Cos a Cos b is a trigonometric formula that isused in trigonometry. Cos a cos b formula is given by, cos a cos b =(1/2)[cos(a + b) + cos(a - b)].We use the cos a cos b formula to find the value of the product of cosine of two different angles. cos a cos b formula can be obtained from the cosine trigonometric identity for sum of angles and difference of angles.
The cos a cos b formula helps in solving integration formulas and problems involving the product of trigonometric ratio such as cosine. Let us understand the cos a cos b formula and its derivation in detail in the following sections.
| 1. | What is Cos a Cos b in Trigonometry? |
| 2. | Derivation of Cos a Cos b Formula |
| 3. | How to Apply cos a cos b Formula? |
| 4. | FAQs on Cos a Cos b |
What is Cos a Cos bin Trigonometry?
Cos a Cos b is the trigonometry identity for two different angles whose sum and difference are known. It is applied when either the two angles a and b are known or when the sum and difference of angles are known. It can be derived using cos (a + b) and cos (a - b) trigonometry identities which are some of the important trigonometric identities. The cos a cos b identity is half the sum of the cosines of the sum and difference of the angles a and b, that is, cos a cos b = (1/2)[cos(a + b) + cos(a - b)].
Derivation of Cos a Cos b Formula
The formula for cos a cos b can be derived using the sum and difference identities of the cosine function. We will use the following cosine identities to derive the cos a cos b formula:
- cos (a + b) = cos a cos b - sin a sin b --- (1)
- cos (a - b) = cos a cos b + sin a sin b --- (2)
Adding equations (1) and (2), we have
cos (a + b) + cos (a - b) = (cos a cos b - sin a sin b) + (cos a cos b + sin a sin b)
⇒ cos (a + b) + cos (a - b) = cos a cos b - sin a sin b + cos a cos b + sin a sin b
⇒ cos (a + b) + cos (a - b) = cos a cos b + cos a cos b - sin a sin b + sin a sin b
⇒ cos (a + b) + cos (a - b) = cos a cos b + cos a cos b [The term sin a sin b got cancelled because of opposite signs]
⇒ cos (a + b) + cos (a - b) = 2 cos a cos b
⇒ cos a cos b = (1/2)[cos (a + b) + cos (a - b)]
Hence the cos a cos b formula has been derived.
Thus, cos a cos b = (1/2)[cos (a + b) + cos (a - b)]
How to Apply Cos a Cos b Formula?
Now that we know the cos a cos b formula, we will understand its application in solving various problems. This identity can be used to solve simple trigonometric problems and complex integration problems. We can follow the steps given below to learn to apply cos a cos b identity. Let us go through some examples to understand the concept clearly:
Example 1: Express cos 2x cos 5x as a sum of the cosine function.
Step 1: We know that cos a cos b = (1/2)[cos (a + b) + cos (a - b)]
Identify a and b in the given expression. Here a = 2x, b = 5x. Using the above formula, we will process to the second step.
Step 2: Substitute the values of a and b in the formula.
cos 2x cos 5x = (1/2)[cos (2x + 5x) + cos (2x - 5x)]
⇒ cos 2x cos 5x = (1/2)[cos (7x) + cos (-3x)]
⇒ cos 2x cos 5x = (1/2)cos (7x) + (1/2)cos (3x) [Because cos(-x) = cos x]
Hence, cos 2x cos 5x can be expressed as (1/2)cos (7x) + (1/2)cos (3x) as a sum of the cosine function.
Example 2: Solve the integral ∫ cos x cos 3x dx.
To solve the integral ∫ cos x cos 3x dx, we will use the cos a cos b formula.
Step 1: We know that cos a cos b = (1/2)[cos (a + b) + cos (a - b)]
Identify a and b in the given expression. Here a = x, b = 3x. Using the above formula, we have
Step 2: Substitute the values of a and b in the formula and solve the integral.
