Sin Cos Formula

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Trigonometry Sin Cos Formula

We know that the part of math called trigonometry is the part of math that deals with triangles. Trigonometry is the part of math that deals with the relationship between the three sides and the three angles. Trigonometry helps us to find the remaining sides and angles of a triangle when some of its sides and angles are given. This problem is solved by using some ratios of the sides of a triangle with respect to its acute angles. These ratios of acute angles are called the basic trigonometric ratios. In this article let us study various trigonometry sin cos formulas and basic trig ratios.

Basic Trigonometric Ratios Formulas

There are six basic trigonometric ratios for the right angle triangle. They are Sin, Cos, Tan, Cosec, Sec, Cot that stands for Sine, Cosecant, Tangent, Cosecant, Secant respectively. Sin and Cos are basic trig ratios that tell about the shape of a right triangle. 

A right-angled triangle is a triangle in which one of the angles is a right-angle i.e it is of 900. The hypotenuse of a right-angled triangle is the longest side, which is the one opposite side to the right angle. The adjacent side is the side which is between the angle to be determined and the right angle. The opposite side is opposite to the angle to be determined..

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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

  • The Cosine of the Angle(cos A) = the length of the adjacent side / the length of the hypotenuse

  • The Tangent of the Angle(tan A) = the length of the opposite side /the length of the adjacent side

  • The Cosecant of the Angle(cosec A) = the length of the hypotenuse / the length of the opposite side

  • The Secant of the Angle(sec A) = the length of the hypotenuse / the length of the adjacent side

  • The Cotangent of the Angle(cot A) = the length of the adjacent side / the length of the opposite side

Reciprocal of Trigonometric Identities

The Reciprocal Identities are given as:

  • 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

Basic Trigonometric Identities for Sine and Cos

These formulas help in giving a name to each side of the right triangle Let’s learn the basic sin and cos formulas.

  • 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

Sin (A/2)= ± \[\sqrt{\frac{1−CosA}{2}}\]

  • If A/2 is in the first or second quadrants, the result will be positive.

  • If A/2 is in the third or fourth quadrants, the result will be negative.

Cos(A/2) = ±1 \[\sqrt{\frac{1+CosA}{2}}\]

  • If A/2 is in the first or fourth quadrants, the will be positive.

  • If A/2 is in the second or third quadrants, the result will be negative.

Double and Triple Angle Formulas

  • Sin 2A = 2Sin A Cos A

  • Cos 2A = Cos2A – Sin2A = 2 Cos2A- 1 = 1- Sin2A

  • Sin 3A = 3Sin A – 4 Sin3A

  • Cos 3A = 4 Cos3A – 3CosA

  • Sin4A = (3/8)−(1/2)cos(2A)+(1/8)cos(4A)

  • Cos4A = cos4A – 6cos2A sin2A + sin4A

  • Sin2A = 1–Cos(2A) / 2

  • Cos2A = 1+Cos(2A) / 2

Sum and Difference of Angles

  • 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) = cos Acos Bcos C- cos Asin Bsin C – sin Acos Bsin C – sin A sin B cos C

  • Sin A + Sin B = 2Sin(A+B)/2 Cos(A−B)/2

  • Sin A – Sin B = 2Sin(A−B)/2Cos(A+B)/2

  • Cos A + Cos B = 2Cos(A+B)/2 Cos(A−B)/2

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

Solved Examples

Example 1: Find the length of side x in the diagram below:

[Image to be added Soon]

Solution: The angle is 600

Hypotenuse = 13cm

 adjacent side = x cm

We have,

cos(60) = adjacent / hypotenuse

therefore, cos(60) = x / 13

therefore, x = 13 × cos(60) = 6.5

therefore the length of side x is 6.5cm.

FAQ (Frequently Asked Questions)

1. What are Trigonometric Ratios?

Answer: Trigonometry is the branch of mathematics that deals with specific functions of angles and their application to calculations. The ratios formed by the sides of a right triangle are called trigonometric ratios. Trigonometric ratios are derived from the sides of a right-angled triangle. There are six functions of an angle commonly used in trigonometry. Their names are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc).Trigonometric ratios apply to a right angle triangle only. It is a special triangle in which one angle is 90° and the other two are less than 90°. Also, each side of the triangle has a name. They are hypotenuse, perpendicular, and base.

2. What are the Six Trigonometric Functions?

Answer: 

Trigonometry means the science of measuring triangles. Trigonometric functions can be simply defined as the functions of an angle of a triangle i.e. the relationship between the angles and sides of a triangle is given by these basic trig functions.

The Six Main Trigonometric Functions are as Follows:

  • Sine (sin)

  • Cosine (cos)

  • Tangent (tan)

  • Secant (sec)

  • Cosecant (csc)

  • Cotangent (cot)

These functions are used to relate the angles of a triangle with the sides of that triangle where the triangle is the right-angled triangle. Trigonometric functions are important when studying triangles. Each function relates the angle to two sides of a right triangle.