3 answers

How do you solve #2sin^2x = 1 + cosx# for #0° <= x <= 180°#?

Question:

How do you solve #2sin^2x = 1 + cosx# for #0° <= x <= 180°#?

Answers

# x=60^@, or, x= 180^@#.

Explanation:

#2sin^2x=1+cosx#.

# :. 2(1-cos^2x)=1+cosx#.

#:. 2(1+cosx)(1-cosx)-(1+cosx)=0#.

#:. (1+cosx){2(1-cosx)-1}=0#.

#:. (1+cosx)(1-2cosx)=0#.

#:.

Cosx=-1, or, cosx=1/2#.

# cosx=-1, 0^@<=x<=180^@"gives "x=180^@, and, #

# cosx=1/2" with "x" in the given range gives "x=60^@#.

.

The solutions are #x=60^@,180^@#.

Explanation:

You can use this altered Pythagorean identity:

#sin^2=1-cos^2x#

Now here's the actual problem. The strategy is to get everything in terms of #cosx#, then factor it like a quadratic. Here's what that looks like:

#2sin^2x=1+cosx#

#2(1-cos^2x)=1+cosx#

#2-2cos^2x=1+cosx#

#-2cos^2x+2=1+cosx#

#-2cos^2x-cosx+2=1#

#-2cos^2x-cosx+1=0#

#2cos^2x+cosx-1=0#

#2cos^2x+2cosx-cosx-1=0#

#color(red)(2cosx)(cosx+1)-cosx-1=0#

#color(red)(2cosx)(cosx+1)-color(blue)1(cosx+1)=0#

#(color(red)(2cosx)-color(blue)1)(cosx+1)=0#

#cosx=1/2,cosx=-1#

Here's a unit circle to remind us of some cosine values:

enter image source here

This means that:

#x=60^@,color(red)cancelcolor(black)(300^@),180^@#
#qquadqquad qquadqquad quad uarr#
#"not in the desired interval"#

Therefore, the solutions are #x=60^@,180^@#.

.

#x=60^@, or, x=180^@#.

Explanation:

# 2sin^2x=1+cosx#.

#:. (1-2sin^2x)+cosx=0#.

#:. cos2x+cosx=0#.

#:. 2cos((2x+x)/2)cos((2x-x)/2)=0#.

#:. 2cos(3/2x)cos(1/2x)=0#.

Since, #0^@<=x<=180^@, 0^@<=1/2x<=90^@, and, #

# 0^@<=3/2x <= 270^@#.

#:.

Cos(3/2x)=0 rArr 3/2x=90^@, 270^@#.

# rArr x=60^@, 180^@#.

# cos(1/2x)=0 rArr 1/2x=90^@, or, x=180^@#.

Altogether, #x=60^@, or, x=180^@#, as Before!

.

