How do you calculate sin22.5 without a calculator?
Use the trigonometric identity:[math]cos(A+B)=cosA cosB - sinA sinB[/math]For [math]A=B=\theta[/math] we have[math]cos(2\theta) = cos^2\theta - sin^2\theta = (1- sin^2\theta) - sin^2\theta = 1 - 2sin^2\theta [/math]For [math]\theta = 22.5^\circ, [/math]we have [math]cos(2\theta) = cos(45^\circ) = \frac{1}{\sqrt2}[/math]Thus[math]\frac{1}{\sqrt2} = 1 - 2sin^2\theta[/math][math]\implies 2sin^2\theta = 1 - 1/\sqrt2[/math][math]\implies sin\theta =\sqrt{ \frac{1 - 1/\sqrt2}{2}}[/math]
How do I calculate 4^(2/5) without calculator?
Note that 4^(2/5) = 16^(1/5).My answer is based on simple observation:Arithmetic Mean >= Geometric Mean >= Harmonic Mean.and GM is approx (AM + HM)/2.Take five numbers: 2, 2, 2, 3/2 and 4/3.Arithmetic Mean = 1.76667Geometric Mean = 16^(1/5)Harmonic Mean = 1.71429Average of Arithmetic Mean and Geometric Mean = 1.74048So, approximate value of 16^(1/5) = 1.74048Actual Value of 16^(1/5) = 1.7411011265922So, the difference is around 0.00062. Reasonably good approximation.If you try with different number, you can get still better approximation.
How do I find the value of tan 10° without using a calculator?
Thanks for the A2A.I have learnt something very recently, which can be used to solve ur question.I m going to use Taylor’s series expansion. By this method, u won’t be using a calculator for finding tan10 (tan of 10 degrees), but u will need a calculator for multiplying those big decimal terms ( unless u r a superbrain!!). Ok, have a look at my solution.Suggestion: read Taylor series expansion and McLaurin series expansion once.Pls study the solution carefully to understand it fully!!
How do I calculate sin 38 without a calculator?
sin x = x - x^3/3! + x^5/5! -.... but first you would have to convert 38 degrees to radians and take sufficient terms to get an accurate answer Not sure if it would the calculation any easier but you could take the series about pi/4 = 45 degrees so would have sin[pi/4 -x ] where x= 7 pi/180 1/Sqrt[2] + (x-pi/4)/sqrt[2] -(x-pi/4)^2/2 sqrt[2]..etc these are the formulas , that are used to calculate the tables
How do I find the exact value of sin 20° without using calculator?
You should be able to find it using the trig identity: [math]\sin(3x)= 3\sin(x) - 4\sin^{3}(x)[/math](I assume this is derived from the identity: sin(x+y) = sin(x)cos(y) + cos(x)sin(y), but used twice. To be honest, I just looked it up.)Now that we know this, make x=20.[math]\sin(60)= 3\sin(20) - 4\sin^{3}(20)[/math]Then make two substitutions. [math]\sin(60)=\frac{\sqrt{3}}{2}[/math] and y = sin(20)[math]\frac{\sqrt{3}}{2} = 3y - 4y^{3}[/math]And then with some manipulation:[math]y^{3} - \frac{3}{4}y + \frac{\sqrt{3}}{8} = 0[/math]All that remains is to solve for y. Solving cubics by hand is a pain, but I will point you here: How can I solve an equation of the third degree? Then I will wave my hands around a bit and solve it here: Computational Knowledge EngineYou get 3 solutions. One negative (not correct) the other two are approximately .34 and .64.Which one is it? sin(30) =.5, and because we know the sine function is increasing up to 90 degrees the solution is approximately .34.So, what is the exact solution? According to Wolfram Alpha:This should yield a real number, but I’m not about to simplify that mess for you.Suffice to say, it can be done, but it’s unsurprisingly a huge headache.
Sin(x)=0.38 ...I can't solve for the answer on my calculator?
make sure your mode is in degrees.. then do second sin (or arcsin) of .38 and you should get 22.334
How do you solve a trig function without using a calculator?
