Mathematics, 26.12.2019 23:31 wolfsilvereye10

Need on these 2 questions and my last 2 uploads !

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Answer from: emily5400

19. $\frac{3\pi}{2}$

20. The work is the answer

Step-by-step explanation:

19. We begin with the equation (for my ease, I will swap theta with x)

$sin(x)+1=cos^{2} (x)$

First, we can use the Pythagorean Identity to substitute in $1-sin^{2} (x)$ for $cos^{2} (x)$

This gives us

$sin(x)+1=1-sin^{2} (x)$

Next, we can subtract 1 from each side

$sin(x)=-sin^{2} (x)$

Now, we can divide each side by sin(x)

1=-sin(x)

Now we move the negative to the other side

sin(x)=-1

Now we can do the inverse of sin to find our value

$x=sin^{-1} (-1)$

With this inverse, we are looking for an angle that has the sin (or the y value) of -1. As this is an inverse sin function, it has the domain restriction of

$[-\frac{\pi }{2} ,\frac{\pi }{2}]$ .

From this information, we find that $x=-\frac{\pi}{2}$

But this answer does not fit the initial interval that we were given. Luckily, $\frac{3\pi}{2}$ is the same location as $\frac{\pi}{2}$ , so that is our answer.

20. We begin with the expression sin(360-x) and we have to prove that it is equal to -sin(x)

First, we need to use the Difference of 2 Angles Trig Identity. This says:

$sin(\alpha -\beta )=sin(\alpha) cos(\beta) -cos(\alpha) sin(\beta)$

When we apply this to our expression, we get

sin(360) cos(x) -cos(360) sin(x)

Next, we can simplify our the two finite aspects

$sin(360)=0\\\\cos(360)=1$

When we plug in these two values, we get

(0) cos(x) -(1) sin(x)

This simplifies to -sin(x) which is what we were trying to verify.

Answer from: Quest

the value of v.w= -12.

step-by-step explanation:

we are given three vector r,v and w as:

r = < 5, -5, -2> ; v = < 2, -8, -8> ; w = < -2, 6, -5>

we know that r,v and w could also be represented as:

r=5i-5j-2k

v=2i-8j-8k

and w= -2i+6j-5k

where i,j and k are the unit vectors of x,y and z-axis respectively.

also we know that it has the following property:

i.i=1 ; j.j=1; k.k=1

and i.=j.i=i.k=k.i=j.k=k.j=0

i.e. when we multiply two vectors or we find it's dot product we need to only multiply the components of x coordinates with each other; y with each other and z with each other.

hence, v.w= (2i-8j- -2i+6j-5k)

= 2×(-2)+(-8)×(6)+(-8)×(-5)

= -4-48+40

= -12

hence, v.w= -12

Answer from: Quest

i can't see the picture : /

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