# #84 5.05 Precalculus questions

*label*Mathematics

*timer*Asked: May 15th, 2015

*account_balance_wallet*$5

### Question Description

## Tutor Answer

? cos^{4}x - sin^{4}x = cos(2x) Factor the left side as the difference of two squares ? (cos²x - sin²x)(cos²x + cos²x) = cos(2x) Use the identity cos²q + sin²q = 1 to replace the second parentheses on left by 1. ? (cos²x - sin²x)(1) = cos(2x) ? cos²x - sin²x = cos(2x) Use the identity cos(2q) = cos²q - sin²q to replace the left side. Ö cos(2x) = cos(2x) ------------------------------------------------------------ For all the others you first have to know that sin^{-1}s means "the angle with the smallest absolute value whose sine is s" cos^{-1}c means "the angle with the smallest absolute value whose cosine is c" tan^{-1}t means "the angle with the smallest absolute value whose tangent is t" And in cases where there are two angles which are the smallest in absolute value, one positive and one negative, we always choose the positive angle. ----------------------------------------- How do you find cos(sin^{-1}()) First we look only at what is in the parentheses. We see this: sin^{-1}(). This means an angle whose sine is . The sine is negative in quadrants III and IV. An angle in quadrant IV will have a smaller absolute value if rotated clockwise as a negative angle than a positive angle in quadrant III, so sin^{-1}() is an angle in quadrant IV, so we draw a radius vector in quadrant IV to represent the angle sin^{-1}() Next we draw a perpendicular from the end of that radius vector directly to the x-axis, like this: Next we indicate the horizontal side of the resulting right triangle as x, the vertical side as y, and the radius vector or hypotenuse as r. Now that angle's sine is . We know that the sine of an angle is or , so we select the numerator of the fraction to be y or the opposite, and the denominator of the fraction to be r or the hypotenuse, which is the radius vector. Since the r or hypotenuse is ALWAYS POSITIVE, we must make the y-value negative, which is correct because y goes downward below the x-axis. So we have y = -5 and r = 13. We label these: Now we must find x by the Pythagorean theorem x² + y² = r² x² + (-5)² = (13)² x² + 25 = 169 x² = 169 - 25 x² = 144 ___ x = ±Ö144 x = ±12 and since x goes to the right we use the positive value and x = 12. So we label x as 12: Now go back to the original expression: cos(sin^{-1}()) We have drawn the angle sin^{-1}() and found the values of x the adjacent, y the opposite, and r the hypotenuse, and since , cos(sin^{-1}()) = =

*flag*Report DMCA

Brown University

1271 Tutors

California Institute of Technology

2131 Tutors

Carnegie Mellon University

982 Tutors

Columbia University

1256 Tutors

Dartmouth University

2113 Tutors

Emory University

2279 Tutors

Harvard University

599 Tutors

Massachusetts Institute of Technology

2319 Tutors

New York University

1645 Tutors

Notre Dam University

1911 Tutors

Oklahoma University

2122 Tutors

Pennsylvania State University

932 Tutors

Princeton University

1211 Tutors

Stanford University

983 Tutors

University of California

1282 Tutors

Oxford University

123 Tutors

Yale University

2325 Tutors