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• * '''Tangent''': The tangent of angle $A$, denoted $\tan (A)$, is defined as the r .../math>, denoted $\cot (A)$, is defined as the reciprocal of the tangent of $A$. <cmath>\cot (A) = \frac{1}{\tan (x)} = \frac{\textrm{adj
7 KB (1,124 words) - 07:36, 29 September 2021
• ...meter $\overline{AB}$ is constructed inside the square, and the tangent to the semicircle from $C$ intersects side $\overline{AD}</ ...h> and [itex]C$ are externally tangent to each other, and internally tangent to circle $D$. Circles $B$ and $C$ are con
13 KB (1,953 words) - 10:01, 6 September 2021
• ...eometry)|tangent]] to the circle at $A$ and $\angle AOB = \theta$. If point $C$ lies on $\overline{OA}$ and <mat ...}\ \frac{1}{1+\sin\theta} \qquad \textbf {(E)}\ \frac{\sin \theta}{\cos^2 \theta}[/itex]
13 KB (1,948 words) - 13:08, 19 February 2020
• ...h>9[/itex], respectively. The equation of a common external [[tangent line|tangent]] to the circles can be written in the form $y=mx+b$ with $...ath> and the x-axis, so [itex]m=\tan{2\theta}=\frac{2\tan\theta}{1-\tan^2{\theta}}=\frac{120}{119}$. We also know that $L_1$ and $L_2< 2 KB (251 words) - 11:55, 28 April 2019 • ...rnally [[tangent (geometry)|tangent]] to [itex] w_2$ and internally tangent to $w_1.$ Given that $m^2=\frac pq,$ where $...etween their centers is [itex]r_1 + r_2$, and if they are internally tangent, it is $|r_1 - r_2|$. So we have
12 KB (2,000 words) - 13:17, 28 December 2020
• Let $\angle CAD = \angle BAE = \theta$. Note by Law of Sines on $\triangle BEA$ we have <cmath>\frac{BE}{\sin{\theta}} = \frac{AE}{\sin{B}} = \frac{AB}{\sin{\angle BEA}}</cmath>
12 KB (1,929 words) - 23:42, 24 December 2020
• ...The circle of radius $9$ has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord. ...ed by faces $OAB$ and $OBC.$ Given that $\cos \theta=m+\sqrt{n},$ where $m_{}$ and $n_{}$ are intege
6 KB (1,000 words) - 07:37, 7 September 2018
• ...ersection. By the problem condition, however, the circle $P$ is tangent to $BC$ at point $N$. ...Moreover, since $AD$ is the only chord, $BC$ must be tangent to the circle $P$.
14 KB (2,516 words) - 22:37, 19 December 2020
• ...frac{\tan\alpha -\tan\beta}{1+\tan\alpha \tan\beta}[/itex]. Therefore, the tangent of the acute angle between the medians from A and B will be $\frac{2-1 By the tangent subtraction identity, [itex]\frac{2r-\frac{r}{2}}{1+2r \cdot \frac{r}{2}}=\ 8 KB (1,319 words) - 13:00, 6 September 2020 • ...1}{10}$. Now to find $\tan{\theta}$, we find $\cos{2\theta}$ using the Pythagorean Identity, and then use the tangent double angle identity. Thus, $\tan{\theta} = 10-3\sqrt{11}$. Substituting into the original sum,
3 KB (537 words) - 17:49, 2 September 2020
• ...[/itex], and then we found $AP$, the segment $OB$ is tangent to the circles with diameters $AO,CO$. ...a} = 4\cos^3{\theta} - 3\cos{\theta}[/itex], and since we have $\cos{\theta} = \frac {4}{5}$, we can solve for $a$. The rest then foll
8 KB (1,270 words) - 23:25, 30 July 2021
• ...have lengths $AB=13, BC=14,$ and $CA=15,$ and the [[tangent]] of angle $PAB$ is $m/n,$ where $m_{}$ an real theta = 29.66115; /* arctan(168/295) to five decimal places .. don't know other w
6 KB (978 words) - 22:31, 28 May 2021
• ...The x-axis and the line $y = mx$, where $m > 0$, are tangent to both circles. It is given that $m$ can be written in the form ...me positive reals $a$ and $b$. These two circles are tangent to the $x$-axis, so the radii of the circles are $a$
5 KB (834 words) - 22:22, 8 July 2021
• ...es that $e^{i\theta}=\cos(\theta)+i\sin(\theta)$ for all $\theta$. He also discovered the power series for the [[tangent function|arctangent]], which is
3 KB (500 words) - 21:28, 15 September 2008
• ...o the extension of [[leg]] $CB$, and the circles are externally tangent to each other. The length of the radius either circle can be expressed as ...s. As $\overline{AF}$ and $\overline{AD}$ are both [[tangent]]s to the circle, we see that $\overline{O_1A}$ is an [[angle bi
11 KB (1,750 words) - 18:39, 15 March 2021
• Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so t Now, let $\theta$ be the angle subtended by a diameter of the base of the cone at the
7 KB (1,214 words) - 18:49, 29 January 2018
• ...[/itex] is tangent to the circle at $A$ and $\angle AOB = \theta$. If point $C$ lies on $\overline{OA}$ and <mat label("$\theta$",(0.1,0.05),ENE);
6 KB (978 words) - 11:28, 3 August 2021
• label("$$\sin \theta = \frac{3}{5}$$",B-(.2,-.1),W); ...$CD$ tangent to $O_a$ $M$, and the point tangent to $O_b$ $N$. Since $\triangle CO_aM$ and
6 KB (951 words) - 16:31, 2 August 2019
• label("$$\theta$$",(7,.4)); Let $x = CA$. Then $\tan\theta = \tan(\angle BAF - \angle DAE)$, and since $\tan\angle BAF = \f 3 KB (513 words) - 14:35, 7 June 2018 • ...[itex]T$ lies on $\omega$ so that line $CT$ is tangent to $\omega$. Point $P$ is the foot of the perpendicul label("$$\theta$$",C + (-1.7,-0.2), NW);
7 KB (1,134 words) - 22:07, 5 October 2020

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