# public Khan /khan-exercises

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Fix some typos and make sure that there are no divs captured by the s…

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commit af2d1ae4854858248935c867b9061112f6ddce03 1 parent 2166c4e
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2  exercises/corresponding_angles_2.html
 @@ -46,7 +46,7 @@ 46 46   47 47   48 48 
49 -i

Corresponding angles are equal to one another. Watch this video to understand why.

49 +

Corresponding angles are equal to one another. Watch this video to understand why.

50 50 
51 51 

The \color{BLUE}{\text{blue angle}} and the \color{ORANGE}{\text{orange angle}} are corresponding angles. Therefore, we can set them equal to one another.

52 52 

\color{BLUE}{Ax + B} = \color{ORANGE}{Cx + D}

2  exercises/factoring_polynomials_2.html
 @@ -14,7 +14,7 @@ 14 14  randRangeNonZero( -10, 10 ) 15 15  randRangeNonZero( -10, 10 ) 16 16   17 - "" + (randRangeNonZero(-1, 1) * randRangeNonZero(2, 5)), 17 + "" + (randRangeNonZero(-1, 1) * randRangeNonZero(2, 5)) 18 18  randRange(0, 1) 19 19  1 20 20  SQUARE*A*B
2  exercises/multiplying_mixed_numbers_1.html
 @@ -37,7 +37,7 @@ 37 37   38 38   39 39 
40 -
Multiply, reduce to lowest terms, and turn into a mixed number: 40 +

Multiply, reduce to lowest terms, and turn into a mixed number:

41 41 

\displaystyleWHOLE_1\ fraction( NUM_1, DENOM_1, false, true ) \times WHOLE_2\ fraction( NUM_2, DENOM_2, false, true )

42 42 
43 43 

(WHOLE_3 + (NUM_3_SIMP/DENOM_3_SIMP))

38  exercises/pythagorean_identities.html
 @@ -178,7 +178,7 @@ 178 178 

So, IDENT = EQUIV

179 179   180 180 
181 - Plugging into our expression, we get 181 +

Plugging into our expression, we get

182 182 
183 183 

\qquad 184 184  (IDENT)(FUNC) @@ -195,10 +195,10 @@ 195 195 

196 196 
197 197 
198 - To make simplifying easier, let's put everything 198 +

To make simplifying easier, let's put everything 199 199  in terms of \sin and \cos.  200 - FUNC = FUNC_SIMP 201 - , so we can plug that in to get 200 + FUNC = FUNC_SIMP, 201 + so we can plug that in to get

202 202 
203 203 

\qquad 204 204  (EQUIV)(FUNC) @@ -306,7 +306,7 @@ 306 306  Pythagorean Theorem.

307 307 
308 308 
309 - Dividing both sides by \cos^2\theta, we get 309 +

Dividing both sides by \cos^2\theta, we get

310 310 

\qquad \dfrac{\sin^2\theta}{\cos^2\theta}  311 311  + \dfrac{\cos^2\theta}{\cos^2\theta}  312 312  = \dfrac{1}{\cos^2\theta}

@@ -317,7 +317,7 @@ 317 317  = EQUIV

318 318 
319 319 
320 - Plugging into our expression, we get 320 +

Plugging into our expression, we get

321 321 
322 322 

\qquad 323 323  (IDENT)(FUNC) @@ -335,12 +335,12 @@ 335 335 

336 336 
337 337 
338 - To make simplifying easier, let's put everything 338 +

To make simplifying easier, let's put everything 339 339  in terms of \sin and \cos. 340 340  We know EQUIV  341 341  = EQUIV_SIMP 342 - and FUNC = FUNC_SIMP 343 - , so we can substitute to get 342 + and FUNC = FUNC_SIMP, 343 + so we can substitute to get

344 344 
345 345 

\qquad  346 346  \left(EQUIV\right) @@ -359,11 +359,11 @@ 359 359 

360 360 
361 361 
362 - To make simplifying easier, let's put everything 362 +

To make simplifying easier, let's put everything 363 363  in terms of \sin and \cos. 364 364  We know EQUIV  365 365  = EQUIV_SIMP, so we can substitute 366 - to get 366 + to get

367 367 
368 368 

\qquad 369 369  \left(EQUIV\right) @@ -475,7 +475,7 @@ 475 475  Pythagorean Theorem.

476 476 
477 477 
478 - Dividing both sides by \sin^2\theta, we get 478 +

Dividing both sides by \sin^2\theta, we get

479 479 

\qquad \dfrac{\sin^2\theta}{\sin^2\theta}  480 480  + \dfrac{\cos^2\theta}{\sin^2\theta}  481 481  = \dfrac{1}{\sin^2\theta}

@@ -486,7 +486,7 @@ 486 486  = EQUIV

487 487 
488 488 
489 - Plugging into our expression, we get 489 +

Plugging into our expression, we get

490 490 
491 491 

\qquad 492 492  (IDENT)(FUNC) @@ -504,12 +504,12 @@ 504 504 

505 505 
506 506 
507 - To make simplifying easier, let's put everything 507 +

To make simplifying easier, let's put everything 508 508  in terms of \sin and \cos. 509 509  We know EQUIV  510 510  = EQUIV_SIMP 511 - and FUNC = FUNC_SIMP 512 - , so we can substitute to get 511 + and FUNC = FUNC_SIMP, 512 + so we can substitute to get

513 513 
514 514 

\qquad  515 515  \left(EQUIV\right) @@ -528,11 +528,11 @@ 528 528 

529 529 
530 530 
531 - To make simplifying easier, let's put everything 531 +

To make simplifying easier, let's put everything 532 532  in terms of \sin and \cos. 533 - We know EQUIV  533 + We know EQUIV 534 534  = EQUIV_SIMP, so we can substitute 535 - to get 535 + to get

536 536 
537 537 

\qquad 538 538  \left(EQUIV\right)

 @@ -107,7 +107,7 @@ 107 107 
108 108 

The key at the bottom of the pictograph shows that each symbol represents plural( VALUE_PER_IMG, "badge" ).

