Cosine Formula

Cosine Formula

Students must retain each and every formula that is given in their Mathematics books. They must know it will be easier for them if they will have formulas all retained in their brains. According to the Cosine Formula rule in Trigonometry, the square of the length of any side of a given triangle equals the sum of the squares of the other sides minus twice the product of the other two sides multiplied by the Cosine Formula of the angle that separates them. The law of cosines and Cosine Formula are other names for the cosine rule. Students must know that Mathematics plays an important role in their school curriculum if they want to pursue a career in Mathematics.

Assuming that a, b, and c are the side lengths of a triangle ABC;

a2 = b2 + c2 – 2bc cos ∠x

b2 = a2 + c2 – 2ac cos ∠y

c2 = a2 + b2 – 2ab cos ∠z

The Cosine Formula is trigonometric cosine function formula. One of the six trigonometric functions, the cosine function (often abbreviated “cos”) is the ratio of the neighbouring side to the hypotenuse. Numerous identities and formulas linked to trigonometry can be used to derive numerous cosine function formulations. Students could study the Cosine Formula and some examples with solutions.

The cosine (cos) function is discussed in the Cosine Formula. Students could assume a right-angled triangle where x is the acute angle. The cosine equation is thus written as cos x = (adjacent side) / (hypotenuse), where “adjacent side” refers to the side next to angle x and “hypotenuse” refers to the longest side of the triangle (the side across from the right angle). There are other different trigonometric formulas that define the cosine function in addition to this standard formula, as seen in the following illustration.

Students are aware that the reciprocal relationship between the cosine function (cos) and the secant function (sec) exists. Specifically, cos x = a / b, then sec x = b / a. So, using one of the reciprocal identities, the Cosine Formula is,

cos x = 1 / (sec x)

Students must have regular and apt practice with every question before their examination so that they will not face any problems while solving any question. Hence, accuracy and precision is the most important aspect of Mathematics and a student must acknowledge the importance of formulas and practice. 

Formula for Cosine

A trigonometric identity discusses the connection between sin and cos. It states that for any value of x, sin2x plus cos2x equals 1. This can be resolved for cos x.

Students could consider sin2x + cos2x = 1.

Sin2x from both sides are subtracted,

cos2x = 1 – sin2x

square roots on each side,

cos x = ± √(1 – sin2x)

The relationships between the cofunctions sin, cos, sec, csc, tan, and cot are defined by the cofunction identities. One of the cofunction identities should be used.

cos x = sin (90o – x) (OR)

cos x = sin (π/2 – x)

Examples of Cosine x Formula

Every trigonometric function has a sum/difference formula that deals with the sum of angles (x + y) and the difference of angles (x – y). The formulas for the Cosine Formula function’s sum and difference are,

cos(x + y) = cos (x) cos(y) – sin (x) sin (y)

cos (x – y) = cos (x) cos (y) + sin (x) sin (y)

In Trigonometry, there are double-angle formulas that deal with angles that are twice as large. Depending on the information at hand, one of the following double-angle cos formulae may be used to solve the problem. It is as follows:

cos 2x = cos2(x) – sin2(x)

cos 2x = 2 cos2(x) − 1 

cos 2x = 1 – 2 sin2(x)

cos 2x = [(1 – tan2x)/(1 + tan2x)]

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FAQs (Frequently Asked Questions)

1. How does Cosine Formula work?

The cosine function is connected to the Cosine Formula. The following are some crucial Cosine Formula:

cos x = (adjacent side) / (hypotenuse)

cos x = 1 / (sec x)

cos x = ± √(1 – sin2x)

cos x = sin (π/2 – x)

2. What purposes does Cosine Formula Serve?

There are several Cosine Formulas, as students have learned in the course of navigating through the curriculum. They could select one of them to demonstrate a trigonometric identity or determine the cosine function’s value using the facts at hand. The Cosine Formulas can be used to determine the triangle’s missing angles or sides. Calculus also employs the Cosine Formula. To learn more about this topic, students can visit the Extramarks educational website.