Compound angle formula

It may be helpful to think about the graphs of sine and cosine. At x = 0, sine starts at 0 and goes up to 1, while cosine starts at 1 and goes down. Think about what happens on the negative side though. On the sine graph, for negative x's you are getting negative y's, as it passes below the x-axis, while the cosine keeps giving you positive y's as it starts down toward the x-axis. Sine graph: Cosine graph: 1) sec x = 1/(cos x), so the first equation simplifies to (cos x)/ (cos x), which equals 1. 2) 1 - tan^2 x = 1 - (sin^2 x)/(cos^2 x). Making common denominators and making one fraction => (cos^2 x - sin ^2 x)/(cos^2 x). Since we know cos^2 x + sin^2 x = 1, then sin^2 x = 1 - cos^2 x. Substition => (cos^2 x - (1 - cos^2 x))/(cos^2 x) = (2cos^2 x - 1)/(cos^2 x). On the right side of original equation, 2 - sec^2 x = 2 - (1/(cos^2 x)). Making common denominators and one fraction => (2cos^2 x -1)/(cos^2 x). Now we have both sides the same, therefore the identity is proven. 0:21, Sal says that he is assuming we already know a bunch of properties, like the fact that the sin(a+b) = sin a + cos b. I don't recall seeing this shown or proved anywhere earlier in the Trig material. Did I just forget it? Or has this not been shown yet? A few times, I've seen the videos ordered such that a video that assumes we already know something is placed before the video that explains it. That gets confusing. ;u; If I did just forget the video where this is shown earlier, can someone point me to ...

Cos(a - b) In trigonometry, cos(a - b) is one of the important trigonometric identities, that finds application in finding the value of the cosine trigonometric function for the difference of angles. The expansion of cos (a - b) helps in representing the cos of a compound angle in terms of trigonometric functions sine and cosine. Let us understand the cos(a-b) identity and its proof in detail in the following sections. 1. 2. 3. 4. 5. Proof of Cos(a - b) Formula The proof of expansion of cos(a-b) formula can be given using the geometrical construction method. Let us see the stepwise derivation of the formula for the To prove: cos (a - b) = cos a cos b + sina sinb Construction: Draw a line OX in a plane and let us rotate it about O in the anti-clockwise direction to a point Z, making an acute ∠XOZ = a, from starting position to its finalposition. Again, rotate the line, this time in the backward direction, starting from the position OZtill it reaches a point Y, thus making out an Next, take a point P on OY, and draw PQ and PR perpendiculars to OX and OZ respectively. Draw Now, from the cos (a - b) = OQ/OP = (OS+SQ)/OP = OS/OP + SQ/OP = OS/OP + TR/OP = OS/OR ∙ OR/OP + TR/PR ∙ PR/OP = cos a cos b + sin ∠TPR sinb = cos a cos b + sin a sin b, (since we know, ∠TPR = a) Therefore, cos ( a - b) = cos a cos b + sin a sin b. How to Apply Cos(a - b)? The expansion of cos(a - b) can be used to find the value of the cosine trigonometric function for angles that can be represented as the...

Sin3x Sin3x gives the value of the sine trigonometric function for triple angle. On the other hand, sin^3x is the whole cube of the sine function. Sin3x is a triple angle identity in trigonometry. The expansion of sin3x formula can be derived using the angle addition identity of the sine function and it involves the term sin^3x (sin cube x). It is a specific case of compound angles identity of the sine function. The formula for Sin3x identity helps in solving various trigonometric problems. In this article, we will discuss the formulas and concepts of sin3x and sin^3x. We will also understand the derivation of these formulas, sin3x graph, and application with the help of solved examples for a better understanding of the concept. 1. 2. 3. 4. 5. 6. 7. Graph of Sin3x The behavior of the graph of sin3x is similar to that of the trigonometric function sin x. The angle in consideration in sin3x is thrice the angle in the function sin x. We know that for a function sin bx, the period is 2π/|b| which implies the period of sin3x is 2π/3. Hence, the graph of sin3x is narrower than the graph of sin x as the period of sin3x is one-third the period of sin x (Period of sin x is 2π) Now, let us plot the graph of sin3x by taking some points on the graph and joining them. Let us consider a few points for y = sin3x and y = sin x and plot them. • When x = 0, 3x = 0 ⇒ sin x = 0, sin3x = 0 • When x = -π/6, 3x = -π/2 ⇒ sin x = -1/2, sin3x = -1 • When x = π/6, 3x = π/2 ⇒ sin x = 1/2, sin3x = 1 •...

