88... USING the formula sin[A-B]= sinA cosB -COS A sin B FIND THE valueof sin 15
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Write sin(15) as the difference of two angles
The equation to use is sin(15) = sin(45 - 30) Using 45 and 30 works well because 45 and 30 degree angles are parts of triangles with well-defined ratios. For example, the 45 degree angle is part of a 45:45:90 triangle with ratios of 1:1:square_root(2). Similarly, the 30 degree angle features in a 30:60:90 triangle with ratios of 1:square_root(3):2.
Expand the expression using the angle difference identity
Here, sin( 45 - 30 ) = sin(45) * cos(30) + cos(45) * sin(30).
Calculate the individual elements
In geometry, sin is defined as the ratio of the side opposite the angle to the hypotenuse. Therefore, sin(45) = 1 / square_root(2) and sin(30) = 1 / 2. Similarly, cos is defined as the ratio of the side adjacent to the angle to the hypotenuse. Hence cos(45) = 1 / square_root(2), and cos(30) = square_root(3) / 2.
Plug these values into the equation
This gives sin(45) * cos(30) + cos(45) * sin(30) = ( 1 / square_root(2) ) * ( sqare_root(3) / 2 ) + ( 1 / square_root(2) ) * ( 1 / 2 ).
Simplify the expression
Using the common denominator of 2 * square_root(2), simplify to: [ square_root(3) + 1 ] / [ 2* square_root(2) ].
Use a calculator to find the result
Use a calculator to compute the expression and find the answer, which is 0.259.
The equation to use is sin(15) = sin(45 - 30) Using 45 and 30 works well because 45 and 30 degree angles are parts of triangles with well-defined ratios. For example, the 45 degree angle is part of a 45:45:90 triangle with ratios of 1:1:square_root(2). Similarly, the 30 degree angle features in a 30:60:90 triangle with ratios of 1:square_root(3):2.
Expand the expression using the angle difference identity
Here, sin( 45 - 30 ) = sin(45) * cos(30) + cos(45) * sin(30).
Calculate the individual elements
In geometry, sin is defined as the ratio of the side opposite the angle to the hypotenuse. Therefore, sin(45) = 1 / square_root(2) and sin(30) = 1 / 2. Similarly, cos is defined as the ratio of the side adjacent to the angle to the hypotenuse. Hence cos(45) = 1 / square_root(2), and cos(30) = square_root(3) / 2.
Plug these values into the equation
This gives sin(45) * cos(30) + cos(45) * sin(30) = ( 1 / square_root(2) ) * ( sqare_root(3) / 2 ) + ( 1 / square_root(2) ) * ( 1 / 2 ).
Simplify the expression
Using the common denominator of 2 * square_root(2), simplify to: [ square_root(3) + 1 ] / [ 2* square_root(2) ].
Use a calculator to find the result
Use a calculator to compute the expression and find the answer, which is 0.259.
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