109 109 
110 - Find person( PERSON + 1 )'s row in the table: 110 +

Find person( PERSON + 1 )'s row in the table:

111 111 
112 112  person( PERSON + 1 )FULL_IMAGE 113 113 
 @@ -64,7 +64,7 @@ 64 64 
65 65 

The key at the bottom of the pictograph shows that each full symbol represents plural( VALUE_PER_IMG, "badge" ).

66 66 
67 - Find person( PERSON + 1 )'s row in the table: 67 +

Find person( PERSON + 1 )'s row in the table:

68 68 
69 69  person( PERSON + 1 )FULL_IMAGE 70 70  HALF_IMAGE @@ -119,7 +119,7 @@ 119 119   120 120 

121 121 
122 - Who has NUM_SYMBOLS plural( "symbol", VALUE / VALUE_PER_IMG ) next to his( PERSON + 1 ) name? 122 +

Who has NUM_SYMBOLS plural( "symbol", VALUE / VALUE_PER_IMG ) next to his( PERSON + 1 ) name?

123 123 
124 124  ???FULL_IMAGE 125 125  HALF_IMAGE @@ -151,7 +151,7 @@ 151 151 
152 152 

The key at the bottom of the pictograph shows that each full symbol represents plural( VALUE_PER_IMG, "badge" ).

153 153 
154 - Find person( PERSON1 + 1 )'s and person( PERSON2 + 1 )'s rows in the table: 154 +

Find person( PERSON1 + 1 )'s and person( PERSON2 + 1 )'s rows in the table:

155 155 
156 156  person( PERSON1 + 1 )FULL_IMAGE 157 157  HALF_IMAGE
 @@ -191,11 +191,11 @@ 191 191  though. 192 192 

193 193 
194 - To simplify this formula to something we can use, we try 194 +

To simplify this formula to something we can use, we try 195 195  the sine addition/subtraction identity: 196 196  \sin(x \pm y)  197 - = \sin x \cdot \cos y \pm \cos x \cdot \sin y 198 -
In this case, we have 197 + = \sin x \cdot \cos y \pm \cos x \cdot \sin y

198 +

In this case, we have

199 199 

\qquad \sin(T_ANG 200 200  OP S_ANG) = 201 201 
@@ -319,11 +319,11 @@ 319 319  though. 320 320 

321 321 
322 - To simplify this formula to something we can use, we try 322 +

To simplify this formula to something we can use, we try 323 323  the cosine addition/subtraction identity: 324 324  \cos(x \pm y)  325 - = \cos x \cdot \cos y \mp \sin x \cdot \sin y 326 -
In this case, we have 325 + = \cos x \cdot \cos y \mp \sin x \cdot \sin y

326 +

In this case, we have

327 327 

\qquad \cos(T_ANG  328 328  OP S_ANG) =
329 329  \qquad\qquad @@ -410,19 +410,19 @@ 410 410  are, though. 411 411 

412 412 
413 - To simplify this formula to something we can use, we try 413 +

To simplify this formula to something we can use, we try 414 414  the sine double-angle identity: 415 - \sin(2x) = 2 \sin (x) \cos (x) 416 -
In this case, we have 415 + \sin(2x) = 2 \sin (x) \cos (x)

416 +

In this case, we have

417 417 

\qquad \sin(2 \cdot T_ANG) = 418 418  2 \sin(T_ANG) 419 419  \cos(T_ANG) 420 420 

421 - (To cut down on the number of identities you have to 421 +

(To cut down on the number of identities you have to 422 422  memorize, you can derive this quickly from the  423 423  angle addition identity for sine) 424 424  [ 425 - Show how] 425 + Show how]

426 426 
427 427  Start with the sine angle addition identity: 428 428 

\qquad \sin(x + y) = \sin(x) \cdot \cos(y) @@ -511,19 +511,19 @@ 511 511  are, though. 512 512 

513 513 
514 - To simplify this formula to something we can use, we try 514 +

To simplify this formula to something we can use, we try 515 515  the cosine double-angle identity: 516 - \cos(2x) = \cos^2 (x) - \sin^2 (x) 517 -
In this case, we have 516 + \cos(2x) = \cos^2 (x) - \sin^2 (x)

517 +

In this case, we have

518 518 

\qquad \cos(2 \cdot T_ANG) = 519 519  \cos^2(T_ANG) - 520 520  \sin^2(T_ANG) 521 521 

522 - (To cut down on the number of identities you have to 522 +

(To cut down on the number of identities you have to 523 523  memorize, you can derive this quickly from the  524 524  angle addition identity for cosine) 525 525  [ 526 - Show how] 526 + Show how]

527 527 
528 528  Start with the cosine angle addition identity: 529 529 

\qquad \cos(x + y) = \cos(x) \cdot \cos(y)