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 θ.
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 θ = Perpendicular/ Hypotenuse = Opposite/Hypotenuse
Cos θ = Base/ Hypotenuse = Adjacent/Hypotenuse
Tan θ = Perpendicular/Base = Opposite/Adjacent
Cosec θ = Hypotenuse/Perpendicular = Hypotenuse/Opposite
Sec θ = Hypotenuse/Base = Hypotenuse/ Adjacent
Cot θ = Base/Perpendicular = Adjacent/Opposite
If A + B = 90°,that is A and B are complementary to each other then:
Sin (A) = Cos (B)
Cos (A) = Sin (B)
If A + B = 180° then:
Sin (A) = Sin (B)
Cos (A) = - Cos (B)
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 – Sin2A = 2Cos2 – 1 = 1 − 2Sin2A
Sin3A = 3 SinA – 4 Sin3A
Cos3A = 4 Cos3A – 3 CosA
Sin4A = 4 Cos3A SinA – 4 CosA Sin3A
Cos4A = Cos4A – 6 Cos2A Sin2A + Sin4A
Sin2A = (1 – 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 − 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) = SinACosBCosC + CosASinBCosC + CosACosBSinC − SinASinBSinC
Cos(A + B + C) = CosACosBCosC − SinASinBCosC − SinACosBSinC − SinACosBSinC − CosASinBSinC
SinA + SinB = 2Sin(A + B/2).Cos(A − B/2)
SinA – SinB = 2Sin(A − B/2).Cos(A + B/2)
CosA + CosB = 2Cos(A + B/2).Cos(A − B/2)
CosA + CosB = −2 Sin(A + B/2).Sin(A − 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, ½, 1/√2, √3/2, and 1 for angles 0°, 30°, 45°, 60° and 90° and the Cos ratio will follow the exact opposite pattern i.e. 0, ½, 1/√2, √3/2, and 1 at 90°, 60°, 45°,30° and 0°. 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.
Q1. How can we Convert Cos to Sin?
Ans: Since the value of Cosθ = Sin (90° – θ)
Means that if θ is equal to 35 degrees, then Cos 35° = Sin (90° – 35°) = Sin 55°.
Also Sin θ = Cos(90° – θ)
Q2. In the given ∆ ABC, right-angled at B and side AB = 24 cm, BC = 7 cm. Find
Sin A, Cos A
Sin C, Cos C
ABC, right-angled triangle at B = ∠B = 90°
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 Sin² 30° + Cos² 30°?
Ans: We know that the value of Sin² θ + Cos² θ = 1
Therefore Sin² 30° + Cos² 30° = 1
Q4. Find the value of Cos 105°.
Ans: We can break 105° into 60° and 45° Since those values are relatively easy to find the Cosine of. Therefore the value of Cos 105° = Cos(60° + 45°)= Cos 60° Cos 45° − Sin 60° Sin 45° , using the unit circle we obtain, = 1/2 ∙ 1/√2 − √3/ 2 ∙ 1/√2 = 1/ 4 (√2 − √6).