How do you find the exact value of $\mathrm{sin}(\frac{x}{2})$ and $\mathrm{cos}(\frac{x}{2})$ if $\mathrm{cos}x=-\frac{1}{4}$ and $\pi <x<2\pi $?

Patricia Bean
2022-07-21
Answered

How do you find the exact value of $\mathrm{sin}(\frac{x}{2})$ and $\mathrm{cos}(\frac{x}{2})$ if $\mathrm{cos}x=-\frac{1}{4}$ and $\pi <x<2\pi $?

You can still ask an expert for help

autarhie6i

Answered 2022-07-22
Author has **18** answers

Use the trig identity: $\mathrm{cos}x=1-2{\mathrm{sin}}^{2}(\frac{x}{2})$

$-\frac{1}{4}=1-2{\mathrm{sin}}^{2}(\frac{x}{2})\to {\mathrm{sin}}^{2}(\frac{x}{2})=\frac{5}{8}=0.625$

$\mathrm{sin}x/2=-0.79$

Next, use trig identity: ${\mathrm{cos}}^{2}(\frac{x}{2})-{\mathrm{sin}}^{2}(\frac{x}{2})=\mathrm{cos}x$

${\mathrm{cos}}^{2}(\frac{x}{2})=-\frac{1}{4}+\frac{5}{8}=\frac{3}{8}\to \ast \frac{\mathrm{cos}x}{2}=-0.61\ast $

Check

${\mathrm{sin}}^{2}(\frac{x}{2})+{\mathrm{cos}}^{2}(\frac{x}{2})=\frac{5}{8}+\frac{3}{8}=\frac{8}{8}=1$

$-\frac{1}{4}=1-2{\mathrm{sin}}^{2}(\frac{x}{2})\to {\mathrm{sin}}^{2}(\frac{x}{2})=\frac{5}{8}=0.625$

$\mathrm{sin}x/2=-0.79$

Next, use trig identity: ${\mathrm{cos}}^{2}(\frac{x}{2})-{\mathrm{sin}}^{2}(\frac{x}{2})=\mathrm{cos}x$

${\mathrm{cos}}^{2}(\frac{x}{2})=-\frac{1}{4}+\frac{5}{8}=\frac{3}{8}\to \ast \frac{\mathrm{cos}x}{2}=-0.61\ast $

Check

${\mathrm{sin}}^{2}(\frac{x}{2})+{\mathrm{cos}}^{2}(\frac{x}{2})=\frac{5}{8}+\frac{3}{8}=\frac{8}{8}=1$

asked 2020-11-09

When working with bivariate data, which of these are useful when deciding whether it’s appropriate to use a linear model?

I. the scatterplot

II. the residuals plot

III. the correlation coefficient

A) I only

B) II only

C) III only

D) I and II only

E) I, II, and III

I. the scatterplot

II. the residuals plot

III. the correlation coefficient

A) I only

B) II only

C) III only

D) I and II only

E) I, II, and III

asked 2022-04-01

We have,

asked 2022-07-26

Find the inverse and the contrapositive of the statement.

$n\to \sim m$

What is the inverse?

What is the contrapositive?

$n\to \sim m$

What is the inverse?

What is the contrapositive?

asked 2021-06-19

Use a graphing calculator to construct a 95% confidence interval for a sample of size 30 from a uniform distribution over the interval (0, 1). Take a class poll to determine the percentage of confidence intervals that contain the true mean. Discuss the result in class.

asked 2022-07-02

How do you write down 5 numbers where the mean, median, and mode are all 6?

asked 2021-05-12

Let ${X}_{1},{X}_{2},...,{X}_{6}$ be an i.i.d. random sample where each ${X}_{i}$ is a continuous random variable with probability density function

$f(x)={e}^{-(x-0)},x>0$

Find the probability density function for${X}_{6}$ .

Find the probability density function for

asked 2022-01-28

How do you find the standard deviation of a set of numbers?