Javascript required
Skip to content Skip to sidebar Skip to footer
Home / Find All Solutions of Cos 3x

Find All Solutions of Cos 3x

Cos 3x

Cos 3x is a triple angle identity in trigonometry. It is a specific case of compound angles identity of the cosine function. cos 3x gives the value of cosine trigonometric function for triple angle. The expansion of cos 3x can be derived using the angle addition identity of cosine. Cos 3x identity helps in solving various trigonometric problems. Let us understand the formula of cos 3x, its derivation, graph and application along with examples.

1. What is Cos 3x Identity?
2. Derivation of cos 3x
3. Graph of cos 3x
4. Derivative and Integral of Cos 3x
5. How to Apply cos 3x Identity?
6. FAQs on cos 3x

What is Cos 3x Identity?

Cos 3x is an important identity in trigonometry which is used to determine the value of the cosine function for an angle that is thrice the measure of angle x. It can be expressed in terms of the cos x. The behavior of the function cos 3x is similar to that of cos x. As the period of cos x is 2π, the period of cos 3x is 2π/3, that is, the cycle of cos 3x repeats itself after every 2π/3 radians. Now, let us see the formula for cos 3x.

Cos 3x Formula

The trigonometric formula for cos 3x is given by, cos 3x = 4 cos3x - 3 cos x. Now, we will go through the derivation of the cos 3x formula.

cos 3x formula

Derivation of cos 3x

We will use the angle addition formula of the cosine function to derive the cos 3x identity. We know that the angle 3x can be written as 3x = 2x + x. We will use the following trigonometric identities to derive cos 3x:

  • cos (a + b) = cos a cos b - sin a sin b
  • sin 2x = 2 sin x cos x
  • cos 2x = 1 - 2sin2x
  • sin2x + cos2x = 1

We will use the above identities to prove the cos 3x identity. Using the angle addition formula for cosine function, we have

cos 3x = cos (2x + x)

= cos 2x cos x - sin 2x sin x [Because cos (a + b) = cos a cos b - sin a sin b]

= (1 - 2sin2x) cos x - (2 sin x cos x) sin x

= cos x - 2sin2x cos x - 2sin2x cos x

= cos x - 4sin2x cos x

= cos x - 4 (1 - cos2x) cos x [Because sin2x + cos2x = 1 ⇒ sin2x = 1 - cos2x]

= cos x - 4 cos x + 4 cos3x

= 4 cos3x - 3 cos x

Hence we have derived cos 3x = 4 cos3x - 3 cos x using the angle addition identity for cosine function.

Cos 3x Graph

The graph of cos 3x is similar to the graph of cos x. Since the angle in cos 3x is thrice the angle in cos x, the graph of cos 3x is narrower than cos x and hence, the period of cos 3x is also one-third the period of the function cos x. Also, for a function cos bx, the period is given by 2π/|b|. Therefore the period of cos 3x is 2π/3. We can plot the graph of cos 3x by taking some points on the graph and joining them. Let us consider a few points for y = cos 3x and y = cos x and plot them.

  • When x = 0, 3x = 0 ⇒ cos x = 1, cos 3x = 1
  • When x = -π/3, 3x = -π ⇒ cos x = 1/2, cos 3x = -1
  • When x = π/3, 3x = π ⇒ cos x = 1/2, cos 3x = -1
  • When x = 2π/3, 3x = 2π ⇒ cos x = -1/2, cos 3x = 1
  • When x = -2π/3, 3x = -2π ⇒ cos x = -1/2, cos 3x = 1

Given below is the graph of cos 3x and cos x:

cos 3x graph

Derivative and Integral of Cos 3x

The differentiation of cos 3x can easilty be done using the formula d[cos(ax + b)]/dx = -asin(ax + b). The formula ∫cos(ax + b) dx = (1/a) sin(ax + b) + C can used to obtain the integral of cos 3x. Therefore, we have d(cos 3x)/dx = -3sin 3x and ∫cos 3x dx = (1/3) sin 3x + C.

Hence the derivative of cos 3x is -3 sin 3x and the integral of cos 3x is (1/3) sin 3x + C.

How to Apply cos 3x Identity?

