Let f(θ)=sinθcosθ(0 ≤θ≤(π)/(2)) (a) Set your calculator in the radian mode and complete the tabl

Let f(θ)=sinθcosθ(0 ≤θ≤(π)/(2)) (a) Set your calculator in...

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in understanding periodic phenomena and angular relationships. Their properties on specific intervals, particularly within the first quadrant, are crucial when evaluating expressions and understanding behaviors such as growth, decay, and symmetry.

Product of Trigonometric Functions

Analyzing the product of trigonometric functions, like sin ? cos ?, often involves finding equivalent expressions, such as through the use of double-angle identities. These identities simplify complex products and facilitate the evaluation or optimization of trigonometric expressions.

Numerical Evaluation of Functions

Numerical evaluation involves using tools like calculators to approximate function values to a desired precision. This process is essential when constructing tables of values or when an exact symbolic evaluation is either not possible or impractical, particularly when working under specific modes such as radians.

Optimization and Maximum Value

Finding extremal values, like the maximum of a function, is a core concept in calculus and analysis. Evaluating a function at several key points over a certain interval or employing analytical techniques helps in understanding where and when the function attains its maximum or minimum.

Arithmetic Mean-Geometric Mean Inequality (AM-GM Inequality)

The AM-GM Inequality is a fundamental result in algebra which provides a relationship between the arithmetic mean and the geometric mean of non-negative numbers. This inequality is a powerful tool for establishing bounds on products and is often applied to simplify and prove inequalities involving trigonometric functions.

Domain and Range Considerations

Understanding the domain and range of functions is critical when applying inequalities and evaluating maximum or minimum values. These considerations ensure that the functions are evaluated over the correct intervals and that any applied inequalities are valid within the appropriate bounds.

Transcript

00:01 Give us this function f of theta is equal to sine a theta times cosine theta and they want us to do a couple of things with it so first they wanted us to fill out this chart here and round to the two decimal places or the second decimal place so i went ahead and make the chart already and so then in part b they asked us was the largest value that f of theta takes and well that would be right here right here at point five so zero point so now they want us to show that sine of theta times cosine theta is less than equal to one half on zero theta to pi half.

00:40 And they tell us to use this hint from an earlier problem that the square root of a times v is less than equal to a plus b over 2.

00:52 So let's use that fact.

00:56 Well, what we can do to start is just plug in sign and cosine.

01:02 So we have sine of theta, cosine of theta.

01:09 We square root this, and then over here, this is going to be cosine theta plus sine theta all over two.

01:18 Okay.

01:20 Well, now, if this value over here is defined, and the one on the right is, then we can also square each side, since these are strictly larger than zero.

01:37 And in doing this it would give us the following now so sign of theta so just remember if this is able to be negative then we have some issues but at least this time we don't have to worry about that so we have sine of theta cosine of theta is less than are equal to so this would be all over four and then let's expand that numerator there so that gives us cosine square theta plus two cosine theta sine theta and then plus sine squared theta well cosine theta sine theta is one so this is going to be sine of theta cosine of theta less than or equal to one plus two cosine theta sine theta over four okay so let's multiply that four over actually so just multiply the 4 over, and then i'm going to erase it right here.

02:58 So now we can subtract 2 cosine theta over, and doing that will give us 2, sine theta, cosine theta, is less than equal to 1, divide each side by 2.

03:10 That gives us sine theta, cosine theta, is less than equal to 1⁄2.

03:15 So we proved what they wanted us to, so we can put a little proofbox in the smiley face.

03:19 And actually, maybe we should say why this is true for 0 to pi half.

03:24 All right, so first we needed to ensure that this was going to be larger than zero, which it is, since sine of theta and cosine theta are going to both be positive in this interval here.

03:37 So over here is zero to pi half.

03:41 So both sine and cosine are positive.

03:42 So multiplying those together, we'll give us a positive number.

03:46 Taking the square root of a positive number is still positive.

03:48 And then over here, adding two positive numbers divided by two also gives us the same.

03:56 Value or a positive value so that's good so that was probably the extra bit that we needed to kind of explain this a little bit better all right so now they want us to figure out is sine of theta cosine theta less than or equal to one -half for all values of theta all right well we've already shown this for zero to pi half so actually let's put this unit circle up again.

04:31 All right, so we already did this one.

04:33 So let's just kind of put this into sections and see.

04:38 Well, if we're in this quadrant, so if we're in quad 2, which is going to be pi half to pi it will be true since so true since of in this quadrant cosine of theta is less than zero so if sine of theta cosine theta is less than zero, well that's definitely less than or equal to one half.

05:21 So this is going to check out.

05:25 And for similar reasons, the fourth quadrant over here will check out.

05:32 Because for quad four, so this is going to be from 3 pi halves to 2 pi halves to 2.

05:45 So this is also true since, well, in that area, sine of theta is less than zero, but cosine is positive.

05:54 So again, sine of theta, cosine of theta, is going to be less than zero since, or actually, i should have put equal to here as well as here, because it can be zero on that interval.

06:10 And then again, this is going to be less than or equal to one half.

06:16 So that one also checks.

06:19 All right, now we just have quad three.

06:23 So quad three.

06:30 So this is the interval pi, two, three, pi haps.

06:38 So in this interval, sine of theta and cosine of theta, these are both less then 0.

06:48 So they're both negative.

06:50 So in this case, we can go ahead and use what we did before for this to see if that would help us.

07:04 Okay...