Sin Cos Formulas: Basic Trigonometric Ratios and Identities

Sin Cos Formula: We all know that trigonometry is the branch of mathematics that deals with triangles. Engineering, astronomy, physics, and architectural design all benefit from trigonometric concepts. Let's look at some fundamental trigonometry sin cos formulae and trigonometric ratios in this article.

Check out: Who is the father of mathematics?

Key Terms: Trigonometric Ratio, right-angled triangle, half angle formula , trigonometric identities, double and triple formula, Sine, Cosecant, Tangent, Cosecant, Secant, Cotangent

---

Basic Trigonometric Ratios Formulas

[Click Here for Sample Questions]

• The right-angle triangle has six fundamental trigonometric ratios: Sine, Cosecant, Tangent, Cosecant, Secant and Cotangent.
• Sin, Cos, Tan, Cosec, Sec and Cot are the abbreviations for Sine, Cosecant, Tangent, Cosecant, Secant and Cotangent, respectively. 
• The basic trigonometric ratios Sin and Cos describe the form of a right triangle.
• A right-angled triangle is one in which one of the angles is a right angle, i.e. it has a 90-degree angle. 
• The hypotenuse is the side that lies opposite the right angle and it is the longest side of a right-angled triangle.
• The opposite side of the angle to be calculated is the perpendicular and the adjacent side is the base of the right-angled triangle. 

Right-angle triangle

In any right-angled triangle, for any angle:

• The Sine of the Angle (sin A) = the length of the opposite side / the length of the hypotenuse = BC / AB
• The Cosine of the Angle (cos A) = the length of the adjacent side / the length of the hypotenuse = AC / AB
• The Tangent of the Angle (tan A) = the length of the opposite side /the length of the adjacent side= BC / AC
• The Cosecant of the Angle (cosec A) = the length of the hypotenuse / the length of the opposite side = AB / BC
• The Secant of the Angle (sec A) = the length of the hypotenuse / the length of the adjacent side = AB / AC
• The Cotangent of the Angle (cot A) = the length of the adjacent side / the length of the opposite side = AC / BC

Also Read:

---

Reciprocal of Trigonometric Identities

[Click Here for Sample Questions]

• cosec A = 1/sin A
• sec A = 1/cos A
• cot A = 1/tan A
• sin A = 1/cosec A
• cos A = 1/sec A
• tan A = 1/cot A

​Read more : Cot Tan Formula​

---

Basic Trigonometric Identities for Sin and Cos

[Click Here for Sample Questions]

• cos²(A) + sin²(A) = 1
• If A + B = 180° then:

          sin(A) = sin(B)

          cos(A) = -cos(B)

• If A + B = 90° then:

          sin(A) = cos(B)

          cos(A) = sin(B)

---

Half-Angle Formulas for Sin and Cos

[Click Here for Sample Questions]

• Sin (A/2)= ± \(\sqrt{[(1-Cos A)/2]}\)
  • If A/2 is in the first or second quadrants, the value will be positive.
  • If A/2 is in the third or fourth quadrants, the value will be negative.
• Cos(A/2) = ±\(\sqrt{[(1+Cos A)/2]}\)
  • If A/2 is in the first or fourth quadrants, the value will be positive.
  • If A/2 is in the second or third quadrants, the value will be negative.

Also Read:

---

Double and Triple Angle Formulas for Sin and Cos

[Click Here for Sample Questions]

• Sin 2A = 2Sin A Cos A
• Cos 2A = Cos2 A – Sin2 A = 2 Cos2A - 1 = 1 - Sin2 A
• Sin 3A = 3Sin A – 4 Sin3 A
• Cos 3A = 4 Cos3 A – 3CosA
• Sin4 A = (3/8)−(1/2)cos(2A)+(1/8)cos(4A)
• Cos4 A = cos4 A – 6cos2 A sin2 A + sin4 A
• Sin2A = [1–Cos(2A)] / 2
• Cos2A = [1+Cos(2A)] / 2

---

Sum and Difference of Angles for Sin and Cos

[Click Here for Sample Questions]

• 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)
• Sin(A+B+C) = SinA⋅CosB⋅CosC + CosA⋅SinB⋅CosC + CosA⋅CosB⋅SinC − SinA⋅SinB⋅SinC
• Cos(A + B +C) = CosA CosB CosC - CosA SinB SinC – SinA CosB SinC – SinA SinB CosC
• Sin A + Sin B = 2 Sin[(A+B)/2] Cos[(A−B)/2]
• Sin A – Sin B = 2 Sin[(A−B)/2] Cos [(A+B)/2]
• Cos A + Cos B = 2 Cos[(A+B)/2] Cos [(A−B)/2]
• Cos A – Cos B = - 2 Sin[(A+B)/2] Sin [(A−B)/2]

Also Read: Difference between Trigonometry and Geometry​

---

Things to Remember

• Basic Sin Cos formula is cos²(A) + sin²(A) = 1
• Sin Cos formula for half angle-

• Sin (A/2)= ±\(\sqrt{[(1-Cos A)/2]}\)
• Cos(A/2) = ±\(\sqrt{[(1+Cos A)/2]}\)

• Sin Cos formula for double angle-

• Sin 2A = 2Sin A Cos A
• Cos 2A = Cos2 A – Sin2 A = 2 Cos2A - 1 = 1 - Sin2 A

• Sin Cos formula for triple angle-

• Sin 3A = 3Sin A – 4 Sin3 A
• Cos 3A = 4 Cos3 A – 3CosA

• Sum of angles for Sin and Cos 

• Sin(A + B) = Sin(A).Cos(B) + Cos(A).Sin(B)
• Cos(A+B) = Cos(A)⋅Cos(B) − Sin(A)⋅Sin(B)
• Sin(A+B+C) = SinA⋅CosB⋅CosC + CosA⋅SinB⋅CosC + CosA⋅CosB⋅SinC − SinA⋅SinB⋅SinC
• Cos(A + B +C) = CosA CosB CosC - CosA SinB SinC – SinA CosB SinC – SinA SinB CosC
• Sin A + Sin B = 2 Sin[(A+B)/2] Cos[(A−B)/2]
• Cos A + Cos B = 2 Cos[(A+B)/2] Cos [(A−B)/2]

• Differences of angles for Sin and Cos 

• Sin(A−B) = Sin(A)⋅Cos(B) − Cos(A)⋅Sin(B)
• Cos(A−B) = Cos(A)⋅Cos(B) + Sin(A)⋅Sin(B)
• Sin A – Sin B = 2 Sin[(A−B)/2] Cos [(A+B)/2]
• Cos A – Cos B = - 2 Sin[(A+B)/2] Sin [(A−B)/2]