IM Alg2.6.13 Lesson: Amplitude and Midline

Match each equation to its graph. Be prepared to explain how you know which graph belongs with each equation.
Suppose a windmill has a radius of 1 meter and the center of the windmill is (0,0) on a coordinate grid.
Write a function describing the relationship between the height [math]h[/math] of [math]W[/math] and the angle of rotation [math]\theta[/math]. Explain your reasoning.[br]
Describe how your function and its graph would change if the windmill blade has length 3 meters.[br]
Describe how your function and its graph would change if the windmill blade has length 0.5 meter.[br]
Test your predictions using graphing technology.
[size=150]A windmill has radius 1 meter and its center is 8 meters off the ground. The point [math]W[/math] starts at the tip of a blade in the position farthest to the right and rotates counterclockwise. [br][br][/size]Write a function describing the relationship between the height [math]h[/math] of [math]W[/math], in meters, and the angle [math]\theta[/math] of rotation.
Graph your function using technology.
How does it compare to the graph where the center of windmill is at [math]\left(0,0\right)[/math]?[br]
What would the graph look like if the center of the windmill were 11 meters off the ground? Explain how you know.[br]
Here is the graph of a different function describing the relationship between the height y, in feet, of the tip of a blade and the angle of rotation θ made by the blade.
Describe the windmill.

IM Alg2.6.13 Practice: Amplitude and Midline

For each trigonometric function, indicate the amplitude and midline.
[math]y=2\sin\left(\theta\right)[/math]
[math]y=\cos\left(\theta\right)-5[/math]
[math]y=1.4\sin\left(\theta\right)+3.5[/math]
Here is a graph of the equation y=2sin(θ)-3. Indicate the midline on the graph.
Use the graph to find the amplitude of this sine equation.[br]
[size=150]Select [b]all [/b]trigonometric functions with an amplitude of 3.[/size]
The center of a windmill is 20 feet off the ground and the blades are 10 feet long.
