Cos3x is a triple angle identity in trigonometry. It is a specific case of compound angles identity of the cosine function. Cos3x gives the value of cosine trigonometric function for triple angle. The expansion of cos3x can be derived using the angle addition identity of cosine and it includes the term cos cube x (cos^3x). Cos^3x gives the value of the cube of the cosine function. Cos3x and cos^3x formula help in solving various trigonometric problems.
Let us understand the formula of cos3x and cos^3x, their derivation, and application along with solved examples for a better understanding. We will also explore the graph of the cos3x function to understand its behavior.
1. What is Cos3x in Trigonometry? 2. Cos3x Formula 3. Derivation of Cos3x Formula 4. Graph of Cos3x 5. Cos^3x (Cos Cube x) 6. Cos^3x Formula 7. How to Apply Cos3x Formula? 8. FAQs on Cos3x
Cos3x 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 cos3x is similar to that of cos x. As the period of cos x is 2π, the period of cos3x is 2π/3, that is, the cycle of cos3x repeats itself after every 2π/3 radians. Now, let us see the formula for cos3x.
The trigonometric formula for cos3x is given by, cos3x = 4cos^3x – 3cos x = 4 cos3x – 3 cos x. Now, we will go through the derivation of the cos3x formula in the next section using the angle sum formula of the cosine function.
We will use the angle addition formula of the cosine function to derive the cos3x formula. We know that the angle 3x can be written as 3x = 2x + x. We will use the following trigonometric identities to derive cos3x:
- 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 cos3x identity. Using the angle addition formula for cosine function, we have
cos3x = 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
= 4cos^3x – 3cosx
Hence we have derived cos3x = 4 cos3x – 3 cos xusing the angle addition identity for cosine function.
The graph of cos3x is similar to the graph of cos x. Since the angle in cos3x is thrice the angle in cos x, the graph of cos3x is narrower than cos x and hence, the period of cos3x 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 cos3x 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 = cos3x 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 cos3x and cos x:
Cos^3x is the formula for the whole cube of the cosine function. Cos cube x formula is used to solve complex integration problems, where we can substitute cos^3x with its formula and simplify the integral. We can derive the cos cub x formula using the cos3x formula by interchanging the terms and simplifying it. In the next section, let us derive the formula of cos^3x and understand it.
The formula for cos^3x is given by cos^3x = (1/4) cos3x + (3/4) cosx. We can derive this formula using the cos3x formula. We know that cos3x = 4cos^3x – 3 cosx which on adding 3 cosx on both sides of the integral can be written as cos3x + 3cosx = 4cos^3x – 3 cosx + 3cosx ⇒ cos3x + 3cosx = 4cos^3x. This can be simplified further as cos^3x = (1/4) cos3x + (3/4) cosx. Also, we know that the reciprocal identity of cosine function is cosx = 1/secx. Therefore, we can also write the cos cube formula as cos^3x = 1/sec^3x. Therefore, the formula for cos^3x are:
- cos^3x = (1/4) cos3x + (3/4) cosx = ⇒ cos3x = (1/4) cos3x + (3/4) cosx
- cos^3x = 1/sec^3x ⇒ cos3x = 1/sec3x
To understand the application of cos3x, 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 cos3x formula. If 3x = 135°, x = 135°/3 = 45°. Hence, using the cos3x 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 cos3x identity.
Important Notes on Cos 3x
- The formula of cos3x is cos3x = 4 cos^3x – 3 cos x
- The derivative of cos3x is -3 sin 3x and the integral of cos3x is (1/3) sin3x + C
- The period of cos3x is 2π/3.
- The most commonly used formula of cos cube x is cos^3x = (1/4) cos3x + (3/4) cosx which is used for simplifying complex integration problems.
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