When is cos(x) rational?

Number System is the method or way to represent numbers with the help of symbols. These symbols range from 0-9. A number is a numerical value that may be used to count, measure, and do arithmetic calculations. Natural numbers, whole numbers, rational and irrational numbers, and so on are all types of numbers.

• Rational Number: A rational number is a number that can be stated as the quotient or fraction p/q of two integers, with a numerator of p and a non-zero denominator of q, in mathematics. For example, 5, 6, 0,-95, 2/9, etc.
• Irrational Numbers: Real numbers that cannot be expressed as a ratio are referred to as irrational numbers. Irrational numbers, on the other hand, are real numbers that are not rational numbers. For example, √2, √3, √5, √11, √21, π(Pi), etc.

Cosine Function

In a triangle, the cos function (or cosine function) is the ratio of the neighboring side to the hypotenuse.

Cos is the ratio of the length of the neighboring side to the length of the longest side, the hypotenuse, in a right triangle. Assume a triangle ABC is formed, with AB as the hypotenuse and as the hypotenuse-base angle.

For the given triangle ABC,

cos α = a / b = BC / AC 

Let α be a counterclockwise angle measured along the arc of the unit circle from the x-axis. The horizontal location of the arc termination is thus cos α.

Formulae of Cosine Function:

• cos x = 1 / sec x
• sin²x + cos²x = 1
• cos (x + y) = cos x cos y – sin x sin y
• cos (x – y) = cos x cos y + sin x sin y
• cos 2x = cos²x – sin²x = 2 cos²x – 1 = 1 – 2 sin²x

When is cos(x) rational?

Solution:

The only rational values of cos(r.π) where r is a rational number occur at,

r = 0,

r = ± 1 / 3,

r = ± 1 / 2,

r = ± 2 / 3 and

r = ± 1.

where the values of cos(r.π) are 1, 1/2, 0, -1/2 and -1, respectively.

Similar Problems

Problem 1: For a given triangle the value of cos x is given as 4/5, Find the value of sin x.

Solution:

As we know that for a given right angle triangle the value of cos x is given as base/hypotenuse. Also, the value of sin x is given by perpendicular/hypotenuse.

here in the question, we have been given the base and hypotenuse, we need to find the perpendicular length.

Also we know that for a right-angled triangle : (base)² + (perpendicular)² = (hypotenuse)².

So we get the value of perpendicular as  √(25 – 16) = 3.

Hence the value of sin x will be 3/5.

Problem 2: For a given triangle the value of tan x is given as 24/7. Find the value of cos x for the given triangle.

Solution:

As we know that for a given right angle triangle the value of tan x is given as perpendicular/base. Also, the value of cos x is given by base/hypotenuse.

In the given question, we have been given the base and perpendicular, we need to find the hypotenuse length.

base = 7

perpendicular = 24

Also we know that for a right-angled triangle : (base)² + (perpendicular)² = (hypotenuse)².

Hence we get the value of hypotenuse as: √{576+49} = 25.

Hence the value of cos x will be 7/25.

Problem 3: For a given triangle the value of cos 27 is a. What will be the value of sin 63?

Solution: 

We know that cos (θ) = sin (π/2 − θ). 

Here in this problem θ = 27 .

So the value of sin (π/2 − θ) = sin (π/2 − 27) = sin(63) = a.

Hence the value of sin 63 is a.

Problem 4: The distance of the rooftop from the ground is 12 m. A ladder is placed from the ground inclined at an angle of 53 degrees to the rooftop. Find the length of the ladder.

Solution:

Given that the height of the rooftop is 12 m.

Also, the inclination of the ladder to the ground is 53 degrees, So the ladder is at an angle of 37 from the perpendicular.

Hence we can write the height as the cosine component of the hypotenuse i. e ladder length. we get :

hypotenuse x cos (37 )=height.

Hypotenuse = height / cos(37)

Hypotenuse= 15 m.

Hence the length of the ladder is 15 m.