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) = senA⋅cosB⋅cosC+cosA⋅sin⋅cosC+cosA⋅cosB⋅sinC−sinA⋅sinB⋅sinC

• Cos(A + B +C) = cos A Cos B Cos 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:

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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.

Learn Sin Cos Formula with Vedantu 

Mathematics is an art and it is the science and study of quality, structure, space, and change of any object. Mathematicians seek out patterns, formulate new conjectures, and establish the truth by rigorous deduction from appropriately chosen axioms and definitions.

It is the science of numbers, quantities, and shapes, how it is measured, and the relations between them. With knowledge of Mathematics, you can actually study the science that revolves around numbers, shapes, and patterns, how things can be counted, how particular things are organized. This subfield of Mathematics with alphabets is usually called algebra and there are lots more in Math.

When you are constructing a house or any building measuring lengths, widths, and angles, estimating project costs, and consolidating it all together are a few instances that show how important Math is for performing construction projects. Wherever you go grocery shopping or any form of buying things you calculate the price per unit, weigh produce, figure out percentage discounts, and estimate the final cost, you're using Math to make your shopping experience easier. So these basic concepts are really useful to store in your mind to perform your day-to-day activities effectively and easily. 

Sin is a term that is equal to the side opposite the angle that you are conducting the functions on over the hypotenuse which is the longest side in the triangle. Cos is adjacent over the hypotenuse. And tan is opposite over adjacent, which means tan is sin/cos.

Sin Cos formulas are always based on the sides of the given right-angled triangle. Sin and Cos are basic trigonometric functions along with tan functions, in Trigonometry which is a part of Mathematics. The sine of any angle to be measured is equal to the ratio of opposite side and hypotenuse whereas the cosine of an angle is equal to the ratio of adjacent side and hypotenuse.