If z=2 3 (x-12) and x=39, then what is z?

Help me with trigonometry

poizon [28]

Answer:

See below

Step-by-step explanation:

It has something to do with the Weierstrass substitution, where we have

$\int\, f(\sin(x), \cos(x))dx = \int\, \dfrac{2}{1+t^2}f\left(\dfrac{2t}{1+t^2}, \dfrac{1-t^2}{1+t^2} \right)dt$

First, consider the double angle formula for tangent:

$\tan(2x)= \dfrac{2\tan(x)}{1-\tan^2(x)}$

Therefore,

$\tan\left(2 \cdot\dfrac{x}{2}\right)= \dfrac{2\tan(x/2)}{1-\tan^2(x/2)} = \tan(x)=\dfrac{2t}{1-t^2}$

Once the double angle identity for sine is

$\sin(2x)= \dfrac{2\tan(x)}{1+\tan^2(x)}$

we know $\sin(x)=\dfrac{2t}{1+t^2}$ , but sure, we can derive this formula considering the double angle identity

$\sin(x)= 2\sin\left(\dfrac{x}{2}\right)\cos\left(\dfrac{x}{2}\right)$

Recall

$\sin \arctan t = \dfrac{t}{\sqrt{1 + t^2}} \text{ and } \cos \arctan t = \dfrac{1}{\sqrt{1 + t^2}}$

Thus,

$\sin(x)= 2 \left(\dfrac{t}{\sqrt{1 + t^2}}\right) \left(\dfrac{1}{\sqrt{1 + t^2}}\right) = \dfrac{2t}{1 + t^2}$

Similarly for cosine, consider the double angle identity

Thus,

$\cos(x)= \left(\dfrac{1}{\sqrt{1 + t^2}}\right)^2- \left(\dfrac{t}{\sqrt{1 + t^2}}\right)^2 = \dfrac{1}{t^2+1}-\dfrac{t^2}{t^2+1} =\dfrac{1-t^2}{1+t^2}$

Hence, we showed $\sin(x) \text { and } \cos(x)$

======================================================

$5\cos(x) =12\sin(x) +3, x \in [0, 2\pi ]$

Solving

$5\,\overbrace{\frac{1-t^2}{1+t^2}}^{\cos(x)} = 12\,\overbrace{\frac{2t}{1+t^2}}^{\sin(x)}+3$

$\implies \dfrac{5-5t^2}{1+t^2}= \dfrac{24t}{1+t^2}+3 \implies \dfrac{5-5t^2 -24t}{1+t^2}= 3$

$\implies 5-5t^2-24t=3\left(1+t^2\right) \implies -8t^2-24t+2=0$

$t = \dfrac{-(-24)\pm \sqrt{(-24)^2-4(-8)\cdot 2}}{2(-8)} = \dfrac{24\pm 8\sqrt{10}}{-16} = \dfrac{3\pm \sqrt{10}}{-2}$

$t=-\dfrac{3+\sqrt{10}}{2}\\t=\dfrac{\sqrt{10}-3}{2}$

Just note that

$\tan\left(\dfrac{x}{2}\right) = \dfrac{3\pm 8\sqrt{10}}{-2}$

and $\tan\left(\dfrac{x}{2}\right)$ is not defined for $x=k\pi , k\in\mathbb{Z}$

6 0

3 years ago

What is the slope of this equation = -0.3x + 3.7.

meriva

Answer:

- 0.3

Step-by-step explanation:

Slope Intercept Form:

y = mx + b

Slope:

m = slope

Equation:

- 0.3x + 3.7

Slope = - 0.3

Hope This Helps :)

5 0

3 years ago

A cube has a surface area of 96 square centimeters. How long is each face of the cube?

netineya [11]

Answer:

I believe the answer would be 4

Step-by-step explanation:

I'm going to use (a) to represent the side length in my explanation. side length is equivalent to the square root of the surface area over 6. The 6 is coming from there being 6 faces on a cube. So to solve this equation, you would first write it out. a=96/6. Next you would divide 96/6, giving you 16. and then lastly, you would find the square root of 16, which is 4. giving you a final answer of 4.

Hope this helps, and you understand it more!

6 0

3 years ago

What percent is shaded in the grid? Answer:

Keith_Richards [23]

I can’t see a picture of the grid

4 0

3 years ago

URGENT!! Need help can someone explain please!!!

forsale [732]

Answer:

The answer to your question is 1 = 69°

Step-by-step explanation:

To solve this problem remember that the sum of the internal angles in a triangle equals 180°.

Let the third angle of the triangle be x.

39° + 30° + x = 180

Solve for x

x = 180 - 39 - 30

x = 111°

Angle x and angle 1 are supplementary so their sum equals 180°.

111° + 1 = 180

1 = 180 - 111

1 = 69°

5 0

3 years ago

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