cos x cos 3x = (1/2)[cos (x + 3x) + cos (x - 3x)]
⇒ cos x cos 3x = (1/2)[cos (4x) + cos (-x)]
⇒ cos x cos 3x = (1/2)cos (4x) + (1/2)cos (x) [Because cos(-x) = cos x]
Step 3: Now, substitute cos x cos 3x = (1/2)cos (4x) + (1/2)cos (x) into the intergral ∫ cos x cos 3x dx. We will use the integral formula of the cosine function ∫ cos x dx = sin x + C
∫ cos x cos 3x dx = ∫ [(1/2)cos (4x) + (1/2)cos (x)] dx
⇒ ∫ cos x cos 3x dx = (1/2) ∫ cos (4x) dx + (1/2) ∫ cos (x) dx
⇒ ∫ cos x cos 3x dx = (1/2) [sin (4x)]/4 + (1/2) sin (x) + C
⇒ ∫ cos x cos 3x dx = (1/8) sin (4x) + (1/2) sin (x) + C
Hence, the integral ∫ cos x cos 3x dx = (1/8) sin (4x) + (1/2) sin (x) + C using the cos a cos b formula.
Important Notes on cos a cos b
- cos a cos b = (1/2)[cos (a + b) + cos (a - b)]
- It is applied when either the two angles a and b are known or when the sum and difference of angles are known.
- The cos a cos b formula helps in solving integration formulas and problems involving the product of trigonometric ratio such as cosine
Related Topics on cos a cos b
- sin(a + b)
- sin of 2 pi
- cos (a + b)
- sin (a - b)
FAQs on Cos a Cos b
What is cos a cos b Formula in Trigonometry?
Cos a Cos b is the trigonometry identity for two different angles whose sum and difference are known. The cos a cos b identity is half the sum of the cosines of the sum and difference of the angles a and b, that is, cos a cos b = (1/2)[cos(a + b) + cos(a - b)].
How Do you Derive cos a cos b Identity?
cos a cos b can be derived using the sum and difference identities of the cosine function. It can be derived by adding the cos (a + b) and cos (a - b) formulas.
What is the Formula for 2 cos a cos b?
We know that cos a cos b = (1/2)[cos (a + b) + cos (a - b)]. Multiply both sides of the equation cos a cos b = (1/2)[cos (a + b) + cos (a - b)] by 2, we have 2 cos a cos b = cos (a + b) + cos (a - b)]. Hence, the formula for 2 cos a cos b is cos (a + b) + cos (a - b).
What is the Formula for cos a cos b?
cos a cos b is one of the important trigonometric formulas used in trigonometry. The formula for cos a cos b is cos a cos b = (1/2)[cos(a + b) + cos(a - b)].
How to Prove cos a cos b Formula?
cos a cos b can be proved using the sum and difference identities of the cosine function. It can be proved by adding the cos (a + b) and cos (a - b) formulas.
What are the Applications of cos a cos b Formula?
The cos a cos b formula helps in solving integration formulas and problems involving the product of trigonometric ratio such as cosine. This identity can be used to solve simple trigonometric problems and complex integration problems.
FAQs
What is the proof of cos a )+ COS B? ›
Proof of Cos A + Cos B Formula
½ [cos(α + β) + cos(α - β)] = cos α cos β, for any angles α and β. ⇒ Cos A + Cos B = 2 cos ½(A + B) cos ½(A - B) Hence, proved.
Cos a cos b formula is given by, cos a cos b = (1/2)[cos(a + b) + cos(a - b)]. We use the cos a cos b formula to find the value of the product of cosine of two different angles.
How do you find the value of cos B? ›Hence, cos B equals a/c. In other words, the cosine of an angle in a right triangle equals the adjacent side divided by the hypotenuse: Also, cos A = sin B = b/c.
What is an example for cos? ›Examples on Cosine Formulas
Example 1: If sin x = 3/5 and x is in the first quadrant, find the value of cos x. Answer: cos x = 4/5. Example 2: If sin (90 - A) = 1/2, then find the value of cos A. Answer: cos A = 1/2.
In trigonometry, the Cosine Rule says that the square of the length of any side of a given triangle is equal to the sum of the squares of the length of the other sides minus twice the product of the other two sides multiplied by the cosine of angle included between them.
What is cos with A and B function? ›Cos(a + b) Compound Angle Formula
The cosine of the sum of two angles is equal to the product of the cosines of the individual angles minus the product of their sines. In other words, cos(a + b) = cos(a)cos(b) – sin(a)sin(b).
Pythagorean theorem: a2 + b2 = c2. Sines: sin A = a/c, sin B = b/c. Cosines: cos A = b/c, cos B = a/c.
What is cos in an equation? ›The cosine function (or cos function) in a triangle is the ratio of the adjacent side to that of the hypotenuse. The cosine function is one of the three main primary trigonometric functions and it is itself the complement of sine(co+sine).
