course Mth 164 I did the online problems while at home with 2 sick kids the past 2 days. I will submit those separately tonight. I did that so I could get the assignment done faster - won't make it a habit. Assignment #4Section 7.3 3. Suppose that f and g are two functions with the same domain. If f(x) = g(x) for every x in the domain, the equation is called an identity. Otherwise it is called a conditional equation. 9. Rewrite in terms of sine and cosine: tan(theta)*csc(theta)= Tan = y/x = sin(theta)/cos(theta) sin(theta)/cos(theta)*(1/sin(theta) = 1/cos(theta) 11. Multiply cos(theta)/1-sin(theta) by 1+sin(theta)/1+sin(theta) = Cos(theta)*(1+sin(theta))/(1-sin^2(theta)) = cos(theta)(1+sin(theta))/(cos(theta))*(cos(theta)) = 1+sin(theta)/cos(theta) 13. Rewrite over a common denominator: (sin(theta)+cos(theta)/cos(theta))+(cos(theta)-sin(theta)/sin(theta)): mult. Left by sin(theta)/sin(theta), mult. Rt. by cos(theta)/cos(theta)= sin^2(theta)+cos^2(theta)/sin(theta)cos(theta) = 1/sin(theta)cos(theta) 15. Multiply and simplify: (sin(theta)+cos(theta)+sin(theta)+cos(theta)-1)/sin(theta)cos(theta)= (Sin^2(theta)+2sin(theta)cos(theta)+cos^2(theta) -1)/sin(theta)cos(theta)= (sin^2(theta)+1-sin^2(theta)-1) +2sin(theta)cos(theta))/sin(theta)cos(theta) = 2 Establish each identity: 19. csc(theta)*cos(theta) = cot(theta) 1/(sin(theta))*cos(theta) = cos(theta)/sin(theta) = cot (theta) 21. 1 + tan^2(-theta) = sec^2(theta) 1+(-tan(theta)*(-tan(theta)) = 1+ tan^2(theta) = sec^2(theta) 23. cos(theta)*(tan(theta)+cot(theta)) = csc(theta) Cos(theta)*(sin(theta)/cos(theta))+(cos(theta)+sin(theta)) = sin(theta) + cos^2(theta)/sin(theta) = (sin(theta))/(sin(theta))*(sin(theta)/1+(1-sin^2(theta)/sin(theta) = 1/sin(theta) 27. (sec(theta)-1)(sec(theta)+1) = tan^2(theta) = sec^2(theta) = tan^2(theta) Tan^2(theta) +1-1 = tan^2(theta) tan^2(theta) = tan^2(theta) 33. (sin(theta)+cos(theta))^2 +(sin(theta)-cos(theta))^2 = 2 2sin^2(theta)+2cos^2(theta) = 2 Sin^2(theta)+cos^2(theta) = 1 Sin^2(theta)+cos^2(theta) = 1 = 2 45. (sec(theta)+csc(theta))+(sin(theta)+cos(theta)) = 2tan(theta) ((1/cos(theta))/(1/sin(theta))+(sin(theta))/(cos(theta)) = (sin(theta)/cos(theta))/(sin(theta)/cos(theta)) = 2(sin(theta)/cos(theta)) = 2tan(theta) 49. (1-sin v)/cos v + cos v/(1-sin v ) = 2 sec v (1-sin v)/cos v + (cos v/(1-sin v)*(cos v/cos v) = 2 sec v (1-sin v)^2 + cos^2/((1-sin v)(cos v)) = (1/cos v)*(1/cos v ) = 2/cos v = (1/cos v) +(1/cos v) = 2/cos v 51. (sin(theta))/(sin(theta)-cos(theta)) = 1/(1-cot(theta))= ****could not figure this one out:divide the whole term by 1/sin(theta) ? 