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1. • Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch
of mathematics that studies relationships between side lengths and angles of triangles. The field
emerged in the Hellenistic world during the 3rd century BC from applications
of geometry to astronomical studies.The Greeks focused on the calculation of chords, while
mathematicians in India created the earliest-known tables of values for trigonometric ratios
(also called trigonometric functions) such as sine.
In this right triangle: sin A = a/c; cos A = b/c; tan A = a/b.
• Here Adjacent means Base , Opposite means perpendicular and Hypotenuse is same
Trigonometric ratios are the ratios between edges of a right triangle. These ratios are given
by the following trigonometric functions of the known angle A, where a, b and c refe00r to the
lengths of the sides in the accompanying figure:
• Sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.
Sinθ= opposite/ Hypotenuse =a/c
• Cosine function (cos), defined as the ratio of the adjacent leg (the side of the triangle joining
the angle to the right angle) to the hypotenuse.
Cosθ= Adjacent/ Hypotenuse = b/c
• Tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg.
Tanθ = opposite/Hypotenuse = a/b
The reciprocals of these functions are named the cosecant (csc), secant (sec),
and cotangent (cot), respectively:
• COSθ= 1/sinθ = Hypotenuse/opposite= c/a
• secθ= 1/Cosθ = Hypotenuse/ Adjacent = c/b
• Cotθ= 1/tanθ = Adjacent/opposite = b/c = Cosθ/Sinθ
•
A common use of mnemonics is to remember facts and relationships in trigonometry. For
example, the sine, cosine, and tangent ratios in a right triangle can be remembered by
representing them and their corresponding sides as strings of letters. For instance, a mnemonic

2. is SOH-CAH-TOA:
Sine = Opposite ÷ Hypotenuse
Cosine = Adjacent ÷ Hypotenuse
Tangent = Opposite ÷ Adjacent
• Unit-circle definitions
In this illustration, the six trigonometric functions of an arbitrary angle θ are represented
as Cartesian coordinates of points related to the unit circle. The ordinates
of A, B and D are sin θ, tan θ and csc θ, respectively, while the abscissas
of A, C and E are cos θ, cot θ and sec θ, respectively.

3. Signs of trigonometric
functions in each quadrant. The mnemonic "allscience teachers (are) crazy" lists the functions
which are positive from quadrants I to IV.This is a variation on the mnemonic "All Students Take
Calculus".
• Pythagorean identity
• The Pythagorean trigonometric identity, also called the fundamental Pythagorean
trigonometric identity or simply Pythagorean identity is an identity expressing the Pythagorean
theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of
the basic relations between the sine and cosine functions
x² + y²= 1
Cos²θ + Sin²θ= 1
• Related Identities
Sin²θ = 1-Cos²θ
Cos²θ= 1-sin²θ
tan²θ=sec²θ-1
tan²θ+1 = sec²θ
Cot²θ = Cosec²θ-1
Cot²θ+1 = Cosec²θ

4. • Algebraic values
The unit circle, with some points labeled with their cosine and sine (in this order), and the
corresponding angles in radians and degrees.
All values are carried from Trignometric Table :

5. • Angle sum and difference identities
• These are also known as the angle addition and subtraction theorems(or formulae). The
identities can be derived by combining right triangles such as in the adjacent diagram, or by
considering the invariance of the length of a chord on a unit circle given a particular central
angle. The most intuitive derivation uses rotation matrices (see below).
The angle addition and subtraction theorems reduce to the following when one of the angles is
small (β ≈ 0):
cos(α + β)≈ cos(α) - βsin(α),
cos(α - β)≈ cos(α) + βsin(α),
sin(α + β)≈ sin(α) + βcos(α),
sin(α - β)≈ sin(α) - βcos(α).
• Double-angle identities
• From the angle sum identities, we get
Sin(2θ) = 2SinθCosθ
Cos(2θ) = Cos²θ-Sin²θ , 2Cos²θ-1 , 1-2Sin²θ

6. Cotθ = cot²θ-1= Cotθ - tanθ 2Cotθ 2
• Half-angle identities
• The two identities giving the alternative forms for cos 2θ lead to the following equations:
The sign of the square root needs to
be chosen properly—note that if 2π is added to θ, the quantities inside the square roots are
unchanged, but the left-hand-sides of the equations change sign. Therefore, the correct sign to
use depends on the value of θ.
For the tan function, the equation is:
Then multiplying the numerator
and denominator inside the square root by (1 + cos θ) and using Pythagorean identities leads to:
Also, if the numerator and denominator are both
multiplied by (1 - cos θ), the result is:
This also gives:

7. Similar manipulations for the cot
function give:
Product to Sum Identities

8. • Sum to Product Identities

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