In arithmetic, pure mathematics is the most vital topic to find out. pure mathematics is largely the study of triangles wherever ‘Trigon’ suggests that triangle and ‘metry’ suggests that activity. Also, a pure mathematics formulas list is made on the premise of pure mathematics ratios like trigonometric function, circular function, and tangent. These formulas area unit wont to solve numerous pure mathematics issues.
Taking an associate degree example of the correct angle triangle, a pure mathematics formulas list is created. All the pure mathematics formulas area unit supported pure mathematics identities and pure mathematics ratios. the connection between angles and length of the edges of the Triangle is developed with the assistance of pure mathematics ideas.
Trigonometry formulas list is going to be useful for college kids to resolve pure mathematics issues simply. Below is that the list of formulas supported the triangle and unit circle which may be used as a relation to study pure mathematics.
List of vital pure mathematics Formulas
First allow us to learn basic formulas of pure mathematics, considering a triangle, that has associate degree angle θ, a flank, a facet opposite angle θ and a facet adjacent to angle θ.
So the general trigonometry ratios for a right-angled triangle can be written as;
sinθ = OppositesideHypotenuse
cosθ = AdjacentSideHypotenuse
tanθ = OppositesideAdjacentSide
secθ = HypotenuseAdjacentside
cosecθ = HypotenuseOppositeside
cotθ = AdjacentsideSideopposite
Similarly, for a unit circle, for which radius is 1, and θ is the angle.Then,
sinθ = y/1
cosθ = 1/y
tanθ = y/x
cotθ = x/y
secθ = 1/x
cosecθ = 1/y
Trigonometry Identities and Formulas
Tangent and Cotangent Identities
tanθ = sinθcosθ
cotθ = cosθsinθ
Reciprocal Identities
sinθ = 1/cosecθ
cosecθ = 1/sinθ
cosθ = 1/secθ
secθ = 1/cosθ
tanθ = 1/cotθ
cotθ = 1/tanθ
Pythagorean Identities
sin^{2}θ + cos^{2}θ = 1
1 + tan^{2}θ = sec^{2}θ
1 + cot^{2}θ = cosec^{2}θ
Even and Odd Formulas
sin(-θ) = -sinθ
cos(-θ) = cosθ
tan(-θ) = -tanθ
cot(-θ) = -cotθ
sec(-θ) = secθ
cosec(-θ) = -cosecθ
Cofunction Formulas
sin(90^{0}-θ) = cosθ
cos(90^{0}-θ) = sinθ
tan(90^{0}-θ) = cotθ
cot(90^{0}-θ) = tanθ
sec(90^{0}-θ) = cosecθ
cosec(90^{0}-θ) = secθ
Formulas for twice of angle
sin2θ = 2 sinθ cosθ
cos2θ = 1 – 2sin^{2}θ
tan2θ = 2tanθ1−tan2θ
Half Angle Formulas
sinθ = ±1−cos2θ2−−−−−−√
cosθ = ±1+cos2θ2−−−−−−√
tanθ = ±1−cos2θ1+cos2θ−−−−−−√
Formulas for Thrice of angle
sin3θ = 3sinθ – 4 sin^{3}θ
Cos 3θ = 4cos^{3}θ – 3 cosθ
Tan 3θ = 3tanθ–tan3θ1−3tan2θ
Cot 3θ = cot3θ–3cotθ3cot2θ−1
The Sum and Difference Formulas
Sin (A+B) = Sin A Cos B + Cos A Sin B
Sin (A-B) = Sin A Cos B – Cos A Sin B
Cos (A+B) = Cos A Cos B – Sin A Sin B
Cos (A-B) = Cos A Cos B + Sin A Sin B
Tan (A+B) = TanA+TanB1–TanATanB
Tan (A-B) = TanA–TanB1+TanATanB
The Product to Sum Formulas
Sin A Sin B = ½ [Cos (A-B) – Cos (A+B)]
Cos A Cos B = ½ [Cos (A-B) + Cos (A+B)]
Sin A Cos B = ½ [Sin (A+B) + Sin (A+B)]
Cos A Sin B = ½ [Sin (A+B) – Sin (A-B)]
The Sum to Product Formulas
Sin A + Sin B = 2 sin A+B2 cos A−B2
Sin A – Sin B = 2 cosA+B2 sin A−B2
Cos A + Cos B = 2 cosA+B2 cos A−B2
Cos A – Cos B = – 2 sinA+B2 sin A−B2
Inverse Trigonometric Functions
If Sin θ = x, then θ = sin^{-1 }x = arcsin(x)
Similarly,
θ = cos^{-1}x = arccos(x)
θ = tan^{-1 }x = arctan(x)
Also, the inverse properties could be defined as;
sin^{-1}(sin θ) = θ
cos^{-1}(cos θ) = θ
tan^{-1}(tan θ) = θ
Values for Trigonometry ratios:
Degrees |
0^{0} |
30^{0} |
45^{0} |
60^{0} |
90^{0} |
180^{0} |
270^{0} |
360^{0} |
Radians |
0 |
π/6 |
π/4 |
π/3 |
π/2 |
π |
3π/2 |
2π |
Sinθ |
0 |
1/2 |
1/2–√ | 3–√/2 |
1 |
0 |
-1 |
0 |
Cosθ |
1 |
3–√/2 | 1/2–√ |
1/2 |
0 |
-1 |
0 |
1 |
Tanθ |
0 |
1/3–√ |
1 |
3–√ |
∞ |
0 |
∞ |
0 |
Cotθ |
∞ |
/3–√ |
1 |
1/3–√ |
0 |
∞ |
0 |
∞ |
Secθ |
1 |
2/3–√ |
/2–√ |
2 |
∞ |
-1 |
∞ |
1 |
Cosecθ |
∞ |
2 |
/2–√ |
2/3–√ |
1 |
∞ |
-1 |
∞ |