Fill out the table showing the vertical position of [math]P[/math] after the windmill has rotated through the given angle.[br][br]Write an equation for the function [math]f[/math] that describes the relationship between the angle of rotation [math]\theta[/math] and the vertical position of the point [math]P[/math], [math]f\left(\theta\right)[/math], in feet.
[size=150]The measure of angle [math]\theta[/math], in radians, satisfies [math]\sin\left(\theta\right)<0[/math][/size][size=150]. If [math]\theta[/math] is between 0 and [math]2\pi[/math] what can you say about the measure of [math]\theta[/math]?[/size]
[size=150]Which rotations, with center [math]O[/math], take [math]P[/math] to [math]Q[/math]? Select [b]all[/b] that apply.[br][img]data:image/png;base64,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[/img][/size]
[size=150]The picture shows two points [math]P[/math] and [math]Q[/math] on the unit 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IJ/BhoTdLliw0cOBAS8Ji8ImobNmy1KxZM0sCVBGIwZeXqmptjxzpBD7yM23aNKGw9u7dKy0wbcHHEdUyn/379wsrJTJh4Hfo0KFJA3qyYVX5P3TokDCuqSp+lfFimXPgB6PQJUqUCLqG+3bnoWHDhmKRi93xBsZ3/PhxYRIr8Jrbv6tWrUqPPfaYtDyxChBWnpzOf6wnVIpNmzaRzAebaTC/KRNmwYIF4tw8vN/LhFPtFwuIsDZbdToq4l+yZInIO/KPz1NPPUUFCxYMumZ32aZMmSIe3iNHjlQqszVr1tD8+fOVpiFbJ8ZMFA58lQkLaz4w6iETxg6/McGP5SH0PizXHDhwIPRy1P/YY1+yZMmofty4yV19Oamrfrc3cqNbV9/IF47nzpMnD12/ft24FPNb265+zJyHeJAFHzbrMSWigx2zkKIQgx8qkcj/nXi3N1LXFfwrV65Q7ty5qV+/fkZWY34nLPjYedegQYOYAnLDA4NvXupOaXvkSFfwkbcJEyYIa04XL140JbyEBB/vKtD2VkxYm5JqnJ4YfHMCdFLbI0c6g//rr78KM9zYVWrGJST4GHCCcUtdHYNvrmac1PbIkc7gI3+wx4d9+1evXo0pwIQDf8eOHULbY4RWV8fgx64Zp7U9cqQ7+BjcwwaeUaNGxRRgwoGPkXzMfersGPzYteO0tkeOdAcfecS6FBwXF2spb0KBj+k+vNtjrllnx+BHrx03tD1y5AXwL126RGnTphUmuaJJMaHA79ChgzhDL5pAdLjH4EevBTe0PXLkBfCRzwEDBlD+/PmjCjFhwD9x4oTQ9ji1VHfH4EeuIbe0PXLkFfDPnTsn2jrW8kdyCQM+VjcVK1Yskhy0us7gR64Ot7Q9cuQV8JFXnMz7pz/9KaIgEwZ8nKM3evToiILQ6QaDH7423NT2yJGXwMeOPYxnYc1KOJcQ4BvWcFQbuwwnYCvXGPzwUnNT2yNHXgIf+S1fvjx17do1rDATAvzmzZtTnTp1wgpAx4sMfvJacVvbI0deA3/cuHHisI5wp0r5HnzsN0aXB0cTe8Ux+Mlrym1t70XwjUG+cMdx+x78t956S8xr3rhxI3lr0vQKgx9cMTpoey+Cjzw/+eST4kyEYIkmwCm7NWrUoLZt24aWW+v/DH5w9eig7ZEjr3X1kef3339f9Hih4QOdrzX+sWPHRKGx995LjsH/vbZ00fbIkRfBx9JdbNzBST2BztfgDxs2LOYKpkBh6PKbwf+9JnTR9siRF8FHvp999lmqXLny70Iln3f1sYABZ5F5zTH4d2pMJ22PHHkV/A0bNoie78GDB5NQ8K3GP3r0qCgstuF6zTH4d2pMJ22PHHkVfOQ9V65cQRt3fAs+1iljb7IXHYNPpJu29zr4Tz/9tBjhN3jwLfhYtNOoUSOjnJ76ZvCJdNP2Xgd/+vTplClTpiQOfAs+1uaPHz8+qaBe+pHo4Ouo7b0OvjHDBdnC+RL8ffv2ifd7fHvRJTr4Omp7tCMvv+Mj/wUKFBDn8uG3L8GHpofG96pLZPB11fZoS14HHwvZjD0rvgQf7/Z4x/eqS2TwddX2fgB/zpw5Yvn67du3/anxMZof7fQR3R8IiQq+ztreD+DD2pSxR993Gt9oPJjH96pLVPB11vZ+AB9lePDBBwkrWn0HPk7ZyZcvn1eZF/lORPCNB/aqVau0rTuvv+NDsM899xw98sgj/gMfAxj169fXtvGYyVgigq+7tke9+QF8vALj3H3fafxKlSpR3759zfClrZ9EA98L2t4v4Bvr9o8cOULbtm3TjoEUsjkyzGRnzZqVZs6cKRtcK/+JBr4XtD0aiB80/pkzZ8QA3+rVq/0D/qeffpo0aqkVyZKZSSTwvaLt/QI+ypEhQwaxqtU3Gn/27NkCfDyZvewSCXyvaHs/gV+2bFnq3r27fzT+yy+/LE4V9TL0yHuigO8lbe8n8GE4tm7duv4BHyP6FStW9Dr3CQO+l7S9n8AfPny4mM/3TVcfB2s+88wzDL5GEnjppZfCGir1mrb3E/gLFy6klClT0tatWzVqKXeyYmlU//7776dRo0ZpVxjZDCVCV99r2h516IdRfZTDMK+FB4BuThp8wxru0qVLdSuLdH78Dr4XtT0q0S/gX716VQyCjxkzRrptqg4gDf7OnTtFYXTsvsgKy+/ge1Hb+wl8lAVHbg8aNEi2aSr3Lw3+xo0bBfgHDhxQnjnVCfgZfK9qe9S5XzQ+yoLDN3v16qW6KUvHLw3+mjVrBPhYwed152fwvart0ab8BH7hwoWpc+fO2qEiDf6iRYsE+OfPn9euMLIZ8iv4Xtb2fgO/VKlS1KZNG9mmqdy/NPjGqj0vGceMJEW/gu9lbe838KtWrUpNmjSJ1ARduy4NPqbxUqVK5VqG7UwY0y0ff/yxnVG6Fpcxj+91bQ8B4nCXDz/80DVZ2pnwo48+SjVr1rQzSlvikgZ/8ODBYqTSltRdjgTg+6WBGeB7XdujSQD8FStWuNw67Eke5rMrVKgQd2SnT58mfOxy0uD37t3b0yfrBgrOb+DDhiHOetP5dJ1A+Uf67SfwW7RoQSVLloxUVKnrHTp0EKszJ0+eHPdDQBp8HCmEc8P94PwGPs5IqF69uuerxk/gd+rUiYoUKWJbnQB+PNzxKVGiBFl9CKRYsGAByXwwWJEnTx6pMDLxO+kX4xX9+/f3RVmqVasmGkOfPn08X56xY8cSepZOtgVVaeF8/ezZs9taFuyVMeBPnTq1pYdAinTp0pHMBwN72HggE0ZXv2nTpqU0adL4oiyoEzQGXWUtky8/1QvAtJsXyMcAP/S7cePGFGimO1JXQ7qr3759e1u7LpEy5sR1v3T1jZH8Bx54wAmxKU/DT139v/3tb4RNbXY5QA17FoHAN2jQgObNm0eXL182nYw0+M8//zzlzZvXdAI6e/QL+BjJz58/f9htuTrLP1Le/AQ+zq4oWrRopKJKXQf02bJlE9BjwHDSpElSsAcmJg0+3omRuB+cH8A3tD3gL1OmjB+qxVfTeaiX0qVLx10vgB7rAQC7HdN60uAPHTqU0qdPH3dBdIjAD+Ab8/bGPL4Oco03D37S+Dh6q0qVKvGKxLJmj5SwNPh44uD9AgYBve6++uorwjZjrzpD22Pe3k/gnzp1irZs2eLVagnKN6zp1KtXL+iaXX82b94s3u0D4wOfONI7lpMG/9133xXgywwkxMqEW/e9vlbf0PaQn5/A99PuPJxNqcqiNLr/WLthsDhixAjxWmH8j8aVNPjLli0T4J87dy5avJ6452XwA7U9g69vcytevDhhJkyVM977oeXxEDD7/i8N/rp16wT4OILL687L4Adqe9QDa3w9WyOm8jClp8oB+EKFCokBd/QAzDpp8PFegXf83bt3m01DW39eBT9U20PADL6ezezee++lfv36KcscuvXgESs2ZZw0+Hv27BEJffbZZzLpaOnXq+CHansIl8HXsomJlZQwQKPCAXpM4WK6EFpfxkmDjyO3YBNs1qxZMulo6deL4IfT