What is the identity of cos2x? ›Cos2x is one of the double angle trigonometric identities as the angle in consideration is a multiple of 2, that is, the double of x. Let us write the cos2x identity in different forms: cos2x = cos2x - sin2x. cos2x = 2cos2x - 1.
What is the rule of the triangle? ›The rule of the sides of a triangle is that the sum of the lengths of any two sides of a triangle is always greater than the length of the third side. This rule is also known as the triangle inequality theorem. This implies that we cannot have a triangle with lengths 3, 4, 9 as 3 + 4 = 7 < 9.
What is cos in algebra? ›
The cosine (often abbreviated “cos”) is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. And the tangent (often abbreviated “tan”) is the ratio of the length of the side opposite the angle to the length of the side adjacent.
What is cos used for in math? ›Cos function is the ratio of adjacent side and hypotenuse. It helps us to find the length of the sides of the triangle, irrespective of given angle.
Is cos a real function? ›Definition. Cosine is an even function function and is periodic with period 2π . The cosine function has a domain of all real numbers, and its range is −1≤cosx≤1 − 1 ≤ cos x ≤ 1 .
What are the 3 formulas for the law of cosines? ›- a2 = b2 + c2 - 2bc·cosA.
- b2 = c2 + a2 - 2ca·cosB.
- c2 = a2 + b2 - 2ab·cosC.
Inverse cosine is also known as arccosine. It is the inverse of cos function. Also, sometimes abbreviated as 'arccos'. It is used to measure the unknown angle when the length of two sides of the right triangle are known.
Why is cos a negative function? ›The cosine of an obtuse angle is negative because of the range of the cosine function which is between 1 and -1. Therefore, when the cosine function completes its half cycle, it is at the middle of 1 and -1, that is 0.
What are the 3 Pythagorean theorem? ›A set of 3 positive numbers that satisfy the formula of the Pythagoras' theorem that is expressed as a 2 + b 2 = c 2 , where a, b, and c are positive integers, are called Pythagorean triples.
Does cos rule apply to all triangles? ›The Cosine Rule can be used in any triangle where you are trying to relate all three sides to one angle. If you need to find the length of a side, you need to know the other two sides and the opposite angle.
Can you do trigonometry without a calculator? ›However, trigonometry sums can be solved without the help of a calculator too. And though unbelievable, it's not as hard as it seems to be. All you have to do is use the Trigonometry table and you will be able to crack up most of the answers in a short time and let me tell you that it's even fun.
What are the six trig functions? ›Review all six trigonometric ratios: sine, cosine, tangent, cotangent, secant, & cosecant.
Is 2cosx the same as cos2x? ›
Summary: 2cosx is twice the cosine of angle x and lies in the range of [-2 , 2] whereas, cos 2x is the cosine of the angle 2x, two times the angle x and it lies between [-1 , 1].
What does cos2x turn into? ›Cos2x Formula in terms of Sine function is cos2x = 1 - 2sin2x. It can be proved using the trigonometry identity cos2x + sin2x = 1. Thus, Cos2x Formula in terms of Sine is cos2x = 1 - 2sin2x.
What is the rule for Sin2x? ›Sin2x is twice the product of the sine function and cosine function which is given as sin2x = 2sin x cosx. It can also be expressed in terms of Tangent Function as well.
What is the 45 45 90 rule? ›The 45-45-90 triangle rule states that the three sides of the triangle are in the ratio 1:1:√2. So, if the measure of the two congruent sides of such a triangle is x each, then the three sides will be x, x and √2x. This rule can be proved by applying the Pythagorean theorem. So, AB:AC:BC =x:x:√2x or 1:1:√2.
What is the 3 4 5 rule right triangle? ›The 3:4:5 triangle is the best way I know to determine with absolutely certainty that an angle is 90 degrees. This rule says that if one side of a triangle measures 3 and the adjacent side measures 4, then the diagonal between those two points must measure 5 in order for it to be a right triangle.
What is square class 9? ›Definition. Square is a regular quadrilateral, which has all the four sides of equal length and all four angles are also equal. The angles of the square are at right-angle or equal to 90-degrees. Also, the diagonals of the square are equal and bisect each other at 90 degrees.
What does cos mean in slang? ›'Cos is an informal way of saying because.
