Express 5 cos ? + 6 sin ? in the form R cos(?

Express 5 cos ? + 6 sin ? in the form R cos(? - ?) where R > 0 and 0 < ? < ?/2. State (a) the maximum value of 5 cos ? + 6 sin ? and the least positive value of ? which gives this maximum, (b) the minimum value of 5 cos ? + 6 sin ? and the least positive value of ? which gives this minimum.

Transcript

00:02 We have to express 5 cosine of theta and 6xx9 theta in the form of r of cosine of theta minus beta, or cosine of theta minus beta where r is positive and beta is an acute angle.

00:15 When we have to state the maximum of this with the least possible value of theta, which gives the maximum and the wide severance of minimum and the corresponding theta, which gives the minimum.

00:28 So remember that any equation of the form, a, sine, theta plus b, cosine of theta can be condensed into a single trigonometric function by multiplying and dividing by root of the coefficients of sine, root of the square of the, root of sum of a square of coefficient of sine and cosine by multiplying.

00:51 And multiplication and division by a single number is totally allowed mathematically because after all they cancel each other and we again get the same equation.

00:59 So we're not changing anything.

01:00 So what we do is we take, we keep the numerator as it is, but the denominator is something which we take inside and we distribute it with border terms like this.

01:14 And now depending on how we want to convert it, since we want cost of theta minus alpha, which means that we should have cosic, cosby plus sine a sign b kind of thing.

01:24 So now we now we assume that this particular expression is cosine of beta.

01:31 So b over, i'm not going to work any more in terms of a and b.

01:36 Let's substitute a is 6 and b is 5 because a is the coefficient of sign.

01:41 That will be 6 square plus 5 square.

01:44 6 over 6 square plus 5 square.

01:47 Let's find that rather than writing that again and again.

01:49 So 36 plus 25 is 61.

01:52 So that will be 6 over 61, root 61 i mean, sine theta plus b, which is 5 over root 61 cosine of theta.

02:03 So this is going, this is how it's going to look like.

02:06 And this will become root of 61, root 6 over root 61, sign theta plus 5 over root 61 times cost theta.

02:15 As i was saying, now i'm going to assume that 5 over root 61 is going to be cosine of beta.

02:22 And if i do that, let's find the value of sign of beta from here...

Question

Express 5 cos ? + 6 sin ? in the form R cos(? - ?) where R > 0 and 0 < ? < ?/2. State (a) the maximum value of 5 cos ? + 6 sin ? and the least positive value of ? which gives this maximum, (b) the minimum value of 5 cos ? + 6 sin ? and the least positive value of ? which gives this minimum.

Please give Ace some feedback