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 và it includes the term cos cube x (cos^3x). Cos^3x gives the value of the cube of the cosine function. Cos3x & cos^3x formula help in solving various trigonometric problems.
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Let us understand the formula of cos3x and cos^3x, their derivation, và application along with solved examples for a better understanding. We will also explore the graph of the cos3x function lớn 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 khổng lồ Apply Cos3x Formula? |
| 8. | FAQs on Cos3x |
What is Cos3x in Trigonometry?
Cos3x is an important identity in trigonometry which is used to lớn 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 khổng lồ 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.
Cos3x Formula
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.
Derivation of Cos3x Formula
We will use the angle addition formula of the cosine function khổng lồ 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 lớn derive cos3x:
sin2x + cos2x = 1We will use the above identities to lớn 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
= (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
= cos x - 4 cos x + 4 cos3x
= 4 cos3x - 3 cos x
= 4cos^3x - 3cosx
Hence we have derived cos3x = 4 cos3x - 3 cos x using the angle addition identity for cosine function.
Cos3x Graph
The graph of cos3x is similar khổng lồ 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 và y = cos x and plot them.
When x = 0, 3x = 0 ⇒ cos x = 1, cos 3x = 1When x = -π/3, 3x = -π ⇒ cos x = 1/2, cos 3x = -1When x = π/3, 3x = π ⇒ cos x = 1/2, cos 3x = -1When x = 2π/3, 3x = 2π ⇒ cos x = -1/2, cos 3x = 1When x = -2π/3, 3x = -2π ⇒ cos x = -1/2, cos 3x = 1Given below is the graph of cos3x và cos x:
Cos^3x (Cos Cube 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 & 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 và understand it.
Cos^3x Formula
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) cosxcos^3x = 1/sec^3x ⇒ cos3x = 1/sec3xHow to lớn Apply Cos3x Formula?
To understand the application of cos3x, we will consider an example. We study & memorize the value of the cosine function for some specific angles lượt thích 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 xThe derivative of cos3x is -3 sin 3x & the integral of cos3x is (1/3) sin3x + CThe 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.☛ Related Articles:
Cos3x Examples
Example 1: Prove that the value of cos 180° is equal lớn -1 using the cos3x identity.
Solution: We know that cos3x = 4 cos3x - 3 cos x --- (1)
Assume 3x = 180° ⇒ x = 180°/3 = 60°
Substitute the value of 3x và 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 cos3x formula.
Example 2: Determine the value of cos 540° using the cos3x identity.
Solution: Assume 3x = 540° ⇒ x = 540°/3 = 180°
We know that cos3x = 4 cos^3x - 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 cos3x formula.
Example 3: Evaluate the derivative & integral of cos3x.
Solution: The differentiation of cos3x can easily be done using the formula d
Answer: Hence the derivative of cos3x is -3 sin 3x and the integral of cos3x is (1/3) sin3x + C.
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Cos3x Practice Questions
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FAQs on Cos3x
What is Cos3x in Trigonometry?
Cos3x is a triple angle identity in trigonometry. It can be derived using the angle addition identity of the cosine function. The identity of cos3x is given by cos3x = 4 cos3x - 3 cos x.
What is the Period of cos3x?
Since the angle in cos3x is thrice the angle in cos x, the period of cos3x 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 cos3x is 2π/3.
How vị you Graph cos3x?
Since the angle in cos3x is thrice the angle in cos x, the graph of cos3x is narrower than cos x. We can consider a few points for y = cos3x và y = cos x & join them khổng lồ plot the graph.
When x = 0, 3x = 0 ⇒ cos x = 1, cos3x = 1When x = -π/3, 3x = -π ⇒ cos x = 1/2, cos3x = -1When x = π/3, 3x = π ⇒ cos x = 1/2, cos3x = -1When x = 2π/3, 3x = 2π ⇒ cos x = -1/2, cos3x = 1When x = -2π/3, 3x = -2π ⇒ cos x = -1/2, cos3x = 1What is the Formula for Cos3x?
The trigonometric formula for cos3x is given by, cos3x = 4 cos3x - 3 cos x. We can derive this formula of cos3x using the cos a plus b formula.
What is the Derivative of Cos3x?
The derivative of cos3x is -3 sin 3x, that is, d(cos3x)/dx = -3sin 3x.
How lớn Integrate cos3x?
The formula ∫cos(ax + b) dx = (1/a) sin(ax + b) + C can used lớn obtain the integral of cos3x. The integral of cos3x is (1/3) sin 3x + C, that is, ∫cos3x dx = (1/3) sin 3x + C.
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What is Cos^3x Formula in Trigonometry?
We know that cos3x formula is cos3x = 4 cos^3x - 3 cos x. Using this formula, we can obtain the value of cos^3x. We have the formula of cos cube x as cos^3x = (1/4)(cos3x + 3 cos x).