9hAug69fE8M4GLTx1KlTlWQOG3awRBeuYMGCpkbzjYxYAh8DFgMGDDDi8Oy3F8EPp+1RAQy+fs0QvWKAb2Z6TTb3oSP4+I9DP8w6S+DXr1+fGjZsaDYNbf15DfxI2h4CZvD1a2bTp08XI+12m9BCFx+gBw7mGdfwbcZZAh/r9R966CEz8Wvtx2vgR9L2EDKDr19Tw4BbuXLl/GM0E/ul0YX59ddf9ZO2RI68BH40bc/gS1S6g17RM8biHbs1vh1FsKTxFy9eLMA/fvy4HXlwLQ4vgR9N20OArPFda0YRE8aZ+rCi4xvwd+zYIcD3+tluXgE/lrZn8COy59qNmzdvCkZwHL1vwIf5LKxIev31110TrB0JewX8WNoesmCNb0eLsC+Ow4cPC/A3bdrkL/Br165NOEjQy84L4JvR9qgDBl+vlvjBBx8I8LF23lcav0ePHoQDHr3svAC+GW3P4OvXCl977TXCO/7333/vL/CxPRcHCd66dUs/qZvMke7gm9X2DL7JCnfQGwxp4Ex934EPaDClt2HDBgfFaW9SuoNvVttDKtzVt7dtxBtbpkyZaMaMGf4DH4LBIh5VGxDiFbyZ8DqDL6PtUVYG30yNO+Nn+/btQiliutt3Gh8i7NKli6ff83UGX0bbM/jOAG02FbzfG0dq+xL8RYsWiSfb9evXzcpEK3+6gi+r7SFU1vj6NC3MeLVr105kyJfg42w0vOevXbtWH6lL5ERX8GW1PYrM4EtUvEKvWMYO60xz584VqfgSfJTs4YcfphdeeEGhKNVFrSP4VrQ9JMTgq2snMjEbW3H/+c9/imC+Bb9nz5622P+WEa5dfnUE34q2hzwYfLtaRXzxwO4EzJUbzrfgr1ixQnT3f/zxR6OsnvnWDXyr2h4CZ/D1aHawZAtT8obzLfg//PCDAH/58uVGWT3zrRv4VrU9BM7gu9/srly5IlhYuHBhUmZ8Cz5KCDNBaLReczqBH4+2h9wZfPdb39tvvy1MYuMBYDhfgw8TvRjdv3TpklFeT3zrBH482h7CZvDdb3I4865169ZBGfE1+D///LMwpKnqNNEgSdr4Rxfw49X2EAmDb2PDsBAV7DBC+a1ZsyYotK/BR0lhJshru/V0AT9ebQ/5M/hBvDn+Z+TIkZQ3b95k6foe/I8++kg88WDi2CtOB/Dt0PaQN4PvbqvDkfN9+/ZNlgnfg48SY32ylzbt6AC+HdqewU/Gm6MXtm7dKpTeF198kSzdhAB/4MCBVKRIkWSF1/WC2+Dbpe0hX9b47rWybt26CVNW4XKQEODjLD4McGzcuDGcDLS75jb4dml7CJbBd695wTBmpPMnEwJ8iL5SpUrUuXNn92pBImU3wbdT26PIDL5ExdvoFQvXoOzOnDkTNtaEAX/atGlCEN9++21YQeh00U3w7dT2kCmD707LqlWrVpLhynA5SBjwUfh8+fJ5YseeW+Dbre0Z/HDIqb/2ySefCCX3+eefR0wsocB/8803hV1wrOPX2bkFvt3aHjJmje98S4OJLCxXj+YSCvxffvmFcuTIQcOGDYsmE9fvuQG+Cm0PQTL4zjanLVu2CG2/fv36qAknFPiQBFYyZcmShW7cuBFVMG7edAN8FdoeMmTwnW1JqMdHHnkkZqIJBz52KGXIkIFGjx4dUzhueXAafFXaHvJj8J1rRViog5H8lStXxkw04cA3GmPu3Lnp9u3bMQXkhgenwVel7Q1ZlylTxg0x2p4mznKEYVZdXZs2bcS0tZn8JST4Fy5coFSpUtHEiRPNyMhxP06Cr1LbQ3Cs8Z1pPocOHRLaHidMm3EJCT4E069fPypYsKAZGTnux0nwVWp7CI7Bd6b5PPvss+KAWbOpJSz4WNGUPn16LUf4nQJftbZn8M1iGJ8/mLzGu/38+fNNR6Qt+NeuXSOZD+x+79y5UyoMLItAYBgUkUlLtd9jx44RzB2pTuepp56iqlWr2prOxYsXKfDTu3dvKlmyZNA13FddNhXxf/PNN4RFMSrijidOLEfHvL1MHCdPnhR7V2TC2OE31pMpxccff0wyn6VLl9L7778vFQbxw85elSpVpMPJ5E3W77Jly8TTWzacjP8JEyaIhx7WNMiEi+V3wYIFBIvFxgfWWbEt2vhvfMeKR8f7sC2P/OuUt65du4p6nD59ulS+cAo1jGs4WRYzm+RSxHoyhN7HEwy78GQdrOpC6weeQCobh93+nejqq363N2TC7/iGJOz/xr4TTE0PGTJEOnJtu/qyJbEKPtLBeeMY6Lt165Zsskr8qwbfiXd7QzAMviEJ+787dOgQZCRDJgUGn0i8f2bNmpVefPFFGdkp86safKe0PQTE4KtpJliSi54qXgutOAb/rtSMbbt79+61Ikdbw6gE30ltD6Ew+LY2jaTIsCiqefPmSf9lfzD4ARLDGucnnngi4Io7P1WC76S2h/QYfPvb0JgxY4S2x9HZVh2DHyC5bdu2CYGOGzcu4KrzP1WB77S2h+QYfIDOkE0AAAtxSURBVHvbz65du0QbffXVV+OKmMEPEZ8xt7958+aQO879VQW+09oeEmPw7Ws3sHFfqlQpqlevXtyRMvhhRNikSRMxv3/9+vUwd9VfUgG+G9oekmLw7WsvWJabM2dOOnv2bNyRMvhhRIiVZfnz5yfsdnLDqQDfDW0P2TH49rSgGTNmiC6+mS23ZlJk8CNIyZguwQo3p53d4Lul7SE3Bj/+1rN7924BvZWFOpFSZ/AjSebuaT2YK4VFEied3eC7pe0hMwY/vpbz22+/CaMYds82Mfgx6gUbWYoVK+boUV12gu+mtmfwYzQuE7exqhSGMdAm7HQMfgxpnj9/nu677z5q27ZtDJ/23bYTfDe1PSTCGt96u5g5c6bo4mNzkN2OwTch0bVr14oKQCN2wtkFvtvaHrJi8K21GNizx2vmoEGDrEUQIxSDH0NAxu1Zs2aJisDZ/KqdXeC7re0hJwZfvrVgTClt2rSE6TtVjsGXkCwMEOIpPGfOHIlQ8l7tAF8HbY+SM/hy9X/kyBHKlSsXNW3aVC6gpG8GX1Jg6HoBfrvmU8Mlbwf4Omh7lI3BD1fD4a9hf33RokXp0UcfJYzmq3QMvgXp4tSTlClTimOYLASPGSRe8HXR9igogx+zuoWH//znP+IkqNKlS5MTJt4YfHP1kswXtkRimuXgwYPJ7sV7IV7wddH2kAODb641wN4djijDuX5OOAY/Dik//vjjVKRIkYg2yK1GHQ/4Oml7lJ/Bj90KsDQ8U6ZMhBV6TjkGPw5JX758mcqVK0d//vOf6csvv4wjpuCg8YCvk7ZHqRj84LoN/deuXTsxZgTT1k46Bj9OaWOBT7Vq1cQiHxzvbYezCr5u2h6yYPDDt4j//ve/1LBhQ/rDH/5AsSzbho8hvqsMfnzyE6Fv3rxJjRo1onvuuYfWrVsXd4xWwddN20MQDH7y5gDooCzy5csnbEEk96H+CoNvo4zbt28vum04Wz4eZwV8HbU9ZMDgB7eE48ePU/HixYWRkXiOzgqOVf4fgy8vs6gh+vTpI+CfOnVqVH/RbloBX0dtjzIy+L/XNB7OefLkoRo1aoiTnX+/4/wvBl+BzF955RUBP76tOFnwddX2KDuDf6cFfPjhh8JWI3Z76mC/gcG3QqaJMND4WOGHHoCskwVfV22PcjP4RPPmzRNtAQYwdHEMvsKawLt+6tSpqWbNmiTzPicDvs7ansEn6tmzp4AeZtl1cgy+4to4dOgQVa5cmTJmzGjaPp8M+Dpr+0QGH9abYYwVtu1gzFU3x+A7VCPdu3c3/eQ3C77u2j5RwYdFXWyrxSDeiRMnHGphcskw+HLyisu32QZhFnzdtT2ElWjv+DIP+LgaU5yBGfw4BSgbPLALGGm+3wz4XtD2kE2igI/NWnilw5r7RYsWyTYLx/0z+I6L/E6CxqBPr1696OrVq0G5MAO+F7Q9CpUI4MPgqpVB3KBKd/gPg++wwAOTwzRP7ty5xeedd95JuhULfK9oexTIz+DD1iIMrWLaVhcT60mNKMYPBj+GgFTfvnbtGkHro/Hg7PQ9e/aIo5Sjmev2iraH7PwIPnpooXWmup3YHT+Db7dELca3ffv2JO2BE34i7fTzkraHKPwG/uDBg5N6aarPXrTYlEwFY/BNick5T3hfzJw5M+XNm5dCB/9wJpuXtD2k5hfw0RPDQiz0zKDt0VPzsmPwNaw9dPNhsReNDGu7YQu9UKFC4j+ujRw5UsNch8+S18HHwxbv75A7FuTAeKUfHIOvYS0ag3sbNmygv/zlL0nAo/Hhg22dXnFeBf/kyZNJy22xb/6tt96i7777jnbs2OEV0UfNJ4MfVTzu3DTAR+pffPFFMvAB/5kzZ9zJnGSqXgP/6NGjBHt1kDHOUwzcXs3gS1a+Be8pZMPgCX3gwAHZYFr6DwQf75KYIza0vfFdvXp1Wr58uZb5D8yUV8DHOzzsI0K+JUuWpNmzZwcWQ/xm8JOJxPYLDP7evUlC7d+/fxL4GPjDPn8c741GWr58eS03gRiZ1x18zMU3a9ZMyLJChQrJBlSNcuCbwQ+UhprfDH4A+BjJx8AS3i9xvp/hMAhoaKls2bLR888/T5s3bzZua/GtI/hff/21GCAtUaKEAB69JzMWaRl89U2Kwb8Lvpl5+7Nnz9Ibb7xBZcuWFQ25WLFiNGLECFuP/LZa5bqAf+XKFYLh01q1agkZ4Qisvn370q5du0wXjcE3LSrLHhn8u+DLztvDKAMOfbjvvvtEA8fcM+ys//TTT5YrI56AboMPc9OtWrUSJs9g9gy/cc2KY/CtSE0uDIO/dy+Z0fahYoWmR7ffGARMly6daPT4X69ePRo7dqyUlguNX/a/0+CjGw/NDus00OooN7Q8rkHrx+MY/HikZy4sg793r/QqPWj3rFmz0urVqwlWfvDB7wIFCohP69atk3oC8Ne4cWOaOHGi0tkQ1eBjWnP+/PnUqVMnKly4sAAd9g3q1q0rHnJ4ENjlGHy7JBk5noQHH+vAoa1WrVoVWUoBd4wDHU+fPh1w9c5PPAAAOvzA4UwAzE/jNSJHjhwiHdhkb9GiBU2fPp0w0n3hwoVk8Vi5YCf4169fp/3799OSJUvEQCYWMkFG6MLDtDRmO1QObjL4VlqAXJiEB/+xxx4jjDabddDe0U5xxWm/6BGEc5jDHjduHD355JPiIAnAhA8eClWrVhXxjhkzhlasWEFY4PLLL7+EiybsNSvgYx0DbMlNmTKFevToQbVr1xaWZI184Rv5QtzwB3NUTjgGX72UU9y4cYNkPmiQGKGVCaOrX5zggsa9ePFi0+WBieUhQ4ZE9I97iNNMmWEAFOfAT5gwQWhWgFewYEER3oAP3WrMe+P9GebD8BrRpUsX6t27t8gHHhToVeBhgryhJ4GZh+HDhxPWJXTr1o1gMLJp06YCbExXYuEMuulGGjlz5hSAw0IR9itA0+/bt08cXGKmHHb7wSIx9CjsjteN+E6dOkWfffaZ42WJ9ehIAUOCMh9A8t5770mFkYnfSb9YF469+TJpAhYMaEUKM3ToUAFUpPtmruO1A7sHsTUV6wcALQYM0ZOoWLGiABcPCLw24NUiffr0SRDj8EkcSwWY8+fPT5hDx+Ij9Grq1KkjNiO1bNlSTLGNHz9eQG4mT076wUrJuXPnRpSxk3mJNy2UBYe/xBuPTPiNGzfG4p4Svqsf7SCOcNIDcBjRj+RwD36cdnhIlC5d2ulklaTHXX0lYg2KlMEPWLkXJJkIf/COjwGuSA7v//DjtLPyju90Hs2mx+CblZR1fwy+JPh490R3P9yoNk6AxT1M7TntGHynJW4uPd6Wa05OjvoK3J0nkzC0OhbvTJ48mbZs2UIAHuMeuBZtxF8mDVm/DL6sxJzxz+A7I2epVKyCj0QwV493amNkHN/G/L1UJmzyzODbJEibo2HwbRaoHdHFA35g+tD4GF3v2LFj4GVHfzP4jorbdGIMvmlROefRLvCRY+Pdf9KkSc4VICAlBj9AGBr9ZPA1qgwjK3aCjzgxqIfpPCzdddox+E5L3Fx6DL45OTnqy27wHc18SGIMfohANPnL4GtSEYHZYPADpaHPb57HV18XPI8vOY+vvkqspcAa35rcVIdija9awhbiZ41vQWgOBGGNr17IrPFZ46tvZZIpMPiSArPgncFn8C00G7VBGHy18kXsDD6Dr76VSabA4EsKzIJ3Bp/Bt9Bs1AZh8NXKF7Ez+Ay++lYmmQKDLykwC94ZfAbfQrNRG4TBVytfxM7gM/jqW5lkCgy+pMAseGfwfQR+pUqVLDQB/YIw+OrrRBp89VniFFgCLAHVEmDwVUuY42cJaCgBBl/DSuEssQRUS4DBVy1hjp8loKEEGHwNK4WzxBJQLQEGX7WEOX6WgIYSYPA1rBTOEktAtQT+Dym5JnfrdEi/AAAAAElFTkSuQmCC[/img][br]Explain why the tangent of [math]P[/math] and [math]Q[/math] is 2.

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