math problem i cant use calculator and i need to know as much information about it as possible so,icanlearn,it

Question

find the exact value of 4sin^3 60degrees+cos^4 30degrees sin^5 0degrees leave answer in fractional form. Do not use a calculator. problem 2 : decide if true or false a: cos138 degrees=cos-222degrees second part of problem 2 b: cos^2 68 degrees=1divided by csc^2 22degrees and for problem 2 a:,b: identify the name of these angles

Answer 1

1. sin 60 = cos 30 = √3 / 2 (one of those values you memorize), so 4 sin^3 60 = 4 * 3√3 / 8 = 3√3 / 2. If the second term is multiplied by sin^5 0, it's 0 since sin 0 = 0 and what we just got is the final answer.
(Careful of typo in previous responsder; lost a 3 in the copied answer.)

2a. By thinking around the unit circle, trig function (x) = trig function (x + multiples of 360). Since -222 + 360 = 138, the cosines of those angles are equal.

2b. Remember csc x = 1 / sin x, so you're looking at whether cos^2 68 = sin^2 22. Just as at the beginning, sin x = cos (90 - x), and still equal if you square both numbers.

Name of the angles? Fred? (Ah, supplementary, complementary, all that -- use the definitions in your textbook since these aren't really standardized. And note that it's complEmentary, not complImentary -- you're completing the angle, not saying nice things about it.)

Answer 2

1.
sin(60) = SQRT(3)/2
cos(30) = SQRT(3)/2
sin(0) = 0

Frist term 4sin^3(60) = 4 (SQRT(3)/2)^3 = 3SQRT(3)/2
Second term cos^4(30)sin^5(0) = 0

Answer is SQRT(3)/2

2.a.
Using the absolute value of the angle:
138 + 222 = 360 so yes the cosines are the same

Another way is to convert -122 to a positive angle and this is done by adding 360 to it and getting 138. So in this way of looking at it the angles are the same angle.

b. csc = 1/sin so 1/csc^2(22) = sin^2(22)
cos^(68) = sin^2(22) and yes these are the same since 68+22 is equal to 90

This was new to me. According to Dr. Math there are two words for angles that sum to 360: explementary angles. and conjugate. He says that explementary is probably better since conjugate has too many other meanings.

Two angles are said to be complmentary if they sum to 90 degrees so the angles in part b are complimentary.

Answer 3

sin 60 = sqrt(3)/2
sin^3 60 = [sqrt(3)/2]^3 = 3sqrt(3)/8
cos 30 = sqrt(3)/2
cos^4 60 = [sqrt(3)/2]^4 = 9/ 16
sin(0) = 0
sin^5(0) = 0
4 sin^3 (60) + cos^4(30)sin^5(0) =
4(3sqrt(3)/8) + 9/16(0)
(3/2)sqrt(3)

2)
a)
cos 138 = cos (360 - 138) = -cos 222 (it is negative because (360 - 138) is in 4th quadrant)
b)cos^2 (68) = sin^2(90 - 68) = sin^2 (22) = 1/csc^2 (22)

a) supplementary angles
b)complimentary angles

Answer 4

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Answer 5

Use the table in the following link to solve problem-1:

http://www.easycalculation.com/area/trigonometry-tables.php

Use the identites in the following page to solve problem-2:

http://www.sosmath.com/trig/Trig5/trig5/trig5.html