Why is it called sine? ›In the twelfth century, when an Arabic trigonometry work was translated into Latin, the translator used the equivalent Latin word sinus, which also meant bosom, and by extension, fold (as in a toga over a breast), or a bay or gulf. This Latin word has now become our English “sine.”
What is the value of cos? ›According to this formula, the value of a cosine function of an angle is the length of the adjacent side divided by the length of the hypotenuse side.
How was sine discovered? ›The use of sines (half-chords) was introduced (not “apparently” but definitely) by the Indian mathematicians, who used the Sanskrit word jīva- “bow string” (which is actually a translation of Greek chordē, but which Indians used not for the whole chord, but the half-chord).
Is cosine ever 0? ›
cos x = 0, when x = ±π/2, ±3π/2, ±5π/2, … It means that cos x vanishes when x is an odd multiple of π/2. So, cos x = 0 implies x = (2n + 1)π/2 , where n takes the value of any integer. For a triangle, ABC having the sides a, b, and c opposite the angles A, B, and C, the cosine law is defined.
Who invented trigonometry? ›Trigonometry in the modern sense began with the Greeks. Hipparchus (c. 190–120 bce) was the first to construct a table of values for a trigonometric function.
Is cosine always odd? ›Sine is an odd function, and cosine is an even function. You may not have come across these adjectives “odd” and “even” when applied to functions, but it's important to know them. A function f is said to be an odd function if for any number x, f(–x) = –f(x).
What is the formula of proving trigonometric identities? ›Proving Trigonometric Identities - Basic
\sin^2 \theta + \cos^2 \theta = 1. sin2θ+cos2θ=1. In order to prove trigonometric identities, we generally use other known identities such as Pythagorean identities. Prove that ( 1 − sin x ) ( 1 + csc x ) = cos x cot x .
Therefore, the 2 cos A cos B procedure or formula is, 2 cos A cos B = cos (A + B) + cos (A – B)
What is the formula for sin A * cos B? ›The formula for sin a cos b is given by, sin a cos b = (1/2)[sin(a + b) + sin(a - b)]. The formula for sin a cos b can be applied when the compound angles (a + b) and (a - b) are known, or when values of angles a and b are known.
How do you solve trigonometric proofs easily? ›- Quotient Identities: tan(x) = sin(x)/cos(x) cot(x) = cos(x)/sin(x)
- Reciprocal Identities: csc(x) = 1/sin(x) sec(x) = 1/cos(x) cot(x) = 1/tan(x) sin(x) = 1/csc(x) ...
- Pythagorean Identities: sin2(x) + cos2(x) = 1. cot2A +1 = csc2A. 1+tan2A = sec2A.
Often, complex trigonometric expressions can be equivalent to less complex expressions. The process for showing two trigonometric expressions to be equivalent (regardless of the value of the angle) is known as validating or proving trigonometric identities.
What is cos2x formula in terms of sin? ›What is Cos2x In Terms of sin x? We can express the cos2x formula in terms of sinx. The formula is given by cos2x = 1 - 2sin2x in terms of sin x.
What is cos2x trigonometric identity? ›Cos2x Formula in trigonometry
Since the angle under examination is a factor of 2, or the double of x, the cosine of 2x is an identity that belongs to the category of double angle trigonometric identities. Let us write the identity of cos2x using a few alternative forms: cos2x = cos2x – sin2x. cos2x = 2cos2x – 1.
How is cos written as a function? ›
Any cosine function can be written as a sine function. y = A sin(Bx) and y = A cos(Bx). The number, A, in front of sine or cosine changes the height of the graph. The value A (in front of sin or cos) affects the amplitude (height).
What is the formula for cos A? ›The Cosine of the Angle(cos A) = the length of the adjacent side / the length of the hypotenuse. The Tangent of the Angle(tan A) = the length of the opposite side /the length of the adjacent side. The Cosecant of the Angle(cosec A) = the length of the hypotenuse / the length of the opposite side.
What is sin formula in algebra? ›Sin Cos Tan Formula
Sine θ = Opposite side/Hypotenuse = BC/AC. Cos θ = Adjacent side/Hypotenuse = AB/AC.
The Law of sines gives a relationship between the sides and angles of a triangle. The law of sines in Trigonometry can be given as, a/sinA = b/sinB = c/sinC, where, a, b, c are the lengths of the sides of the triangle and A, B, and C are their respective opposite angles of the triangle.