You need to remember certain exact values of the trigonometric functions for the first quadrant. Try remember the values for sinx, and keep in mind that cosx = sin(90 - x) and tanx = sinx / cosx to calculate the values for those functions. You also need to know in which quadrants each trigonometric function is positive or negative, so you can calculate values for any quadrant. An easy way to remember this is the phrase "All Students Take Chemistry": - ALL the functions are positive in the first quadrant, x. - Only the SINE function is positive in the second quadrant, 180 - x, - Only the COSINE function is positive in the third quadrant, 180 + x, - Only the TANGENT function is positive in the fourth quadrant, 360 - x tan330° = tan(360 - 30)° tan330° = -tan30° tan330° = -1 / √3 csc(-405)° = 1 / sin(-405)° csc(-405)° = -1 / sin405° csc(-405)° = -1 / sin(360 + 45)° csc(-405)° = -1 / sin45° csc(-405)° = -1 / sin(45)° csc(-405)° = -1 / (1 / √2) csc(-405)° = -√2 sin(13π / 6) = sin(2π + π / 6) sin(13π / 6) = sin(π / 6) sin(13π / 6) = ½ sec(11π / 3) = 1 / cos(11π / 3) sec(11π / 3) = 1 / cos(2π + 5π / 3) sec(11π / 3) = 1 / cos(5π / 3) sec(11π / 3) = 1 / cos(2π - π / 3) sec(11π / 3) = 1 / cos(π / 3) sec(11π / 3) = 1 / ½ sec(11π / 3) = 2
How can I get the value of sin 23 degree without using calculator?
The answer is that by the rather vacuous arithmetic of the real numbers, the exact value of [math]\sin 23^\circ[/math] is, drum roll,[math]\sin 23^\circ[/math]I’m not trying to be a wiseguy. There’s not really a good way to write down a better exact value. I’m sure the other answers will tell you any number of ways to calculate an approximation. My point here is that the approximation is not the value. Trigonometry as practiced is exasperating in that we start with nice exact numbers and end up with approximations.[math]23^\circ[/math] is not a constructible angle since it is an integral angle not a multiple of three degrees. That means there’s no expression for the sine and cosine consisting of integers composed via addition, subtraction, multiplication and division and square rooting.[math]\sin 23^\circ=\cos 67^\circ[/math] is an algebraic number, the root of a high degree polynomial equation. For example it satisfies [math]\cos(90 \theta) = 0.[/math] The Chebyshev polynomial of the first kind [math]T_{90}(x)[/math] satisfies [math]\cos(90 \theta)=T_{90}(\cos \theta)[/math] so our equation with root [math]x=\cos 67^\circ [/math] is [math]T_{90}(x) = 0.[/math] Wolfram Alpha is happy to tell us [math]T_{90}(x)[/math]; here’s a taste of our equation:[math]618970019642690137449562112 x^{90} [/math][math]\ \ \ - 13926825441960528092615147520 x^{88} [/math][math] \ \ \ + 151454226681320743007189729280 x^{86} + … [/math][math] \ \ \ + 736290720 x^6 - 2732400 x^4 + 4050 x^2 - 1 = 0[/math][math]T_{90}(x) = 0 [/math] has 90 real roots, many (including [math]\cos 67^\circ)[/math] with no closed form consisting of integers combined with the usual arithmetic operations and square roots.Sometimes we see exact answers written as “Root of
How did some of the early portable calculators like TI-38 vintage 1974 calculate sine and cosine?
Here is an answer from TI (http://ftp://ftp.ti.com/pub/graph-ti/calc-apps/info/cordic.txt)Most practical algorithms in use for transcendental functionsare either polynomial approximations or the CORDIC method. TIcalculators have almost always used CORDIC, the exceptions beingthe CC-40, TI-74 and TI-95 which used polynomial approximations.In the PC world, the popular Intel math coprocessors like the8087 use CORDIC methods, while the Cyrix 83D87 uses polynomialmethods. There are pros and cons to both methods. References 1and 2 give some information on these devices but not down to thedetailed algorithm level.You can read more and get more links at a post I have written on this topic Not Taylor, but Cordic and at a previous Quora question How can we calculate sin and cos without a calculator using the CORDIC algorithm?
How does a computer calculate the sine, cosine, or tangent of a given angle?
The computer does not has mind to memorize the function sine or cosine or other trigonometric functions but it has series expression for any function of the angle, for example:Cos(x)=1-x^2/2!+x^4/4!+……And the error ~0 when number of terms ~ infinitySo, if you ask for cosine of angle =180, the computer will convert it to radian by multiply by pi =(22/7) then divide by 180 then sunsitiute in the series expression above then the computer will choose number of terms as much as possible until it gets an error ~ zeroAnd this is happening for all functine such as logarithmic functions or exponential functionsRegards