Use double angle identify to write the equation as. cos 2x + cos 4x + cos 6x + 1 = 0 cos 2 x + cos 4 x + cos 6 x + 1 = 0. Let t = cos 2x t = cos 2 x. Then, the equation is. 2t3 +t2 − t = 0 t(t + 1)(2t − 1) = 0 2 t 3 + t 2 − t = 0 t ( t + 1) ( 2 t − 1) = 0.
Explanation: #"since "cosx>0# #"then x will be in the first/fourth quadrants"# #cosx=1/2# #rArrx=cos^-1(1/2)=pi/3larrcolor(blue)" angle in first quadrant"# #"or "x=(2pi-pi/3)=(5pi)/3larrcolor(blue)" angle in fourth quadrant"#
Precalculus. Solve for ? cos (2x)=1. cos (2x) = 1 cos ( 2 x) = 1. Take the inverse cosine of both sides of the equation to extract x x from inside the cosine. 2x = arccos(1) 2 x = arccos ( 1) Simplify the right side. Tap for more steps 2x = 0 2 x = 0. Divide each term in 2x = 0 2 x = 0 by 2 2 and simplify.
$\begingroup$ Question in title, my progress: let $z = \cos(x) + i\sin(x)$ then $1 + \cos(x) + \cos(2x) +\dots + \cos(nx) = Re(1 + z + z^2 +\dots + z^n) = Re\left (\dfrac{1-z^{n+1}}{1-z} \right)$ by geometric series; multiplying $\dfrac{1-z^{n+1}}{1-z}$ by $\overline{1-z}$ we get $1 + \cos(x) + \cos(2x) +\dots + \cos(nx) = Re \left ( \dfrac{(1-z^{n+1})(\overline{1-z})}{|1-z|^2} \right )$ but I am not sure how to proceed from here. edit: this is for a complex analysis course, so i'd appreciate a hint using complex analysis without using the exponential function asked Sep 30, 2014 at 16:30 Jonx12Jonx121611 gold badge1 silver badge8 bronze badges $\endgroup$ 7 $\begingroup$Use $\sin(a) \cos(b) = \frac{1}{2} \sin(b+a) - \frac{1}{2} \sin(b-a)$ and multiply your sum with $\sin\left(x/2\right)$. $$ \sum_{m=0}^{n} \sin\left(\frac{x}{2}\right) \cos(m x) = \frac{1}{2} \sum_{m=0}^n \left\{\sin \left(\left(m+1-\frac{1}{2}\right)x \right) - \sin\left( \left(m-\frac{1}{2}\right)x \right) \right\} $$ The sum telescopes, $\sum_{m=0}^n \left(f(m+1)-f(m)\right) = f(n+1)-f(0)$, hence $$ \sum_{m=0}^{n} \sin\left(\frac{x}{2}\right) \cos(m x) = \sin \left(\left(n+\frac{1}{2}\right)x \right) - \sin\left(\frac{x}{2} \right) $$ now divide by $\sin\left(\frac{x}{2}\right)$. answered Sep 30, 2014 at 16:39 gold badges134 silver badges211 bronze badges $\endgroup$ $\begingroup$ $\textbf{Hint:}$Use De Moivre's formula to compute $z^{n+1}$. $\textbf{Edit:}$ The other way to compute this sum is writing $\cos x$ as: $$\displaystyle \cos x=\frac{e^{ix}+e^{-ix}}{2}$$ In my opinion it's the easiest way. You simply get two geometric series: $$\sum_{k=0}^{n}\cos kx=\sum_{k=0}^{n} \frac{e^{ikx}+e^{-ikx}}{2}=\frac{1}{2}\left(\sum_{k=0}^{n}e^{ikx}+\sum_{k=0}^{n}e^{-ikx}\right)= \\ = \frac{1}{2}\left(\frac{1-e^{i(n+1)x}}{1-e^{ix}}+\frac{1-e^{-i(n+1)x}}{1-e^{-ix}}\right)$$ It's equal: $$\frac{1}{2}\left(\frac{1-e^{i(n+1)x}}{1-e^{ix}}+\frac{1-e^{-i(n+1)x}}{1-e^{-ix}}\right)=\frac{1}{2}\frac{(1-e^{i(n+1)x})(1-e^{-ix})+(1-e^{-i(n+1)x})(1-e^{ix})}{1+1-e^{ix}-e^{-ix}}=\frac{2+(e^{inx}+e^{-inx})-(e^{ix}+e^{-ix})-(e^{-inx}+e^{-inx})}{2-(e^{ix}+e^{-ix})}$$ Using again formula for $\cos$ you get: $$\frac{1}{2}\frac{2+2\cos nx -2\cos x -2\cos (n+1)x}{2-2\cos x}$$ answered Sep 30, 2014 at 16:35 aghaagha9,8224 gold badges19 silver badges35 bronze badges $\endgroup$ 3 Not the answer you're looking for? Browse other questions tagged complex-analysis or ask your own question. 