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

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) $\cos (\alpha+\theta)$ (b) $\cos (\alpha-\theta)$

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) $\cos (\alpha+\theta)$
(b) $\cos (\alpha-\theta)$

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

00:01 Hello everybody.

00:03 In this video, i'm going to be showing you how to solve exercise 28 in chapter 9, section 1 of cohen's pre -calculus 7th edition.

00:11 Now in this problem, we're given two angles, alpha and theta, and we're told that for alpha, the sign of alpha is equal to 12th 15th's, and this angle lies in between pi over 2 and pi.

00:22 Whereas for theta, its cosine is equal to 7 .25s, and theta lies in between negative 2 pi, negative 3 pi over 2.

00:31 And with this information, they want us to calculate both quantities, cosine of alpha plus theta, as well as the cosine of alpha minus theta, which i've written in this form just for convenience later on.

00:44 Now, the first step in computing these quantities is to find the cosine of alpha as well as the sign of theta.

00:51 And to do this, we can use a pythagorean identity.

00:54 And this identity states that for some angle t, cosine squared of t plus sine squared of t is equal to 1.

01:05 And so if we went ahead and solved for cosine of t, we could use this identity to show that the cosine of t is equal to 1 minus sine squared of t all squared.

01:23 And similarly, solving for sign, we can show that the sine of t is equal to 1 minus cosine squared of t all square rooted.

01:37 So let's use the first of these two equations we just found to calculate the cosine of alpha.

01:42 We have that the cosine of alpha is equal to 1 minus sine squared of alpha, which is just going to be 12 .13 squared, all square rooted.

02:00 We can go ahead and carry this, taking the square of 1213, which is going to be 144 over 169.

02:09 And now, because we can rewrite 1 as 169 over 169, this becomes 169 minus 144 over 169.

02:22 And now 169 minus 144 is equal to 25.

02:27 And so we end up with a square root of 25 over 169.

02:32 But here, notice that alpha lies in between pi over 2 and pi, which means that alpha is in the second quadrant.

02:39 And this means that its cosine must be negative.

02:42 So we're going to go ahead and take the negative square root of this quantity here, negative 5 over 13, which is the cosine of alpha.

02:53 And now let's use the second.

02:54 These equations for sign to calculate the sign of theta.

02:59 We have that sine theta is equal to 1 minus cosine squared of theta, which is going to be 725 squared.

03:10 This is all square rooted.

03:12 We carry this over in square 7 .25s to find that it is equal to 49 over 625.

03:21 One can be rewritten as 625 over 625.

03:24 So this turns into the fraction 625 minus 49 over 625.

03:35 And now 625 minus 49 is equal to 576.

03:40 So we're taking the square root of 576 over 625.

03:47 And now, as theta lies in between the angles negative 2 pi and negative 3 pi over 2, we can digest this to mean that theta lies in the first quadrant.

03:57 And therefore, its sign must be positive...

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