To understand the application of cos 3x, we will consider an example. We study and memorize the value of the cosine function for some specific angles like 0°, 30°, 45°, 60°, 90°, etc. Now, will determine the value of cos 135° using the cos 3x identity. If 3x = 135°, x = 135°/3 = 45°. Hence, using the cos 3x formula, we have

cos 135° = cos (3 × 45°) = 4 cos3(45°) - 3 cos 45°

= 4 × (1/√2)3 - 3 × (1/√2)

= 2/√2 - 3/√2

= -1/√2

= - √2/2

Therefore, we have obtained the value of cos 135° as - √2/2 using the cos 3x identity.

Important Notes on cos 3x

  • cos 3x = 4 cos3x - 3 cos x
  • The derivative of cos 3x is -3 sin 3x and the integral of cos 3x is (1/3) sin 3x + C
  • The period of cos 3x is 2π/3.

Related Topics on cos 3x

  • Sin of Sin Inverse
  • sin of 2 pi
  • cos a cos b

Cos 3x Examples

  1. Example 1: Prove that the value of cos 180° is equal to -1 using the cos 3x identity.

    Solution: We know that cos 3x = 4 cos3x - 3 cos x --- (1)

    Assume 3x = 180° ⇒ x = 180°/3 = 60°

    Substitute the value of 3x and x in (1)

    cos 180° = 4 cos3(60°) - 3 cos (60°)

    = 4 × (1/2)3 - 3 × (1/2)

    = 4/8 - 3/2

    = 1/2 - 3/2

    = -2/2

    = -1

    Answer: Hence we have proved that the value of cos 180° = -1 using the cos 3x identity.

  2. Example 2: Determine the value of cos 540° using the cos 3x identity.

    Solution: Assume 3x = 540° ⇒ x = 540°/3 = 180°

    We know that cos 3x = 4 cos3x - 3 cos x --- (1)

    Substitute the values of 3x and x in (1), we have

    cos 540° = 4 cos3(180°) - 3 cos (180°)

    = 4 (-1)3 - 3 (-1)

    = -4 + 3

    = -1

    Answer: Hence the value of cos 540° is -1 using the cos 3x identity.

go to slidego to slide

Have questions on basic mathematical concepts?

Become a problem-solving champ using logic, not rules. Learn the why behind math with our certified experts

Book a Free Trial Class

Practice Questions Using Cos 3x

go to slidego to slide

FAQs on cos 3x

What is Cos 3x in Trigonometry?

Cos 3x is a triple angle identity in trigonometry. It can be derived using the angle addition identity of the cosine function. The identity of cos 3x is given by cos 3x = 4 cos3x - 3 cos x.

What is the Period of cos 3x?

Since the angle in cos 3x is thrice the angle in cos x, the period of cos 3x is also one-third the period of the function cos x. For a function cos bx, the period is given by 2π/|b|. Therefore the period of cos 3x is 2π/3.

How do you Graph cos 3x?

Since the angle in cos 3x is thrice the angle in cos x, the graph of cos 3x is narrower than cos x. We can consider a few points for y = cos 3x and y = cos x and join them to plot the graph.

  • When x = 0, 3x = 0 ⇒ cos x = 1, cos 3x = 1
  • When x = -π/3, 3x = -π ⇒ cos x = 1/2, cos 3x = -1
  • When x = π/3, 3x = π ⇒ cos x = 1/2, cos 3x = -1
  • When x = 2π/3, 3x = 2π ⇒ cos x = -1/2, cos 3x = 1
  • When x = -2π/3, 3x = -2π ⇒ cos x = -1/2, cos 3x = 1

What is the Formula for Cos 3x?

The trigonometric formula for cos 3x is given by, cos 3x = 4 cos3x - 3 cos x.

What is the Derivative of Cos 3x?

The derivative of cos 3x is -3 sin 3x, that is, d(cos 3x)/dx = -3sin 3x.

How to Integrate cos 3x?

The formula ∫cos(ax + b) dx = (1/a) sin(ax + b) + C can used to obtain the integral of cos 3x. The integral of cos 3x is (1/3) sin 3x + C, that is, ∫cos 3x dx = (1/3) sin 3x + C.

What is Cos3x Formula in Trigonometry?

We know that cos 3x formula is cos 3x = 4 cos3x - 3 cos x. Using this formula, we can obtain the value of cos3x. We have cos3x = (1/4)(cos 3x + 3 cos x).

Find All Solutions of Cos 3x

Source: https://www.cuemath.com/trigonometry/cos-3x/