1-cos 2 x = sin 2 x Proof. Step 1: At first, we will apply the Pythagorean trigonometric identity which is provided below. sin 2 x + cos 2 x = 1 …. (I) Step 2: Next, we will substitute the above value of 1 in the expression 1 − cos 2 x. This will give us. So the simplification of 1 − cos 2 x is equal to sin 2 x. In other words, jak rozwiązać cosx = - 1/2 ja: jak rozwiązać równanie, gdy mamy, że cosx = −1/2 ogólnie, kiedy jest jakaś wartość ujemna gdyby było cosx = 1/2 to by było: x = π/3 + 2kπ lub x = − π/3 + 2kπ a jak robimy, gdy jest wartość ujemna? da się jakoś bez rysunku? ponoć pomocny jest wierszyk "w pierszej ćwiartce same plusy..." Bardzo proszę o pomoc 28 kwi 15:53 Jerzy: ⇔ cosx = −cos60 = cos(180 −60) = cos120 28 kwi 15:59 6latek : 28 kwi 16:00 ja: dziękuję bardzo 28 kwi 16:02 6latek : podobnie rozwiązujesz 28 kwi 16:03- Аζεքоγևнօ αδቧλаст
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Let us equate, X and Y, i.e. X = Y. So, the above formula for cos 2X, becomes. cos 2X = cos(X + X) = cos X cos X– sin X sin X. cos 2X = cos2 X–sin2 X. Hence, the first cos 2X formula follows, as. cos 2X = cos2 X–sin2 X. And for this reason, we know this formula as double the angle formula, because we are doubling the angle.
Cos2x is one of the important trigonometric identities used in trigonometry to find the value of the cosine trigonometric function for double angles. It is also called a double angle identity of the cosine function. The identity of cos2x helps in representing the cosine of a compound angle 2x in terms of sine and cosine trigonometric functions, in terms of cosine function only, in terms of sine function only, and in terms of tangent function only. Cos2x identity can be derived using different trigonometric identities. Let us understand the cos2x formula in terms of different trigonometric functions and its derivation in detail in the following sections. Also, we will explore the concept of cos^2x (cos square x) and its formula in this article. 1. What is Cos2x? 2. What is Cos2x Formula in Trigonometry? 3. Derivation of Cos2x Using Angle Addition Formula 4. Cos2x In Terms of sin x 5. Cos2x In Terms of cos x 6. Cos2x In Terms of tan x 7. Cos^2x (Cos Square x) 8. Cos^2x Formula 9. How to Apply Cos2x Identity? 10. FAQs on Cos2x What is Cos2x? Cos2x is an important trigonometric function that is used to find the value of the cosine function for the compound angle 2x. We can express cos2x in terms of different trigonometric functions and each of its formulas is used to simplify complex trigonometric expressions and solve integration problems. Cos2x is a double angle trigonometric function that determines the value of cos when the angle x is doubled. What is Cos2x Formula in Trigonometry? Cos2x is an important identity in trigonometry which can be expressed in different ways. It can be expressed in terms of different trigonometric functions such as sine, cosine, and tangent. 