Trigonometri Jumlah Dan Selisih Sudut

Explanation
The value of cos(A+B) can be found using the cosine addition formula, which states that cos(A+B) = cos(A)cos(B) - sin(A)sin(B). Since A and B are acute angles, sin(A) and sin(B) are both positive. Therefore, we can substitute the given values of cos(A) and cos(B) into the formula to find cos(A+B). Plugging in the values, we get cos(A+B) = (4/5)(24/25) - (3/5)(7/25) = 96/125 - 21/125 = 75/125 = 3/5. Therefore, the correct answer is 3/5.

Explanation
The given equation states that A + B = (2/3)π and sin A = 2sin B. We need to find the value of (A - B). From the given equation, we can rearrange it as sin A = 2sin (π - A) since sin (π - A) = sin A. This implies that A = π - A, which means A = π/2. Therefore, (A - B) = π/2 - B. Since we don't have any information about B, we cannot determine its value. Hence, the given answer of π/2 is correct.

Explanation
The given equation cos 2A = 1/3 can be rewritten as 2cos^2(A) - 1 = 1/3. Simplifying further, we have 2cos^2(A) = 4/3. Dividing both sides by 2, we get cos^2(A) = 2/3. Taking the square root of both sides, we have cos(A) = √(2/3). Since sin^2(A) + cos^2(A) = 1, we can substitute the value of cos(A) in this equation to find sin(A). Therefore, sin(A) = √(1 - 2/3) = √(1/3) = 1/√(3) = 1/3√(3). Hence, the answer is 1/3 (V3).

Explanation
In a right triangle ABC, if cos A cos B = 1/2, then cos (A-B) = 1.

Explanation
The given question is asking for the value of sin 2x if tan x = a. To find this, we can use the identity sin 2x = 2sin x cos x. Since tan x = a, we can rewrite it as sin x / cos x = a. Solving for sin x, we get sin x = a cos x. Substituting this into the identity, sin 2x = 2(a cos x)(cos x) = 2a cos^2 x. Since cos^2 x = 1 - sin^2 x, we can rewrite it as sin 2x = 2a(1 - sin^2 x) = 2a - 2a sin^2 x. Dividing both sides by 1 - sin^2 x (which is equal to cos^2 x), we get sin 2x = 2a / (1/cos^2 x) = 2a / (a^2 + 1). Therefore, the correct answer is 2a / (a^2 + 1).