Leadership

Mathematics

Eleven Standard >> Compound angles formula cos(A+B), sin(A+B) & sin(A-B)

Click the green "Start" button for MCQ.
Leadership

 

Compound angles formula cos(A+B), sin(A+B) & sin(A-B)

 

We know the formula for cos(A - B):

cos(A - B) = cos(A) * cos(B) + sin(A) * sin(B)

Now, we'll introduce a new angle, C, such that A = (C + B). In other words, angle A is the sum of angles C and B.

Substituting A = C + B into the formula for cos(A - B):

cos((C + B) - B) = cos(C + B) = cos(C) * cos(B) + sin(C) * sin(B)

Now, we want to find the value of cos(C + B), which is essentially cos(A + B). We can rewrite this equation as:

cos(A + B) = cos(C) * cos(B) + sin(C) * sin(B)

So, by introducing a new angle C and utilizing the formula for cos(A - B), we have derived the compound angle formula for cos(A + B):

cos(A + B) = cos(A) * cos(B) - sin(A) * sin(B)

This is the formula for cos(A + B) derived from the formula for cos(A - B) using the concept of complementary angles.

 

Let's start with the formula for sin(A + B):

sin(A+B)=sin(A)∗cos(B)+cos(A)∗sin(B)

Now, we'll substitute the given information A=\(\frac{\pi}{2}\)​−A, which implies A+A=\(\frac{\pi}{2}\)​:

sin(\(\frac{\pi}{2}\)​)=sin(A)∗cos(B)+cos(A)∗sin(B)

Since sin(\(\frac{\pi}{2}\)​)=1, the equation simplifies to:

1=sin(A)∗cos(B)+cos(A)∗sin(B)

Now, we'll use the formula for cos(A - B):

cos(A−B)=cos(A)∗cos(B)+sin(A)∗sin(B)

Substituting A=\(\frac{\pi}{2}\)−A into the cos(A - B) formula:

cos(\(\frac{\pi}{2}\)​−A−B)=cos(\(\frac{\pi}{2}\)​−A)∗cos(B)+sin(\(\frac{\pi}{2}\)−A)∗sin(B)

Using the trigonometric identity sin(\(\frac{\pi}{2}\)​−x)=cos(x):

cos(\(\frac{\pi}{2}\)−A−B)=cos(A)∗cos(B)+cos(A)∗sin(B)

Simplifying further:

cos(\(\frac{\pi}{2}\)−A−B)=cos(A)∗(cos(B)+sin(B))

Since cos(\(\frac{\pi}{2}\)​−x)=sin(x), we can rewrite the left side:

sin(A+B)=cos(A)∗(cos(B)+sin(B))

Now, using the equation 1=sin(A)∗cos(B)+cos(A)∗sin(B), we can solve for sin(A)∗cos(B):

sin(A)∗cos(B)=1−cos(A)∗sin(B)

Substituting this back into the previous equation:

sin(A+B)=cos(A)∗(1−cos(A)∗sin(B))

This is the expression for sin(A + B) in terms of cos(A - B) and the given relationship A=\(\frac{\pi}{2}\)​−A.

When we replace B with −B, the expression for sin(A+B) in terms of cos(A−B) becomes:

sin(A−B)=cos(A)⋅(1−cos(A)⋅sin(−B))

Remember that sin(−x)=−sin(x), so sin(−B)=−sin(B). Substituting this into the equation:

sin(A−B)=cos(A)⋅(1−cos(A)⋅(−sin(B)))

Leadership
Hand drawn

Hide

Forgot your password?

Close

Error message here!

Hide

Lost your password? Please enter your email address. You will receive a link to create a new password.

Back to log-in

Close