Addition formulae When we add or subtract angles, the result is called a compound angle. For example, \(30^\circ + 120^\circ\) is a compound angle. Using a calculator, we find: \[\sin (30^\circ + 120^\circ ) = \sin 150^\circ = 0.5\] \[\sin 30^\circ + \sin 120^\circ = 1.366\,(to\,3\,d.p.)\] This shows that \(\sin (A + B)\) is not equal to \(\sin A + \sin B\) . Instead, we can use the following identities: \[\sin (A + B) = \sin A\cos B + \cos A\sin B\] \[\sin (A - B) = \sin A\cos B - \cos A\sin B\] \[\cos (A + B) = \cos A\cos B - \sin A\sin B\] \[\cos (A - B) = \cos A\cos B + \sin A\sin B\] • Addition formulae are given in a condensed form: • \[\sin (A \pm B) = \sin A\cos B \pm \cos A\sin B\] • \[\cos (A \pm B) = \cos A\cos B \mp \sin A\sin B\] These formulae are used to expand trigonometric functions to help us simplify or evaluate trigonometric expressions of this form. See how we approach this two-part question: Question 1. By writing \(75^\circ = 45^\circ + 30^\circ\) determine the exact value of \(\sin 75^\circ\) Reveal answer down 1. \(\sin 75^\circ = \sin (45 + 30)^\circ\) Using the formula for \(\sin (A + B)\) \[= \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ\] Using exact values that you should know: \[= \frac\) Reveal answer down 2. Since \(\frac\]

Sin3x Sin3x gives the value of the sine trigonometric function for triple angle. On the other hand, sin^3x is the whole cube of the sine function. Sin3x is a triple angle identity in trigonometry. The expansion of sin3x formula can be derived using the angle addition identity of the sine function and it involves the term sin^3x (sin cube x). It is a specific case of compound angles identity of the sine function. The formula for Sin3x identity helps in solving various trigonometric problems. In this article, we will discuss the formulas and concepts of sin3x and sin^3x. We will also understand the derivation of these formulas, sin3x graph, and application with the help of solved examples for a better understanding of the concept. 1. 2. 3. 4. 5. 6. 7. Graph of Sin3x The behavior of the graph of sin3x is similar to that of the trigonometric function sin x. The angle in consideration in sin3x is thrice the angle in the function sin x. We know that for a function sin bx, the period is 2π/|b| which implies the period of sin3x is 2π/3. Hence, the graph of sin3x is narrower than the graph of sin x as the period of sin3x is one-third the period of sin x (Period of sin x is 2π) Now, let us plot the graph of sin3x by taking some points on the graph and joining them. Let us consider a few points for y = sin3x and y = sin x and plot them. • When x = 0, 3x = 0 ⇒ sin x = 0, sin3x = 0 • When x = -π/6, 3x = -π/2 ⇒ sin x = -1/2, sin3x = -1 • When x = π/6, 3x = π/2 ⇒ sin x = 1/2, sin3x = 1 •...

