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) $\sin (\theta-\beta)$ (b) $\sin (\theta+\beta)$

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) $\sin (\theta-\beta)$
(b) $\sin (\theta+\beta)$

Key Concepts

Trigonometric Angle Sum Identities

The trigonometric angle sum identities, such as sin(? + ?) = sin ? · cos ? + cos ? · sin ? and cos(? + ?) = cos ? · cos ? ? sin ? · sin ?, are essential tools for breaking down the sine and cosine of a sum of angles into expressions involving the individual angles. This simplifies the computation when certain trigonometric values of the individual angles are known.

Pythagorean Identity

The Pythagorean identity, sin²? + cos²? = 1, is fundamental in trigonometry as it allows for deriving one trigonometric function given the other. In problems like these, it facilitates finding the missing sine or cosine values from the provided data, ensuring that all components needed for the angle sum formulas are appropriately determined.

Quadrant and Sign Considerations

Considering the quadrant in which an angle lies is crucial because the sign of the trigonometric functions (sine and cosine) depends on the quadrant. Each quadrant of the unit circle has specific sign conventions that must be applied to the calculated values, ensuring the correct overall sign when using the angle sum formulas.

Transcript

00:02 Hello everybody.

00:03 In this video, i'll be showing you how to solve exercise 27 in chapter 9, section 1 of cohen's pre -calculus 7th edition.

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

00:23 We are also told that the cosine of theta is equal to 7 .25s, where theta lies in between the angles negative 2 -py and negative 3 -py over 2.

00:33 And with this information, they want us to calculate both the sign of theta minus beta as well as the sign of theta plus beta, which i've written as the sign of theta plus or minus beta just for convenience purposes.

00:45 But to calculate both of these quantities, what we first want to do is calculate the signs of both beta and theta.

00:52 And to do this, we can use a pythagorean identity, which states that for an angle t, cosine squared of t plus sine squared of t is equal to 1.

01:06 And if we take this identity and solve for sign, we have that the sign of this angle t is equal to the square root of 1 minus cosine squared of t.

01:24 And so now, let's use this formula to calculate the sign of beta.

01:29 You have that the sign of beta is equal to 1 minus cosine squared of beta, which is going to be negative 3 fifth squared as we're just squaring the cosine of beta.

01:45 Then we're going to square root this whole quantity.

01:48 Carrying this over in square negative three -fifths, we get nine, 20 -fifths, and now one can be represented as 25 over 25.

01:58 So this fraction becomes 25 minus 9 over 25.

02:04 25 minus 9 is 16, and now we take the score root of 16 over 25, which can result in either a positive root or a negative root.

02:16 However, notice that beta sits in between pi and 3 pi over 2.

02:20 Which means that beta is an angle in the third quadrant.

02:24 And so the sign of beta must necessarily be negative.

02:28 So we take the negative root four -fifths.

02:35 Now let's do the same to find the sign of theta.

02:39 We have that the sign of theta is equal to 1 minus the square of the cosine of theta, which is 7 .25s, and we square root this whole quantity.

02:54 Now we carry this over, and 725 squared is 49.

03:00 Over 625.