Compute the indicated quantity using the following data. sinα=(12)/(13) where (π)/(2)<α<π

Compute the indicated quantity using the following data. ...

Compute the indicated quantity using the following data. $$\begin{aligned}&\sin \alpha=\frac{12}{13} \quad \text { where } \frac{\pi}{2}<\alpha<\pi\\&\cos \beta=-\frac{3}{5} \quad \text { where } \pi<\beta<\frac{3 \pi}{2}\\&\cos \theta=\frac{7}{25} \quad \text { where }-2 \pi<\theta<-\frac{3 \pi}{2}\end{aligned}$$ (a) $\sin (\alpha-\beta)$ (b) $\cos (\alpha-\beta)$

Compute the indicated quantity using the following data.
$$\begin{aligned}&\sin \alpha=\frac{12}{13} \quad \text { where } \frac{\pi}{2}<\alpha<\pi\\&\cos \beta=-\frac{3}{5} \quad \text { where } \pi<\beta<\frac{3 \pi}{2}\\&\cos \theta=\frac{7}{25} \quad \text { where }-2 \pi<\theta<-\frac{3 \pi}{2}\end{aligned}$$
(a) $\sin (\alpha-\beta)$
(b) $\cos (\alpha-\beta)$

Key Concepts

Trigonometric Identities

Trigonometric identities provide equations involving trigonometric functions that are true for all angles, such as sin²? + cos²? = 1 and the difference formulas for sine and cosine. These identities are used to manipulate and simplify expressions and solve problems where only partial trigonometric information is provided.

Pythagorean Identity

The Pythagorean identity, sin²? + cos²? = 1, is used to determine the unknown sine or cosine value when one is given. It is essential for recovering the complete set of trigonometric values of an angle, ensuring that the correct magnitude (and with the appropriate sign) is used based on the given information.

Quadrant Considerations

The signs of trigonometric functions depend on the quadrant in which an angle lies. Understanding quadrant considerations is critical for assigning the correct sign to sine, cosine, and other trigonometric values. This ensures that when using identities or difference formulas, the derived values accurately reflect the angle's location on the unit circle.

Sine and Cosine Difference Formulas

The sine and cosine difference formulas, namely sin(? - ?) = sin? cos? - cos? sin? and cos(? - ?) = cos? cos? + sin? sin?, are fundamental tools for calculating the sine and cosine of the difference between two angles. They express the functions of angle differences in terms of the individual sine and cosine values, allowing for evaluation using known quantities.

Transcript

00:01 Hello, everybody.

00:02 In this video, i'm going to be showing you how to solve exercise 26 in chapter 9, section 1 of cohen's pre -calculus 7th edition.

00:11 Now, in this problem, we are given two angles, alpha and beta, and we are told that the sign of alpha equals 12 -13s, with alpha being in between the angles pi over 2 and pi, and the cosine of beta, and the cosine of beta is equal to negative 3 -5s, where beta lies in between the angles pi and 3 pi over 2.

00:28 And with this information, they want us to compete the quantities, sine of alpha minus beta, as well as cosine of alpha minus beta.

00:38 Now, in order to do this, we're going to want to know the cosine of alpha as well as the sine of beta, which we can calculate using the pythagorean identity.

00:47 For an angle t, cosine squared of t plus sine squared of t is equal to one.

01:00 And we can rearrange this equation in two ways.

01:02 First, we can solve the cosine of t, which will be equal to 1 minus sine squared of t, all square rooted.

01:16 And similarly, we can solve for the sign of t and write that the sign of this angle t is equal to 1 minus cosine squared of t.

01:28 That's there.

01:30 All square rooted.

01:32 And so now, using these two identities, we can go ahead and solve for the cosine of alpha.

01:38 We have that the cosine of alpha is equal to 1 minus sine squared of alpha which is just 1213 squared and this will all be under a square root and now under the radical we have 1 minus 144 over 169 which is 12 squared over 13 squared and now one can be written as 169 over 169 so this becomes 169 minus 144 divided by 169 and then we're taking the square root of this fraction 169 minus 144 is 25 which we divide by 169 and take the square root and now recall that this root will result in a positive and a negative root but because the alpha is in between the angles pi over two and pi the cosine of alpha will necessarily be negative and therefore we want to take the negative root negative 5 over 13, which is the cosine of alpha.

02:58 And now similarly, we want to calculate the sign of beta using this equation down here.

03:04 We have that the sign of beta is equal to 1 minus the cosine squared of beta, which is negative 3 5th squared, all square rooted.

03:20 We carry out that square, which is equal to 9 .25s.

03:27 We get 1 minus 9 .25s, all squareded.

03:31 And now 1 .1 .5ths.

03:31 And now 1.

03:31 Is equal to 25 over 25.

03:34 So the term under the square root becomes 25 minus 9 over 25.

03:41 This fraction is still squareded.

03:43 25 minus 9 is 16.

03:46 So we have 16, 20 fifths under the square root...