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2 months ago

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The point (Negative StartFraction StartRoot 2 EndRoot Over 2 EndFraction, StartFraction StartRoot 2 EndRoot Over 2 EndFraction)

is the point at which the terminal ray of angle Theta intersects the unit circle. What are the values for the cosine and cotangent functions for angle Theta?

2 answers:

Leona [12.6K]2 months ago

8 0

The values of cosine Ф and cotangent Ф are and -1.

zzz [12.3K]2 months ago

4 0

The cosine and cotangent values of angle Ф are $\frac{-\sqrt{2} }{2}$ and -1. When the terminal side of an angle intersects the unit circle at point (x, y), then: The x-coordinate corresponds to the cosine of the angle formed with the positive x-axis and the terminal side, while the y-coordinate corresponds to the sine of that angle. Depending on the signs of x and y coordinates, we can determine the quadrant in which the angle resides.

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How many possible values for y are there where y = cos^-1 0?

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Further clarification

Trigonometry is a branch of math focused on the connections between the sides and angles of triangles.

Considering special angles of trigonometric functions, for instance

$\displaystyle sin~0=0\\\\cos~30=\frac{1}{2}\sqrt{3}\\\\tan~45=1\\\\sec~45=\sqrt{2}\\\\etc.$

In the equation y = cos⁻¹ 0, the value of y can be derived as follows:

y = cos⁻¹0

y = arc cos 0

cos y = 0

Thus, the resulting value of y:

$\displaystyle \frac{\pi }{2},\frac{3}{2}\pi,\frac{5}{2}\pi, etc$

Alternatively, it can be expressed as:

$\displaystyle y=\frac{\pi }{2} (2n-1)$ ⇒ y: arithmetic sequence

So there are infinite solutions for y

Learn more

trigonometric identities

Keywords: trigonometric, infinite values,arithmetic sequence

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A flat circular plate has the shape of the region x2 + y2≤1. The plate, including the boundary where x2 + y2 = 1, is heated such

Leona [12618]

Setting both partial derivatives to zero results in a single critical point at $(x,y)=\left(\dfrac12,0\right)$ , located within the unit disk.

At this given point, the derivative value of the Hessian matrix is

$|H|=\begin{vmatrix}T_{xx}&T_{xy}\\T_{yx}&T_{yy}\end{vmatrix}=\begin{vmatrix}2&0\\0&4\end{vmatrix}=8>0$

and the second-order partial derivative with respect to $x$ yields

$T_{xx}\bigg|_{(x,y)=(1/2,0)}=2>0$

This suggests that the critical point represents a local minimum, marking it as the coldest area on the plate with a temperature of $T\left(\dfrac12,0\right)=-\dfrac14$ .

To find the hottest area on the plate, it must be located along the boundary. Let $x=\cos\theta$ and $y=\sin\theta$ , so that

$T(x,y)=T(\theta)=\cos^2\theta+2\sin^2\theta-\cos\theta$
$T(\theta)=\dfrac32-\cos\theta-\dfrac12\cos2\theta$

Thus, the plate's boundary (the circle $x^2+y^2=1$ ) is treated as a single variable function $\theta$ examined over $\theta\in[0,2\pi)$ . A single differentiation gives

$T'(\theta)=\sin\theta+\sin2\theta=0$
$\implies\theta=0,\theta=\dfrac{2\pi}3,\theta=\pi,\theta=\dfrac{4\pi}3$

You will discover that $T(\theta)$ achieves three extrema on the interval $(0,2\pi)$ , with relative maxima occurring at $\theta=\dfrac{2\pi}3$ and $\theta=\dfrac{4\pi}3$ , and a relative minimum at $\theta=\pi$ (and $\theta=0$ , if you wish to include that).

Our minimum has already been identified inside the plate - which you can check to have a lower temperature than at the points noted by $T(\theta)$ - and we identify two maxima at $\theta=\dfrac{2\pi}3$ and $\theta=\dfrac{4\pi}3$ , both showing a maximum temperature of $T=\dfrac94$ .

Reverting to Cartesian coordinates, these points match up with $\left(-\dfrac12,\pm\dfrac{\sqrt3}2\right)$ .

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1 month ago