If angle A is 45 degrees and angle B is 60 degrees.
Find sin(A)cos(B),
find cos(A)sin(B),
find sin(A)sin(B),
and find cos(A)cos(B)
The choises for the first are:
A. 1/2[sin(105)+sin(345)]
B. 1/2[sin(105)-sin(345)]
C. 1/2[sin(345)+cos(105)]
D. 1/2[sin(345)-cos(105)]
You don't need to give me the answer to all of them. Just help explain at least one, so I can figure out the rest on my own.
choices*
To find sin(A)cos(B), we need to find the sine of angle A and the cosine of angle B.
Given:
Angle A = 45 degrees
Angle B = 60 degrees
Step 1: Find sin(A)
The sine of angle A is given by sin(A) = sin(45).
Step 2: Find cos(B)
The cosine of angle B is given by cos(B) = cos(60).
Using the values from the trigonometric functions table or a scientific calculator, we can find the specific values:
sin(45) = √2/2
cos(60) = 1/2
Now, we can substitute these values into the expression sin(A)cos(B):
sin(A)cos(B) = (√2/2)(1/2)
Simplifying this expression:
sin(A)cos(B) = √2/4
From here, you should compare this simplified expression (√2/4) to the provided choices.
To find the value sin(A)cos(B), we can use the trigonometric identity:
sin(A)cos(B) = sin(A)cos(B) + cos(A)sin(B) - cos(A)sin(B)
Now, let's break down each term step by step.
First, sin(A)cos(B):
We know that sin(A) = sin(45°) = 1/√2 and cos(B) = cos(60°) = 1/2.
So, sin(A)cos(B) = (1/√2)(1/2) = 1/(√2 * 2) = 1/(2√2) = 1/(2√2) * (√2/√2) = √2/4.
Now, let's move on to the second term, cos(A)sin(B):
We already know sin(B) = sin(60°) = √3/2 and cos(A) = cos(45°) = 1/√2.
So, cos(A)sin(B) = (1/√2)(√3/2) = √3/(2√2) = √3/(2√2) * (√2/√2) = √6/4.
Finally, we subtract the second term from the first term:
sin(A)cos(B) - cos(A)sin(B) = √2/4 - √6/4 = (√2 - √6)/4.
Now, we need to compare this result with the answer choices given:
A. 1/2[sin(105)+sin(345)]
B. 1/2[sin(105)-sin(345)]
C. 1/2[sin(345)+cos(105)]
D. 1/2[sin(345)-cos(105)]
A quick glance shows that none of the answer choices match the result we obtained.
Therefore, none of the given answer choices is correct for sin(A)cos(B).
You just need to review your formulas:
http://www.sosmath.com/trig/prodform/prodform.html
and recall that for trig functions, 345° is the same as -15°