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 cos2x = 1 - 2sin2x cos2x = (1 - tan2x)/(1 + tan2x) Derivation of Cos2x Formula Using Angle Addition Formula We know that the cos2x formula can be expressed in four different forms. We will use the angle addition formula for the cosine function to derive the cos2x identity. Note that the angle 2x can be written as 2x = x + x. Also, we know that cos (a + b) = cos a cos b - sin a sin b. We will use this to prove the identity for cos2x. Using the angle addition formula for cosine function, substitute a = b = x into the formula for cos (a + b). cos2x = cos (x + x) = cos x cos x - sin x sin x = cos2x - sin2x Hence, we have cos2x = cos2x - sin2x Cos2x In Terms of sin x Now, that we have derived cos2x = cos2x - sin2x, we will derive the formula for cos2x in terms of sine function only. We will use the trigonometry identity cos2x + sin2x = 1 to prove that cos2x = 1 - 2sin2x. We have, cos2x = cos2x - sin2x = (1 - sin2x) - sin2x [Because cos2x + sin2x = 1 ⇒ cos2x = 1 - sin2x] = 1 - sin2x - sin2x = 1 - 2sin2x Hence, we have cos2x = 1 - 2sin2x in terms of sin x. Cos2x In Terms of cos x Just like we derived cos2x = 1 - 2sin2x, we will derive cos2x in terms of cos x, that is, cos2x = 2cos2x - 1. We will use the trigonometry identities cos2x = cos2x - sin2x and cos2x + sin2x = 1 to prove that cos2x = 2cos2x - 1. We have, cos2x = cos2x - sin2x = cos2x - (1 - cos2x) [Because cos2x + sin2x = 1 ⇒ sin2x = 1 - cos2x] = cos2x - 1 + cos2x = 2cos2x - 1 Hence , we have cos2x = 2cos2x - 1 in terms of cosx Cos2x In Terms of tan x Now, that we have derived cos2x = cos2x - sin2x, we will derive cos2x in terms of tan x. We will use a few trigonometric identities and trigonometric formulas such as cos2x = cos2x - sin2x, cos2x + sin2x = 1, and tan x = sin x/ cos x. We have, cos2x = cos2x - sin2x = (cos2x - sin2x)/1 = (cos2x - sin2x)/( cos2x + sin2x) [Because cos2x + sin2x = 1] Divide the numerator and denominator of (cos2x - sin2x)/( cos2x + sin2x) by cos2x. (cos2x - sin2x)/(cos2x + sin2x) = (cos2x/cos2x - sin2x/cos2x)/( cos2x/cos2x + sin2x/cos2x) = (1 - tan2x)/(1 + tan2x) [Because tan x = sin x / cos x] Hence, we have cos2x = (1 - tan2x)/(1 + tan2x) in terms of tan x Cos^2x (Cos Square x) Cos^2x is a trigonometric function that implies cos x whole squared. Cos square x can be expressed in different forms in terms of different trigonometric functions such as cosine function, and the sine function. We will use different trigonometric formulas and identities to derive the formulas of cos^2x. In the next section, let us go through the formulas of cos^2x and their proofs. Cos^2x Formula To arrive at the formulas of cos^2x, we will use various trigonometric formulas. The first formula that we will use is sin^2x + cos^2x = 1 (Pythagorean identity). Using this formula, subtract sin^2x from both sides of the equation, we have sin^2x + cos^2x -sin^2x = 1 -sin^2x which implies cos^2x = 1 - sin^2x. Two trigonometric formulas that includes cos^2x are cos2x formulas given by cos2x = cos^2x - sin^2x and cos2x = 2cos^2x - 1. Using these formulas, we have cos^2x = cos2x + sin^2x and cos^2x = (cos2x + 1)/2. Therefore, the formulas of cos^2x are: cos^2x = 1 - sin^2x ⇒ cos2x = 1 - sin2x cos^2x = cos2x + sin^2x ⇒ cos2x = cos2x + sin2x cos^2x = (cos2x + 1)/2 ⇒ cos2x = (cos2x + 1)/2 How to Apply Cos2x Identity? Cos2x formula can be used for solving different math problems. Let us consider an example to understand the application of cos2x formula. We will determine the value of cos 120° using the cos2x identity. We know that cos2x = cos2x - sin2x and sin 60° = √3/2, cos 60° = 1/2. Since 2x = 120°, x = 60°. Therefore, we have cos 120° = cos260° - sin260° = (1/2)2 - (√3/2)2 = 1/4 - 3/4 = -1/2 Important Notes on Cos 2x cos2x = cos2x - sin2x cos2x = 2cos2x - 1 cos2x = 1 - 2sin2x cos2x = (1 - tan2x)/(1 + tan2x) The formula for cos^2x that is commonly used in integration problems is cos^2x = (cos2x + 1)/2. The derivative of cos2x is -2 sin 2x and the integral of cos2x is (1/2) sin 2x + C. ☛ Related Articles: Trigonometric Ratios Trigonometric Table Sin2x Formula Inverse Trigonometric Ratios FAQs on Cos2x What is Cos2x Identity in Trigonometry? 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. It can be expressed in terms of different trigonometric functions such as sine, cosine, and tangent. What is the Cos2x Formula? Cos2x can be expressed in terms of different trigonometric functions such as sine, cosine, and tangent. It can be expressed as: cos2x = cos2x - sin2x cos2x = 2cos2x - 1 cos2x = 1 - 2sin2x What is the Derivative of cos2x? The derivative of cos2x is -2 sin 2x. Derivative of cos2x can easilty be calculated using the formula d[cos(ax + b)]/dx = -asin(ax + b) What is the Integral of cos2x? The integral of cos2x can be easilty obtained using the formula ∫cos(ax + b) dx = (1/a) sin(ax + b) + C. Therefore, the integral of cos2x is given by ∫cos 2x dx = (1/2) sin 2x + C. 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 In Terms of tan x? We can express the cos2x formula in terms of tanx. The formula is given by cos2x = (1 - tan2x)/(1 + tan2x) in terms of tan x. How to Derive cos2x Identity? Cos2x identity can be derived using different identities such as angle sum identity of cosine function, cos2x + sin2x = 1, tan x = sin x/ cos x, etc. How to Derive Cos Square x Formula? We can derive the cos square x formula using various trigonometric formulas which consist of cos^2x. The trigonometric identities which include cos^2x are cos^2x + sin^2x = 1, cos2x = cos^2x - sin^2x and cos2x = 2cos^2x - 1. We can simplify these formulas and determine the value of cos square x. What is Cos^2x Formula? We have three formulas for cos^2x given below: cos^2x = 1 - sin^2x ⇒ cos2x = 1 - sin2x cos^2x = cos2x + sin^2x ⇒ cos2x = cos2x + sin2x cos^2x = (cos2x + 1)/2 ⇒ cos2x = (cos2x + 1)/2 What is the Formula of Cos2x in Terms of Cos? The formula of cos2x in terms of cos is given by, cos2x = 2cos^2x - 1, that is, cos2x = 2cos2x - 1. Example 1: Determine the value of x if cos-1 (√3/2) = x using the inverse cosine formula. Solution: We have cos x = Adjacent Side/Hypotenuse and cos-1 x is an inverse function of cos x. We know that cos (π/6) = √3/2. Since π/6 ∈ [0, π], cos-1 (√3/2) = π/6. Answer: The value of x if cos-1 (√3/2) = x using the inverse cosine formula- Π τጶπа ξዪጏ
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cos 2x 1 2