1/sin(theta)-cos(theta) does not equal 1/(1-(cos(theta)/sin(theta)) 53. (1-sin(theta))/(1+(sin(theta)) = (sec(theta) tan(theta))^2 (1-2sin(theta) + sin^2(theta))/(1-sin^2(theta)) = (1-sin(theta))*(1-sin(theta))/(1+sin(theta)*(1-sin(theta)) = (1-sin(theta))/(1+sin(theta)) 57. tan(theta) + (cos(theta)+(1+sin(theta)) = sec(theta) (1+sin(theta)*sin(theta))/((1+sin(theta)*cos(theta))+(cos(theta)*cos(theta))/(1+sin(theta))*cos(theta) = 1/cos(theta) (1+sin(theta)*sin(theta) + cos^2(theta))/(1+sin(theta)*cos(theta)) = (Sin+sin^2(theta) +cos^2(theta))/(1+sin(theta)*(cos(theta)) = (1+sin(theta))/((1+sin(theta))cos(theta)) = 1/cos(theta) 63. (tan u cot u)/(tan u + cot u) + 1 = 2 sin^2 u 69. (sec(theta) csc(theta))/(sec(theta)*csc(theta)) = sin(theta) cos(theta) ((1/cos(theta))-(1/sin(theta)))/((1/cos(theta))*(1/sin(theta))) = (1/cos(theta))-(1/sin(theta))*(sin(theta)*(cos(theta)) = Sin(theta) cos(theta) = sin(theta) cos(theta) 75. (sec(theta))/(1-sin(theta)) = (1+sin(theta)/cos^3(theta)) = Sec(theta)/(1-sin(theta))*((1+sin(theta)/(1+sin(theta)) = (1+sin(theta))/(1+sin(theta))(1-sin(theta))(cos(theta)) = (1+sin(theta))/(cos^3(theta)) 81. (sin^3(theta)+cos^3(theta))/(sin(theta)+cos(theta)) = 1-sin(theta)cos(theta) = Sin^2(theta)+cos^2(theta) = 1***** 105. One thing that I do when trying to establish identities is to rewrite the identity using the letters to see if that gives me any insight. For example, I would represent sec(theta) as 1/x instead of 1/cos(theta), and cot(theta) as x/y. Sometimes this helps me to see more clearly the algebra that I need to do to establish the identity(but sometimes it doesnt !) Sec. 7.4 3.a) sin(pi/4)*cos(pi/3) = (sq. rt. of 2)/2*(1/2) = (sq. rt. of 2)/4 b) tan(pi/4)- sin(pi/6) = ½ Find the exact value of: 9. sin(5pi/12) = 75 deg. 225 deg. 150 deg. = sin(5pi/4-5pi/6) = (sq. rt. of 6)/4+(sq. rt. of 2)/4 11. cos(7pi/12) = cos(9pi/12 2pi/12) = (-sq. rt. of 2)/4 (sq. rt. of 6)/4 = -1/4)(sq. rt. of 2 + sq. rt. of 6) 15. tan 15 deg. = tan(21pi/12-20pi/12) = .2679 same as calculator gives in deg. mode = pi/12 = app. .2618 17. sin(17pi/12) = 3pi/4+2pi/3 = sin(a +B) = sin(3pi/4)cos(2pi/3)+cos(3pi/4)sin(2pi/3) = -.9659 21. sin20deg.cos10deg +.cos20deg.sindeg. = sin(12pi/6-11pi/6) = 0-(-1/2) = 1/2 23. cos70deg.cos20deg. sin70deg.sin20 = cos(a+B) = cos(70+20) = cos(90) = 0 27. sin(pi/12)cos(7pi/12) cos(pi/12)sin(7pi/12) = sin(15) sin(75) = Sin(a+B)+(cos(a+B)-cos(a+B) = 1 31. sin a = 3/5, 0 less than a is less than pi/2; cos B = (2(sq. rt. of 5)/5), -pi/2 is less than B is less than 0 cos a = 4/5 sin B = (sq. rt. of 5)/5 A) sin(a +B) = sin(a)cos(B)+cos(a)sin(B) = (6*(sq. rt. of 5)/25 + 4*(sq. rt. of 5)/25) = 2(sq. rt. of 5)/5 ( I got this which is the decimal equivalent, .894427191, of the radical terms added together. The decimal equivalent of the answer in the book, 2*(sq. rt. of 5)/25, is.1788854382. Either I did something wrong or 10(sq. rt. of 5)/25 should reduce to 2(sq. rt. of 5)/5. 