I'm really confused on how to do this. Can someone show me step by step on how to solve this? The directions say to use the Sum-to-Product formulas to find the exact value of the expression.
sin 5pi/4-sin 3pi/4
you want to use these identities
http://www.mathwords.com/s/sum_to_product_identities.htm
especially this one
sinx - siny = 2cos( (x+y)/2 )sin(( x-y)/2 )
sin 5π/4 - sin3π/4
= 2cos(π)sin(π/4)
= 2(-1)(√2/2)
= -√2
To find the exact value of the expression sin(5π/4) - sin(3π/4) using the Sum-to-Product formulas, follow these steps:
Step 1: Recall the Sum-to-Product formulas.
The Sum-to-Product formulas are trigonometric identities that allow us to rewrite the difference of two trigonometric functions as a product of trigonometric functions. The formulas are:
sin(A) - sin(B) = 2 * sin((A - B)/2) * cos((A + B)/2)
cos(A) - cos(B) = -2 * sin((A + B)/2) * sin((A - B)/2)
These formulas will be useful in simplifying the expression sin(5π/4) - sin(3π/4).
Step 2: Substitute the values into the formulas.
In this case, A = 5π/4 and B = 3π/4.
Step 3: Use the Sum-to-Product formulas to simplify the expression.
Applying the first formula, sin(A) - sin(B) = 2 * sin((A - B)/2) * cos((A + B)/2), we can substitute the values into the formula:
sin(5π/4) - sin(3π/4) = 2 * sin((5π/4 - 3π/4)/2) * cos((5π/4 + 3π/4)/2)
Simplifying the angle inside the first sin term gives:
sin((5π/4 - 3π/4)/2) = sin(π/4/2) = sin(π/8)
Similarly, simplifying the angle inside the cos term gives:
cos((5π/4 + 3π/4)/2) = cos(8π/4/2) = cos(2π/8) = cos(π/4)
Substituting the simplified values back into the expression, we get:
2 * sin(π/8) * cos(π/4)
Step 4: Evaluate the remaining trigonometric functions.
To evaluate sin(π/8) and cos(π/4), you can either use a calculator that supports trigonometric functions or refer to a trigonometric table.
Sin(π/8) is approximately 0.3827, and cos(π/4) is approximately 0.7071.
Step 5: Multiply the values to get the exact value.
Multiply the evaluated trigonometric functions to get the exact value:
2 * 0.3827 * 0.7071 = 0.5403
Therefore, the exact value of the expression sin(5π/4) - sin(3π/4) using the Sum-to-Product formulas is 0.5403.