Error Propagation Sine
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constant size. Indeterminate errors have unpredictable size and sign, with equal likelihood of being + or -. RULES FOR ELEMENTARY OPERATIONS (DETERMINATE ERRORS) SUM RULE: uncertainty subtraction When R = A + B then ΔR = ΔA + ΔB error propagation sin DIFFERENCE RULE: When R = A - B then ΔR = ΔA - ΔB PRODUCT RULE: When R error propagation exponential = AB then (ΔR)/R = (ΔA)/A + (ΔB)/B QUOTIENT RULE: When R = A/B then (ΔR)/R = (ΔA)/A + (ΔB)/B POWER RULE: When R = An then (ΔR)/R = n(ΔA)/A
Error Propagation Trig Functions
or (ΔR) = n An-1(ΔA) Memory clues: When quantities are added (or subtracted) their absolute errors add (or subtract). But when quantities are multiplied (or divided), their relative fractional errors add (or subtract). These rules will be freely used, when appropriate. We can also collect and tabulate the results for commonly used elementary functions. Note: Where Δt appears, it uncertainty of sine must be expressed in radians. RULES FOR ELEMENTARY FUNCTIONS (DETERMINATE ERRORS) EQUATION ERROR EQUATION R = sin q ΔR = (dq) cos q R = cos q ΔR = -(dq) sin q R = tan q ΔR = (dq) sec2 q R = ex ΔR = (Δx) ex R = e-x ΔR = -(Δx) e-x R = ln(x) ΔR = (Δx)/x Any measures of error may be converted to relative (fractional) form by using the definition of relative error. The fractional error in x is: fx = (ΔR)x)/x where (ΔR)x is the absolute ereror in x. Therefore xfx = (ΔR)x. The rules for indeterminate errors are simpler. RULES FOR ELEMENTARY OPERATIONS (INDETERMINATE ERRORS) SUM OR DIFFERENCE: When R = A + B then ΔR = ΔA + ΔB PRODUCT OR QUOTIENT: When R = AB then (ΔR)/R = (ΔA)/A + (ΔB)/B POWER RULE: When R = An then (ΔR)/R = n(ΔA)/A or (ΔR) = n An-1(ΔA) The indeterminate error rules for elementary functions are the same as those for determinate errors except that the error term
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Sine Cosine Error Metrology
Mathematics Questions Tags Users Badges Unanswered Ask Question _ Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in https://www.lhup.edu/~dsimanek/scenario/errorman/rules.htm related fields. Join them; it only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Calculate uncertainty of sine function result up vote 1 down vote favorite 1 I have an angle given in degrees: $$\theta_{\min} = 63^{\circ} http://math.stackexchange.com/questions/1045076/calculate-uncertainty-of-sine-function-result \pm 0.5^{\circ}$$ I need to calculate it's sine and still know the uncertainty of the value: $$n = 2\sin(\theta_{\min}) = 1.7820130483767356 \pm ???$$ How do I calculate the value represented by ???? Edit: I cheated and had a look in my friends work. This is how he did it: $$u_C=\sqrt{\left(\dfrac{\partial n}{\partial \theta_\min}u_C(\theta_\min)\right)^2}=\sqrt{\left(2\cos63^\circ\cdot\dfrac{0.5^\circ}{\sqrt{12}}\right)^2}=\sqrt{(0.908\cdot0.144)^2}=0.131$$ But I don't seem to understand that, though I encountered similar thing before. trigonometry error-propagation share|cite|improve this question edited Nov 30 '14 at 15:22 Mathematician171 2,813829 asked Nov 30 '14 at 14:59 Tomáš Zato 184212 add a comment| 3 Answers 3 active oldest votes up vote 2 down vote accepted Let's write your stuff in a cleaner way: $$n_\text{avg} = 2\sin(63°) = 1.7820130483767356$$ $$n = n_\text{avg} \pm^{u}_l \ .$$ Then $$u = 2\sin(63.5°) - 2\sin(63°)$$ $$l = 2\sin(63°) - 2\sin(62.5°)$$ The way your friend does it is via first order Taylor approximation: $$\Delta n \approx \left.\frac{dn}{d\theta}\right|_{\theta=\theta_\text{min}} \cdot \Delta\theta$$ Your buddy uses the absolute value in a sloppy notation. Evaluate the derivative, use $|\Delta\theta| =
a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the http://physics.stackexchange.com/questions/94110/error-propagation-estimations-for-sine-and-cosine workings and policies of this site About Us Learn more about Stack Overflow the company Business Learn more about hiring developers or posting ads with us Physics Questions Tags Users Badges Unanswered Ask Question _ Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. Join them; it only takes error propagation a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Error propagation estimations for sine and cosine up vote 0 down vote favorite My lab manual gives this: $B$ is a function of $A$, Greek are uncertainties... $$B + \beta = error propagation sin \sin(A + \alpha) = \sin(A)\cdot\cos(\alpha) + \sin(\alpha)\cdot\cos(A)$$ --> because $\alpha$ is taken to be (at least relatively) small, $\cos(\alpha) \to 1$, and $\sin(\alpha) \to \alpha$, measured in radians. I see that $\cos(\alpha) \to 1$, but I would have expected that $\sin(\alpha)$ for a small would, by the same logic, go to 0. Has it got something to do with the rate of change of either function near zero? Why is $\cos(\alpha)$ of small $\alpha$ not also proportional or written by relation to $\alpha$? experimental-technique error-analysis calculus share|cite|improve this question edited Jan 17 '14 at 17:15 Kyle Kanos 18.8k103874 asked Jan 17 '14 at 16:22 user37464 684 1 Have a look at Taylor series. Normally you have to keep at least the linear term to have consistent results. –DarioP Jan 17 '14 at 16:33 add a comment| 3 Answers 3 active oldest votes up vote 1 down vote accepted The easiest way is to plot $x$, $\sin(x)$, and $\cos(x)$. Fortunately, Wikipedia has done that for us: From the first
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