33. A)sin(a+B) =sin a cosB + cos a sinB = (4/5)(1/2) + (-3/5)(sq. rt. of 3)/2) = 4-3*(sq. rt. 33. tan a = -4/3, pi/2 is less than a is less than pi; cos B = ½, 0 is less than B is less than pi/2 B) Cos(a+B) = cos a cosB sin a sin B = (-3/5)(1/2) (4/5)(sq. rt. of 3/2) = -3-4(sq. rt. of 3)/10 C) sin( a-B) sin a cos B cos a sin B = (4/5)(1/2)-(-3/5)((sq. rt. of 3)/ 2) = 4+3*(sq. rt. of 3)/10 = (4+3*(sq. rt. of 3))/10 D) tan(a-B) = (tan a tan B)/(1+tan a tan B) = (-4/3)-sq. rt. of 3)/(1+(-4/3)(sq. rt. of 3) = -2.34( same as( 25(sq. rt.of 3)+48)/39 given in book Use the figures to evaluate each function if f(x) = sin x, g(x)= cos x, and h(x) = tan x 39. f(a+B) sin(a+B) = sinacosB+cos a sinB sin a = 1, cos a = sq. of 3, tan y/x = (sq. rt. of 3)/3 cos B = 1 sin B = sq. rt. of 2(sq. rt.of 2)/3 Sin(1)(1/3)+(sq. rt. of 3)(2 *sq rt. of 2)/3) = 2.04632 Establish each identity: 45. sin(pi/2 + (theta)) = cos(theta) sin(pi/2+(theta)) = cos(theta) Sin(pi/2)cos(theta)+cos(pi/2)sin(theta) = (1)(cos(theta))+(0)(sin(theta)) = cos(theta) 51. tan(pi theta) = -tan tan(a B) = (tan a tanB)/1+tan a tanB = (tan pi theta)/(1+tan pi*(theta)) = -tan(theta)/1 = -tan(theta) 57. sin( a+B)/sin a cos B = 1 + cot a tan B = (sin a cos B)+(cos a sin B)/sin a cos B +sin a cosB = (sin a cos B)(sin a Cos B)+ (cos a/sin a )*(sin B/cos B) ? 63. cot( a + B) = (cot a cot B 1)/(cot B + cot a)= ((cos a/sin a)(cos B/sin B) (sin a sinB)/(sin a sin B))/((sin a/sin a)(cos B/sin B)+(cos a)/(sin a)(sin B/sin B) = (sin a cos B+ cos a sin B)/(sin a sin B) Establish the identity: 69. sin((theta)+ k pi) = (-1) ^k, sin(theta) k any integer (sin(theta))(cos 2pi)+cos(theta)(sin 2pi) = (sin(theta))(1) + (cos(theta))(0) = (-1)^2 = 1 (sin(theta)(-1)+cos(theta)(0) = (-1)^1 = -1 Works for 1, -1 73. sin[sin ^-1 (3/5) cos^-1(-4/5)] = sin(a B) = sin a cos B cos a sin B = (3/5)(-4/5) (4/5)(3/5) = -24/25 79. tan(sin^-1(3/5) + pi/6) = (tan a + tan B)/(1-tan a tanB) = ((5-(sq. rt. of 3)/3-(54/20(sq. rt. of 3) Write as an algebraic expression 83. cos(cos^-1 u +sin^-1 v) 85. sin(tan^-1 u sin^-1 ) -pi/2 is less than or equal to a is less than or equal to pi/2 -1 is less than or equal to u is less than or equal to 1 Tan a = sq. rt. of 1-(cos a/sin a) ^2 = sq. rt. of 1- (v/u)^2 Sin B = sq. rt. of 1- cos^2 v = sq.rt. of 1-v^2 Tan a tan B + sina sin B = (1/u)+ sq rt. of 1-(v/u)^2 sq. rt. of (v/u)^2)(sq. rt. of 1-v^2) " start