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Trigonometry is the branch of mathematics that studies the relationships between angles and sides of triangles or that deal with angles, lengths, and heights of triangles and relations between different parts of circles and other geometrical figures. The Concepts of trigonometry are very useful in practical life as well as finds application in the field of engineering, astronomy, Physics, and architectural design.

These are in total 6 ratios that we study in trigonometry used to tell us about the triangle and Sine and Cos are two of them. We will discuss in detail about the Sin Cos ratio, formula and other concepts.

In mathematics six trigonometric ratios for the right angle triangle are defined i.e Sine, Cosecant, Tangent, Cosecant, Secant respectively. These trigonometric Ratios are real functions which relate an angle of a right-angled triangle to ratios of two of its side lengths. Sin and Cos are basic trigonometric functions that tell about the shape of a right triangle.There are six trigonometric ratios, Sine, Cosine, Tangent, Cosecant, Secant and Cotangent and are abbreviated as Sin, Cos, Tan, Csc, Sec, Cot. These are referred to as ratios because they can be expressed in terms of the sides of a right-angled triangle for a specific angle Î¸.

Adjacent: It is the side adjacent to the angle being taken for consideration.

Opposite: It is the side opposite to angle being taken for consideration.

Hypotenuse: It is the side opposite to the right angle of the triangle or the largest side of the triangle.

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**Sin Î¸ =**Perpendicular/ Hypotenuse = Opposite/Hypotenuse**Cos Î¸ =**Â Base/ Hypotenuse = Adjacent/Hypotenuse**Tan Î¸ =**Perpendicular/Base = Opposite/Adjacent**Cosec Î¸ =**Hypotenuse/Perpendicular = Hypotenuse/Opposite**Sec Î¸ =**Hypotenuse/Base = Hypotenuse/ Adjacent**Cot Î¸ =**Base/Perpendicular = Adjacent/Opposite

If A + B = 90Â°,that is A and B are complementary to each other then:

Sin (A) = Cos (B)

Cos (A) = Sin (B)

If A + B = 180Â° then:

Sin (A) = Sin (B)

Cos (A) = - Cos (B)

Â Â Â Â Â Â Cos2 (A) + Sin2 (A) = 1

The double or triple angle ratios can be converted to Single angle ratio of Sin and Cos using the below mentioned formulae:

Sin2A = 2 SinA.CosA

Cos2A = Cos2A â€“ Sin2A = 2Cos2 â€“ 1 = 1 âˆ’ 2Sin2A

Sin3A = 3 SinA â€“ 4 Sin3A

Cos3A = 4 Cos3A â€“ 3 CosA

Sin4A = 4 Cos3A SinA â€“ 4 CosA Sin3A

Cos4A = Cos4A â€“ 6 Cos2A Sin2A + Sin4A

Sin2A = (1 â€“ Cos2A)/2

Cos2A = (1 + Cos2A)/2

The Sin and Cos ratio of sum and difference of two angles can be converted two product using below mentioned identities:

Sin(A + B) = Sin(A) Cos(B) + Cos(A) Sin(B)

Sin(A âˆ’ B) = Sin(A) Cos(B) âˆ’ Cos(A) Sin(B)

Cos(A + B) = Cos(A) Cos(B) âˆ’ Sin(A) Sin(B)

Cos(A âˆ’ B) = Cos(A) Cos(B) + Sin(A) Sin(B)

Sin(A + B + C) = SinACosBCosC + CosASinBCosC + CosACosBSinC âˆ’ SinASinBSinC

Cos(A + B + C) = CosACosBCosC âˆ’ SinASinBCosC âˆ’ SinACosBSinC âˆ’ SinACosBSinC âˆ’ CosASinBSinC

SinA + SinB = 2Sin(A + B/2).Cos(A âˆ’ B/2)

SinA â€“ SinB = 2Sin(A âˆ’ B/2).Cos(A + B/2)

CosA + CosB = 2Cos(A + B/2).Cos(A âˆ’ B/2)

CosA + CosB = âˆ’2 Sin(A + B/2).Sin(A âˆ’ B/2)

In order to remember the trigonometric ratios values follow the given below steps:

We can the values for Sine ratios,i.e., 0, Â½, 1/âˆš2, âˆš3/2, and 1 for angles 0Â°, 30Â°, 45Â°, 60Â° and 90Â° and the Cos ratio will follow the exact opposite pattern i.e. 0, Â½, 1/âˆš2, âˆš3/2, and 1 at 90Â°, 60Â°, 45Â°,30Â° and 0Â°. While the Tan will be the ratio of Sin and Cos ratio.

The value of Cosec, Sec and Cot is exactly the reciprocal of Sin, Cos and Tan.

FAQ (Frequently Asked Questions)

Q1. How can we Convert Cos to Sin?

Ans: Since the value of CosÎ¸ = Sin (90Â° â€“ Î¸)

Means that if Î¸ is equal to 35 degrees, then Cos 35Â° = Sin (90Â° â€“ 35Â°) = Sin 55Â°.

Also Sin Î¸ = Cos(90Â° â€“ Î¸)

Q2. In the given âˆ† ABC, right-angled at B and side AB = 24 cm, BC = 7 cm. FindÂ

Sin A, Cos A

Sin C, Cos C

Ans:

ABC, right-angled triangle at B = âˆ B = 90Â°

Here it's given: AB = 24 cm and BC = 7 cm

That means, AC = Hypotenuse

According to the Pythagoras Theorem,

The squares of the hypotenuse side are equal to the sum of the squares of the other two sides in a right-angled triangle,

By applying Pythagoras theorem, we get

AC2 = AB2 + BC2

AC2 = (24)2 + 72

AC2 = (576 + 49)

AC2 = 625 cm2

Therefore, AC = 25 cm

Now for Sin A and Cos A.

Sin A = BC/AC = 7/25

Again, the Cosine of an angle is equal to the ratio of the adjacent side and hypotenuse side. Therefore,

Cos A = AB/AC = 24/25

For Sin C and Cos C.

Sin C = AB/AC = 24/25

Cos C = BC/AC = 7/25

Q3. Find the value of the expression SinÂ² 30Â° + CosÂ² 30Â°?

Ans: We know that the value of SinÂ² Î¸ + CosÂ² Î¸ = 1

Therefore SinÂ² 30Â° + CosÂ² 30Â° = 1

Q4. Find the value of Cos 105Â°.

Ans: We can break 105Â° into 60Â° and 45Â° Since those values are relatively easy to find the Cosine of. Therefore the value of Cos 105Â° = Cos(60Â° + 45Â°)= Cos 60Â° Cos 45Â° âˆ’ Sin 60Â° Sin 45Â° , using the unit circle we obtain, = 1/2 âˆ™ 1/âˆš2 âˆ’ âˆš3/ 2 âˆ™ 1/âˆš2 = 1/ 4 (âˆš2 âˆ’ âˆš6).