Sin a Sin b

Sin a Sin b Definitions, Formulas and Explanations

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    Sin a Sin b Definitions and Examples

    Sin A + Sin B

    The sum of two sines is equal to the cosine of their difference multiplied by the product of their amplitudes. The two sines are out of phase with each other if their difference is not an integer multiple of pi.

    What is SinA + SinB Identity in Trigonometry?

    In trigonometry, the sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. The cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.

    The sin a sin b identity states that:

    sin(a + b) = sin a cos b + cos a sin b

    This identity can be derived from first principles using the definition of sine and cosine. It can also be verified using basic algebraic manipulation.

    This identity is useful in solving problems involving angles that are not multiples of 90 degrees. For example, consider finding the value of sin 75 degrees without using a calculator. By applying the sin a sin b identity, we can break down this angle into two smaller angles: 60 degrees and 15 degrees. We know that sin 60 degrees = 0.5 and that cos 60 degrees = 0.866, so we can plug these values into our equation:

    sin(60 + 15) = sin 60 cos 15 + cos 60 sin 15
    = (0.5)(0.2587) + (0.866)(0.9659)
    = 0.1329 + 0.8205
    = 0.9534

    Sin A + Sin B Sum to Product Formula

    When two angles in a triangle have their sides opposite to each other equal in length, the triangle is said to be isosceles. The side opposite the shared angle is the one that’s equal in length, so in the image above, side AC is of equal length to side BC. Angles A and B are therefore of equal measure.

    We can label these angles with letters:
    Angle A = a
    Angle B = b
    Shared Angle C = c

    The Sin A + Sin B Sum to Product Formula states that:
    sin(a) + sin(b) = 2 * sin((a + b)/2) * cos((a – b)/2)

    Proof of SinA + SinB Formula

    This formula states that the sine of the sum of two angles is equal to the product of the sines of those angles. This can be written as: Sin(A + B) = SinA * SinB.

    This formula is useful in many situations, such as calculating the sides of a triangle when two angles and one side are known. It can also be used to find an angle when two sides and one angle are known.

    There are a few things to keep in mind when using this formula. First, it only works for angles that are less than 90 degrees. Second, the order of the angles matters. The sine of A + B is not necessarily the same as the sine of B + A.

    To use this formula, simply substitute in the values for A and B and then calculate the sine of each side. The result will be equal to the product of the sines of A and B.

    How to Apply Sin A + Sin B?

    To apply sin A + sin B, first calculate the value of sin A and sin B. Then, add these values together to get the final answer.

     

    Sin a Sin B

    3D plot

    3D plot

    Contour plot

    Contour plot

    Reduced trigonometric form

    1/2 (cos(a - b) - cos(a + b))

    Alternate form

    -1/4 (e^(-i a) - e^(i a)) (e^(-i b) - e^(i b))

    Roots

    a = π n, n element Z

    b = π n, n element Z

    Integer roots

    b = 0

    a = 0

    Properties as a function

    periodic in a with period 2 π periodic in b with period 2 π

    even

    Series expansion at a = 0

    a sin(b) - 1/6 a^3 sin(b) + 1/120 a^5 sin(b) + O(a^6) (Taylor series)

    Derivative

    d/da(sin(a) sin(b)) = cos(a) sin(b)

    Indefinite integral

    integral sin(a) sin(b) da = -cos(a) sin(b) + constant

    Global minima

    min{sin(a) sin(b)} = -1 at (a, b) = (2 π n_2 - π/2, 2 π n_1 + π/2) for integer n_1 and n_2

    min{sin(a) sin(b)} = -1 at (a, b) = (2 π n_2 + (3 π)/2, 2 π n_1 + π/2) for integer n_1 and n_2

    Global maxima

    max{sin(a) sin(b)} = 1 at (a, b) = (2 π n_2 - π/2, 2 π n_1 - π/2) for integer n_1 and n_2

    max{sin(a) sin(b)} = 1 at (a, b) = (2 π n_2 - π/2, 2 π n_1 + (3 π)/2) for integer n_1 and n_2

    Alternative representations

    sin(a) sin(b) = 1/(csc(a) csc(b))

    sin(a) sin(b) = cos(a + π/2) cos(b + π/2)

    sin(a) sin(b) = cos(-a + π/2) cos(-b + π/2)

    Series representations

    sin(a) sin(b)∝( sum_(k_1=0)^∞ sum_(k_2=0)^∞ (-1)^(k_1 + k_2) (d^(2 k_1) δ(a))/(da^(2 k_1)) (d^(2 k_2) δ(b))/(db^(2 k_2)))/(θ(a) θ(b))

    sin(a) sin(b) = 4 sum_(k_1=0)^∞ sum_(k_2=0)^∞ (-1)^(k_1 + k_2) J_(1 + 2 k_1)(a) J_(1 + 2 k_2)(b)

    sin(a) sin(b) = sum_(k_1=0)^∞ sum_(k_2=0)^∞ ((-1)^(k_1 + k_2) a^(1 + 2 k_1) b^(1 + 2 k_2))/((1 + 2 k_1)! (1 + 2 k_2)!)

    Integral representations

    sin(a) sin(b) = integral_0^1 integral_0^1 cos(a t_1) cos(b t_2) dt_2 dt_1

    sin(a) sin(b) = -(a b ( integral_(-i ∞ + γ)^(i ∞ + γ) e^(-a^2/(4 s) + s)/s^(3/2) ds) integral_(-i ∞ + γ)^(i ∞ + γ) e^(-b^2/(4 s) + s)/s^(3/2) ds)/(16 π) for γ>0

    sin(a) sin(b) = -(( integral_(-i ∞ + γ)^(i ∞ + γ) (2^(-1 + 2 s) a^(1 - 2 s) Γ(s))/Γ(3/2 - s) ds) integral_(-i ∞ + γ)^(i ∞ + γ) (2^(-1 + 2 s) b^(1 - 2 s) Γ(s))/Γ(3/2 - s) ds)/(4 π) for (0<γ<1 and a>0 and b>0)

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