HW11sol - MAC1114 Homework 11 Due Tuesday March 29 cot u...

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Unformatted text preview: MAC1114 - Homework 11 Due Tuesday - March 29 cot u cot v − 1 (hint: use the sum identity for tangent) cot u + cot v 1 1− 1 1 − tan u tan v cot u cot v 1 cot(u + v ) = = tan u+tan v = = 1 cot u cot v 1 tan(u + v ) tan u + tan v cot u cot v 1−tan u tan v cot u + cot v 1. Show that cot(u + v ) = = cot u cot v − 1 cot v + cot u 2. Let f (x) = sin x. Verify the following identity used in calculus: (note the typo) f (x + h) − f (x) = sin x h cos h − 1 h + cos x sin h h f (x + h) − f (x) sin(x + h) − sin x 1 = = sin x cos h + sin h cos x − sin x h h h 1 1 = sin x cos h − sin x + sin h cos x h h sin h cos h − 1 + cos x = sin x h h 2 3. Let sin α = 1 and cos β = 5 . Evaluate each of the six trig functions at α + β , i.e., nd sin(α + β ), 3 cos(α + β ), etc. (hint: only use sum identities for sin and cos, get the rest of them in an easier way). Note thatsin(α + β ) = sin α cos β + cos α sin β , and cos(α + β ) = cos α cos β − sin α sin β , so we need to nd sin β and cos α, then just plug everything in. To get sin β , draw a triangle with angle √ with adjacent√side 2 and hypotenuse 5. Then by the β √ 21 Pythagorean theorem, the opposite side is 21, so sin β = 5 . Similarly, get that cos α = 38 . So: √√ √ 8 21 2 + 2 42 12 + = 35 35 15 √ √ √ √ 8 2 1 21 4 2 − 21 cos(α + β ) = cos α cos β − sin α sin β = − = 35 3 5 15 sin(α + β ) = sin α cos β + cos α sin β = Once you have sin(α + β ) and cos(α + β ), it's easy to get the rest: sin(α + β ) tan(α + β ) = = cos(α + β ) √ 2+2 42 15√ √ 4 2− 21 15 √ 2 + 2 42 √ =√ 4 2 − 21 √ √ 1 4 2 − 21 √ cot(α + β ) = = tan(α + β ) 2 + 2 42 1 15 √ sec(α + β ) = =√ cos(α + β ) 4 2 − 21 1 15 √ csc(α + β ) = = sin(α + β ) 2 + 2 42 It's ne if you rationalized these, but not necessary (denitely don't spend time doing that on a test). 1 ...
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