
and, more general, sum and difference formulas:
sin (α + β) = sin α cos β + cos α sin β
cos (α + β) = cos α cos β - sin α sin β
sin (α - β) = sin α cos β - cos α sin β
cos (α - β) = cos α cos β + sin α sin β.
The rhombus inscribed into a rectangle has side length of 1. The rhombus cuts off of the
rectangle two pairs of equal right triangles. The acute angles of the triangles are A, 90° A, B, 90° - B. The vertices of the rhombus split the sides of the rectangle into segments of
lengths cos A, sin A, cos B, sin B, as shown.
The area of the rhombus is sin(A + B). In the right half of the applet, the triangles rearranged
leaving two rectangles unoccupied. The area of one is sin A × cos B, that of the other cos A × sin
B, proving the "sine of the sum" formula
sin(A + B) = sinA cosB + cosA sinB.
3.
The simplest, truly "proofs without words" for the addition and
subtraction formulas for sine and cosine ,As the other proofs relying on
visual imagery,
0 < α, β, α + β, < 90°, for addition, and
0 < α, β, α - β < 90°, for subtraction.
Formulas for sin(α + β) and cos(α + β)
sin(α + β) = sinα cos β + cos α sin β.
cos(α + β) = cosα cos β - sin α sin β.
Formulas for sin(α - β) and cos(α - β)
sin(α - β) = sinα cos β - cos α sin β.
cos(α - β) = cosα cos β + sin α sin β.
The proofs below of the addition and subtraction formulas for sine and
cosine use the least amount of drawing attributes (5 lines each) and
combine only three triangles all in an essential way. The proofs are by
Leonard M. Smiley, University of Alaska Anchorage. A probably more
professional copies of the proofs are available at the MAA site. They have
also appeared in [R. Nelsen, pp. 42-43]. Naturally, the proofs are only valid
for small angles, the constraints being clear from the diagrams. In fact, all
four proofs use exactly same geometric configuration.
Formulas for sin(α - β)
h = cosα / cosβ. x = h
sin(α - β). x = (sinα - h
sinβ) cosα.
sin(α - β) = sinα cos β - cos α sin β.
Formulas for cos(α - β)
h = cosα / sinβ. x = h cos(α - β). x = (sinα + h cosβ) cosα.
cos(α - β) = cosα cos β + sin α sin β.
Formulas for cos(α + β)
sinα
cosβ
=
cosα
sinβ + cos(α+β)/sinα
cos(α + β) = cosα cos β - sin α sin β.
Formulas for sin(α + β)
4. The first example is from Physics. Suppose that we are given three objects, one with a mass known to
be 2 kg, and are asked to find the unknown masses. Suppose further that experimentation with a meter
stick produces these two balances.
Since the sum of moments on the left of each balance equals the sum of moments on the right (the
moment of an object is its mass times its distance from the balance point), the two balances give this
system of two equations.
The second example of a linear system is from Chemistry. We can mix, under controlled conditions,
toluene
and nitric acid
to produce trinitrotoluene
along with the
byproduct water (conditions have to be controlled very well, indeed— trinitrotoluene is better known
as TNT). In what proportion should those components be mixed? The number of atoms of each
element present before the reaction
must equal the number present afterward. Applying that principle to the elements C, H, N, and O
in turn gives this system.
To finish each of these examples requires solving a system of equations. In each, the
equations involve only the first power of the variables. This chapter shows how to solve any
such system.