Use one of the Pythagorean Identities to find cos a and sin b. Then use the identity sin (a-b)=sin acos b -cos asin b.
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Practice makes perfect
To find the value of sin (a-b), we will start by recalling the formula for the sine of a difference.
sin (a-b)=sin acos b -cos asin b
Therefore, we need to know the values of sin a, cos b, cos a, and sin b. We are given that sin a= 817 and cos b= - 2425. With this information we can find the values of cos a and sin b. Let's start by finding cos a. To do so we will use one of the Pythagorean Identities.
sin ^2 a + cos ^2 a =1In this identity, we can substitute 817 for sin a and solve for cos a.
Let's now determine the sign of cos a. We are told that a is greater than 0 and less than π2. Therefore, if we draw angle a in standard position, its terminal side will be located in the first quadrant.
If the terminal side of an angle in standard position is in the first quadrant, then its cosine is positive. Therefore, we have that cos a= 1517.
Let's now find the value of sin b. We will substitute the given value cos b= - 2425 into the same identity as previously used.
Let's now determine the sign of sin b. We are told that b is greater than π and less than 3π2. Therefore, its terminal side is located in the third quadrant.
If the terminal side of an angle in standard position is in the third quadrant, then its sine is negative. Therefore, we have that sin b= - 725.
Now we have all the information we need to calculate sin (a-b).