Cos(a - b) In trigonometry, cos(a - b) is one of the important trigonometric identities, that finds application in finding the value of the cosine trigonometric function for the difference of angles. The expansion of cos (a - b) helps in representing the cos of a compound angle in terms of trigonometric functions sine and cosine. Let us understand the cos(a-b) identity and its proof in detail in the following sections. 1. 2. 3. 4. 5. Proof of Cos(a - b) Formula The proof of expansion of cos(a-b) formula can be given using the geometrical construction method. Let us see the stepwise derivation of the formula for the To prove: cos (a - b) = cos a cos b + sina sinb Construction: Draw a line OX in a plane and let us rotate it about O in the anti-clockwise direction to a point Z, making an acute ∠XOZ = a, from starting position to its finalposition. Again, rotate the line, this time in the backward direction, starting from the position OZtill it reaches a point Y, thus making out an Next, take a point P on OY, and draw PQ and PR perpendiculars to OX and OZ respectively. Draw Now, from the cos (a - b) = OQ/OP = (OS+SQ)/OP = OS/OP + SQ/OP = OS/OP + TR/OP = OS/OR ∙ OR/OP + TR/PR ∙ PR/OP = cos a cos b + sin ∠TPR sinb = cos a cos b + sin a sin b, (since we know, ∠TPR = a) Therefore, cos ( a - b) = cos a cos b + sin a sin b. How to Apply Cos(a - b)? The expansion of cos(a - b) can be used to find the value of the cosine trigonometric function for angles that can be represented as the...

It may be helpful to think about the graphs of sine and cosine. At x = 0, sine starts at 0 and goes up to 1, while cosine starts at 1 and goes down. Think about what happens on the negative side though. On the sine graph, for negative x's you are getting negative y's, as it passes below the x-axis, while the cosine keeps giving you positive y's as it starts down toward the x-axis. Sine graph: Cosine graph: 1) sec x = 1/(cos x), so the first equation simplifies to (cos x)/ (cos x), which equals 1. 2) 1 - tan^2 x = 1 - (sin^2 x)/(cos^2 x). Making common denominators and making one fraction => (cos^2 x - sin ^2 x)/(cos^2 x). Since we know cos^2 x + sin^2 x = 1, then sin^2 x = 1 - cos^2 x. Substition => (cos^2 x - (1 - cos^2 x))/(cos^2 x) = (2cos^2 x - 1)/(cos^2 x). On the right side of original equation, 2 - sec^2 x = 2 - (1/(cos^2 x)). Making common denominators and one fraction => (2cos^2 x -1)/(cos^2 x). Now we have both sides the same, therefore the identity is proven. 0:21, Sal says that he is assuming we already know a bunch of properties, like the fact that the sin(a+b) = sin a + cos b. I don't recall seeing this shown or proved anywhere earlier in the Trig material. Did I just forget it? Or has this not been shown yet? A few times, I've seen the videos ordered such that a video that assumes we already know something is placed before the video that explains it. That gets confusing. ;u; If I did just forget the video where this is shown earlier, can someone point me to ...

Addition formulae When we add or subtract angles, the result is called a compound angle. For example, \(30^\circ + 120^\circ\) is a compound angle. Using a calculator, we find: \[\sin (30^\circ + 120^\circ ) = \sin 150^\circ = 0.5\] \[\sin 30^\circ + \sin 120^\circ = 1.366\,(to\,3\,d.p.)\] This shows that \(\sin (A + B)\) is not equal to \(\sin A + \sin B\) . Instead, we can use the following identities: \[\sin (A + B) = \sin A\cos B + \cos A\sin B\] \[\sin (A - B) = \sin A\cos B - \cos A\sin B\] \[\cos (A + B) = \cos A\cos B - \sin A\sin B\] \[\cos (A - B) = \cos A\cos B + \sin A\sin B\] • Addition formulae are given in a condensed form: • \[\sin (A \pm B) = \sin A\cos B \pm \cos A\sin B\] • \[\cos (A \pm B) = \cos A\cos B \mp \sin A\sin B\] These formulae are used to expand trigonometric functions to help us simplify or evaluate trigonometric expressions of this form. See how we approach this two-part question: Question 1. By writing \(75^\circ = 45^\circ + 30^\circ\) determine the exact value of \(\sin 75^\circ\) Reveal answer down 1. \(\sin 75^\circ = \sin (45 + 30)^\circ\) Using the formula for \(\sin (A + B)\) \[= \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ\] Using exact values that you should know: \[= \frac\) Reveal answer down 2. Since \(\frac\]