Similar Solved Questions

1 answer
Your organization currently has operations in Ontario, New Brunswick and Alberta. The organization is planning on...
Your organization currently has operations in Ontario, New Brunswick and Alberta. The organization is planning on expanding into the provinces of Manitoba and British Columbia within the next six months. The Manager of Finance has asked you for information on the health care funding methods used in ...
1 answer
An operating unit that is responsible for only costs is commonly referred to as an: Multiple...
An operating unit that is responsible for only costs is commonly referred to as an: Multiple Choice Ο Ο revenue center Ο profit center Ο...
1 answer
Describe the physical and Psychological effects of Marijuana usage.
Describe the physical and Psychological effects of Marijuana usage....
1 answer
20) 20) Suppose the distribution of of serum cholesterol values in undergraduate men is approximately normally...
20) 20) Suppose the distribution of of serum cholesterol values in undergraduate men is approximately normally distributed with mean 190 mg/dL and standard deviation 0 - 40 mg/dL. You take a simple random sample of n 49 individuals from this population. What is the probability of getting a sample me...
1 answer
A steam power plant equipped with an open feed water heater as well as reheat arrangement,...
A steam power plant equipped with an open feed water heater as well as reheat arrangement, as shown in figure 1, supplies 50 kg/s of steam at 80 bar 470°C to the H.P. turbine. For feed heating, a part of steam is extracted at 7 bar and the remainder of the steam is reheated to 350°C in a reh...
1 answer
Oncology d abdominal pain decision which ach aussion is most appropriate a provider to make when...
oncology d abdominal pain decision which ach aussion is most appropriate a provider to make when a pt har uncontrolled diaudea on severe neutropenia ar an expected side effect of treatment. a administer the next to cycle as planned b) delay the start of the next azce, 6 change the medication pro...
1 answer
Question 9 6 pts Coleman's tax rate is 35 percent. The current price of Colemans 12...
Question 9 6 pts Coleman's tax rate is 35 percent. The current price of Colemans 12 percent coupon price, semiannual payment, noncallable bonds with 15 years remaining to maturity is $ 1,153.72. Coleman does not use short-term interest-bearing debt on a permanent basis. New bonds would be privat...
1 answer
You are the new accounting manager at the Barry Transport Company. Your CFO has asked you...
You are the new accounting manager at the Barry Transport Company. Your CFO has asked you to provide input on the company's income tax position based on the following: Pretax accounting income was $60 million and taxable income was $16 million for the year ended December 31, 2021. The differenc...
1 answer
What was the goal of the National Origins Act of 1924?
What was the goal of the National Origins Act of 1924?...
1 answer
Record the appropriate journal entry to reflect the following: Veggies-R-Us, Inc uses the allowance method to...
Record the appropriate journal entry to reflect the following: Veggies-R-Us, Inc uses the allowance method to determine their bad debt expense. The allowance for bad debts is based upon the balance in accounts receivable. Based upon past experience, Veggies-R-Us, Inc. uses the rate of 0.75% (.0075) ...
1 answer
Help me please
1) The time between arrivals of small aircraft at a county airport is exponentially distributed with a mean of one hour.a) What is the probability that more than two aircraft arrive within an hour?b) If 10 separate one-hour intervals are chosen, what is the probability that no interval contains more...
1 answer
KO Saved HelpS [The following information applies to the questions displayed below. Adger Corporation is a...
KO Saved HelpS [The following information applies to the questions displayed below. Adger Corporation is a service company that measures its output based on the number of customers served. The company provided the following fixed and variable cost estimates that it uses for budgeting purposes and th...
2 answers
Hamlet by William Shakespeare in 1 paragraph
What sort of person is Osric? What message does he have for Hamlet? What seems to be the problem with his hat? What is the wager? (No one has been able to explain thisspeech in a way convincing to everyone.)...
1 answer
6. (10 points) The graph of a function fis given below. 6 2 0 2 6...
6. (10 points) The graph of a function fis given below. 6 2 0 2 6 a. For which values of x in the interval (-4,4) is f not continuous? Give the type of discontinuity (ump, infinite or removable) in each case. b. For which values of x in the interval (-4,4) is fnot differentiable? c. Give the value o...
1 answer
Determine the area under the standard normal curve that lies to the right of (a) z...
Determine the area under the standard normal curve that lies to the right of (a) z =-0.44, (b) z = 0.96, (c) z= 1.91, and (d) z= 1.15...
1 answer
Please explain/ show work Harris Fabrics computes its plantwide predetermined overhead rate annually on the basis...
Please explain/ show work Harris Fabrics computes its plantwide predetermined overhead rate annually on the basis of direct labor-hours. At the beginning of the year, it estimated that 34,000 direct labor-hours would be required for the period's estimated level of production. The company also e...
1 answer
Please ASAP! QUESTION 5 There exists a function f such that S2++1 L(f) (s) = s2-5+1...
please ASAP! QUESTION 5 There exists a function f such that S2++1 L(f) (s) = s2-5+1 True False...
1 answer
Mannisto Inc. uses the FIFO inventory cost flow assumption. In a year of rising costs and...
Mannisto Inc. uses the FIFO inventory cost flow assumption. In a year of rising costs and prices, the firm reported net income of $217,775 and average assets of $1,463,010. If Mannisto had used the LIFO cost flow assumption in the same year, its cost of goods sold would have been $39,290 more than u...