Formulas » Cosine Formulas
unacademy_centre

Cosine Formulas

Cosine Formulas with solved example: Explore more about the Cosine Formulas with solved examples.

Table of Content

Cosine Formulas with solved examples

This article gives the formulas of the cosine function that helps to find the values of other trigonometric functions and even the angles and sides of triangles.

Trigonometric functions aim to establish a functional relationship between the ratio of two of its sides and an angle in case of right-angled triangles. Basically, there are six main trigonometric functions and the cosine function relates to the base and hypotenuse of the right-angled triangle and it is represented as Cosθ where theta is the measure of angle.

The following is the list of the cosine formulas:

  • General Formula

cos x = Side adjacent to the angle x / Hypotenuse

  • Reciprocal Formula

cos x = 1/ (sec x) 

  • Sum and Difference Formulas

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

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

  • Double Angle Formulas

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

Cos (x/2) = ((1 + cos(x)) / 2)

  • Triple Angle Formula

Cos 3x = 4 cos3x – 3 cos x

  • Triangle Formulas (where a, b and c refer to the length of the sides of the triangle)

cos A = (b2 + c2 – a2) / 2bc

cos B = (c2 + a2 – b2) / 2ac

cos C = (a2 + b2 – c2) / 2ab

Solved Examples

Question 1. Find the value of cos 2x if the value of sin x is given to be 3/5.

Solution: Applying the formula Cos 2x = 1 – 2*sin2 (x),

We get cos 2x = 1 – 2 (3/5)2

Or cos 2x = 725

Question 2. Find the value of cos 15°.

Solution: Using the sum-difference formula,

Cos 15° = cos (45° – 30°)

Or Cos 15° = Cos 45° x Cos 30° + sin 45° x sin 30°

Or Cos 15° = 2/2 x 3/2 + 2/2 x ½

Or Cos 15° = 6/4 + 2/4

Or Cos 15° = 6 + 2/4

Question 3. In a triangle ABC, AB = 25 cm, BC = 40 cm and AC = 60 cm. Find cos A.

Solution: Let the sides AB, BC and AC be a, b and c respectively.

Using the triangle formula,

     cos A = (b2 + c2 – a2) / 2bc

      Putting the values,

cos A = (402 + 602 – 252) / 2(40)(60)

Or cos A = 4275 / 4800 Or cos A = 61/64

faq

Frequently asked questions

Get answers to the most common queries related to the Cosine Formulas.

What is the minimum and maximum value of a cosine function?

Ans. The Cosine function is a decreasing function whose minimum value is -1 and maximum value is 1.

What is the reciprocal identity of the cosine function?

Ans. The reciprocal function of the cosine function is the secant function.

Does the cosine function hold true only for the right-angled triangles?

Ans. No, it is not necessary for the triangle to be right-angled and the function works in all types of triangles....Read full