Formulas and Identities for Cosine
There are various different identities in trigonometry, some of these identities for cosine trigonometric ratio are discussed as follows:
Reciprocal Identities
- sin θ = 1/ cosec θ
- cosec θ = 1/ sin θ
- cos θ = 1/ sec θ
- sec θ = 1 / cos θ
- cot θ = 1 / tan θ
- tan θ = 1 / cot θ
- cot θ = cos θ / sin θ
- tan θ = sin θ / cos θ
- tan θ.cot θ = 1
Complementary Angles Identities
Pair of angles whose sum is equal to 90° are called complementary angles and the identities of Complementary angles are:
- sin (90° – θ) = cos θ
- cos (90° – θ) = sin θ
- tan (90° – θ) = cot θ
- cot (90° – θ) = tan θ
- sec (90° – θ) = cosec θ
- cosec (90° – θ) = sec θ
Supplementary Angles Identities
Pair of angles whose sum is equal to 180° are called supplementary angles and the identities of supplementary angles are:
- sin (180° – θ) = sin θ
- cos (180° – θ) = – cos θ
- tan (180° – θ) = – tan θ
- cot (180° – θ) = – cot θ
- sec (180° – θ) = – sec θ
- cosec (180° – θ) = – cosec θ
Cosine Formulas Using Pythagorean Identity
One of the trigonometric identities between sin and cos. It represents sin2x + cos2x = 1
sin2x + cos2x = 1
Now Subtracting sin2x from both sides,
cos2x = 1 – sin2x
now square both sides
cos x = ± √(1 – sin2x)
Cosine Formulas with Sum/Difference Formulas
There are sum/difference formulas for every trigonometric function that deal with the sum of angles (x + y) and the difference of angles (x – y).
Formulas of cosine function with sum difference formulaes are,
cos(x + y) = cos (x) cos(y) – sin (x) sin (y)
cos (x – y) = cos (x) cos (y) + sin (x) sin (y)
Formulas for Law of Cosines
This law is used to find the missing sides/angles in a non-right angled triangle. Assume a triangle ABC in which AB = c, BC = a, and CA = b.
The cosine formulas are,
cos A = (b2 + c2 – a2)/(2bc)
cos B = (c2 + a2 – b2)/(2ac)
cos C = (a2 + b2 – c2)/(2ab)
Learn more about, Law of Cosine Formula
Double Angle Formula of Cosine
In trigonometry while dealing with 2 times the angle. There are multiple sorts of double-angle formulas of cosine and from that, we use one of the following while solving the problem depending on the available information.
cos 2x = cos2(x) – sin2(x)
cos 2x = 2 cos2(x) − 1
cos 2x = 1 – 2 sin2(x)
cos 2x = [(1 – tan2x)/(1 + tan2x)]
Half Angle Formula of Cosine
Half angle formula for the cosine is similar to the double angle formula but 2(Angle) is changed to (Angle). The half-angle formula for the cosie is,
cos (x/2) =± √[ (1 + cos x) / 2]
Triple Angle Formula of Cosine
The triple angle formula for the cosine is,
cos 3x = 4cos3x – 3cosx
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What are Cosine Formulas?
Trigonometry is a discipline of mathematics that studies the relationships between the lengths of the sides and angles of a right-angled triangle. Trigonometric functions, also known as goniometric functions, angle functions, or circular functions, are functions that establish the relationship between an angle to the ratio of two of the sides of a right-angled triangle. The six main trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. Trigonometric angles are the Angles defined by the ratios of trigonometric functions. Trigonometric angles represent trigonometric functions. The value of the angle can be anywhere between 0-360°.
As given in the above figure in a right-angled triangle:
- Hypotenuse: The side opposite to the right angle is the hypotenuse, It is the longest side in a right-angled triangle and opposite to the 90° angle.
- Base: The side on which angle C lies is known as the base.
- Perpendicular: It is the side opposite to angle C in consideration.
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