Question

1)5*sin98/(sin49*sin41) 2)8/(sin(-27/пи/4)*cos(31*пи/4))

Answer 1

$5\frac{\sin 98}{\sin 49*\sin 41} = 5\frac{2\sin 49 \cos 49}{\sin 49*\sin 41} =\\ = 10\frac{\cos 49}{\sin 41} = 10\frac{\sin (90-49)}{\sin 41} =10$

 

$\frac{8}{sin(-27\pi/4)*cos(31\pi/4)} = \frac{8}{sin(\pi/4)*cos(-\pi/4)} = \\ =\frac{8}{\sqrt{2}/2*\sqrt{2}/2} =8*2=16$

Answer 2

Let's calculate the values of the given expressions:

1. $\frac{5 \cdot \sin(98)}{\sin(49) \cdot \sin(41)}$

First, we need to convert the angles from degrees to radians:

$\sin(98^\circ)$ is equivalent to $\sin\left(\frac{98\pi}{180}\right)$ $\sin(49^\circ)$ is equivalent to $\sin\left(\frac{49\pi}{180}\right)$ $\sin(41^\circ)$ is equivalent to $\sin\left(\frac{41\pi}{180}\right)$

Using these values, we can evaluate the expression:

$\frac{5 \cdot \sin\left(\frac{98\pi}{180}\right)}{\sin\left(\frac{49\pi}{180}\right) \cdot \sin\left(\frac{41\pi}{180}\right)}$

2. $\frac{8}{\sin\left(-\frac{27}{\pi/4}\right) \cdot \cos\left(\frac{31\pi}{4}\right)}$

In this expression, we have a negative angle and trigonometric functions with fractional arguments. We'll need to evaluate these values step by step.

First, let's convert the angles to radians:

$-\frac{27}{\pi/4}$ is equivalent to $-\frac{27}{\frac{\pi}{4}}$ $\cos\left(\frac{31\pi}{4}\right)$

Now, let's calculate each part separately:

1. Evaluating $\sin(98^\circ)$: $\sin\left(\frac{98\pi}{180}\right) \approx 0.906307787$ (rounded to 9 decimal places)

2. Evaluating $\sin(49^\circ)$: $\sin\left(\frac{49\pi}{180}\right) \approx 0.75470958$ (rounded to 8 decimal places)

3. Evaluating $\sin(41^\circ)$: $\sin\left(\frac{41\pi}{180}\right) \approx 0.656059029$ (rounded to 9 decimal places)

Now, let's substitute these values back into the expression:

1. $\frac{5 \cdot 0.906307787}{0.75470958 \cdot 0.656059029} \approx 14.365$ (rounded to 3 decimal places)

2. For the second expression, we need to calculate $\sin\left(-\frac{27}{\pi/4}\right)$ and $\cos\left(\frac{31\pi}{4}\right)$ separately.

a) Evaluating $\sin\left(-\frac{27}{\pi/4}\right)$: $\sin\left(-\frac{27}{\pi/4}\right) \approx -0.992477875$ (rounded to 9 decimal places)

b) Evaluating $\cos\left(\frac{31\pi}{4}\right)$: $\cos\left(\frac{31\pi}{4}\right) \approx 0.707106781$ (rounded to 9 decimal places)

Now, let's substitute these values back into the expression:

2. $\frac{8}{-0.992477875 \cdot 0.707106781} \approx -11.368$ (