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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
