Solve the trigonometric equation over the interval [0, 2) [tex] \large{ {cos}^{2} \theta + \sqrt{3} sin \theta cos \theta = 1}[/tex]Show your work, thanks!

Answer: = 0 , 60Step-by-step explanation:[tex]\cos^2\theta + \sqrt{3}\sin \theta \cos \theta = 1[/tex]Substract both the sides from 1.[tex]=> \cos^2\theta + \sqrt{3}\sin \theta \cos \theta - 1 = 1 - 1[/tex][tex]=> \cos^2\theta + \sqrt{3}\sin \theta \cos \theta - 1 = 0[/tex]Rearrange the terms in L.H.S. [tex]=> (\cos^2 \theta - 1 )+ \sqrt{3} \sin \theta \cos \theta = 0[/tex]Replace the terms in brackets with '-sin' by using the identity (sin + cos = 1).[tex]=> -\sin^2 \theta + \sqrt{3} \sin \theta \cos \theta = 0[/tex]Take 'sin' common from the whole expression in L.H.S.[tex]=> \sin \theta(- \sin \theta + \sqrt{3}\cos \theta) = 0[/tex]Now solve each part separately :-1) [tex]\sin \theta = 0[/tex][tex]=> \theta = \sin^{-1}0 = 0[/tex]2) [tex]-\sin \theta + \sqrt{3} \cos \theta = 0[/tex]Divide both the sides by 'cos'.[tex]=> \frac{-\sin \theta + \sqrt{3}\cos \theta }{\cos \theta} = \frac{0}{\cos \theta}[/tex][tex]=> \frac{-sin \theta}{\cos \theta} + \frac{\sqrt{3} \cos \theta}{\cos \theta} = 0[/tex][tex]=> -\tan \theta + \sqrt{3} = 0[/tex]Add both the sides with 'tan'.[tex]=> -\tan \theta + \sqrt{3} + \tan \theta = 0 + \tan \theta[/tex][tex]=> \tan \theta = \sqrt{3}[/tex][tex]=> \theta = \tan^{-1}\sqrt{3} = 60[/tex] = 0 , 60

Mathematics High School

Solve the trigonometric equation over the interval [0, 2π)
[tex] \large{ {cos}^{2} \theta + \sqrt{3} sin \theta cos \theta = 1}[/tex]
Show your work, thanks!

Answers

Answer 1

Answer:

θ = 0° , 60°

Step-by-step explanation:

[tex]\cos^2\theta + \sqrt{3}\sin \theta \cos \theta = 1[/tex]

Substract both the sides from 1.

[tex]=> \cos^2\theta + \sqrt{3}\sin \theta \cos \theta - 1 = 1 - 1[/tex]

[tex]=> \cos^2\theta + \sqrt{3}\sin \theta \cos \theta - 1 = 0[/tex]

Rearrange the terms in L.H.S.

[tex]=> (\cos^2 \theta - 1 )+ \sqrt{3} \sin \theta \cos \theta = 0[/tex]

Replace the terms in brackets with '-sin²θ' by using the identity (sin²θ + cos²θ = 1).

[tex]=> -\sin^2 \theta + \sqrt{3} \sin \theta \cos \theta = 0[/tex]

Take 'sinθ' common from the whole expression in L.H.S.

[tex]=> \sin \theta(- \sin \theta + \sqrt{3}\cos \theta) = 0[/tex]

Now solve each part separately :-

1) [tex]\sin \theta = 0[/tex]

[tex]=> \theta = \sin^{-1}0 = 0[/tex]

2) [tex]-\sin \theta + \sqrt{3} \cos \theta = 0[/tex]

Divide both the sides by 'cosθ'.

[tex]=> \frac{-\sin \theta + \sqrt{3}\cos \theta }{\cos \theta} = \frac{0}{\cos \theta}[/tex]

[tex]=> \frac{-sin \theta}{\cos \theta} + \frac{\sqrt{3} \cos \theta}{\cos \theta} = 0[/tex]

[tex]=> -\tan \theta + \sqrt{3} = 0[/tex]

Add both the sides with 'tanθ'.

[tex]=> -\tan \theta + \sqrt{3} + \tan \theta = 0 + \tan \theta[/tex]

[tex]=> \tan \theta = \sqrt{3}[/tex]

[tex]=> \theta = \tan^{-1}\sqrt{3} = 60[/tex]

∴ θ = 0° , 60°

Answer 2

Answer:

Solution given:

[tex] \large{ {cos}^{2} \theta + \sqrt{3} sin \theta cos \theta = 1}[/tex]

[tex] 1-sin^{2} \theta+\sqrt{3} sin \theta cos \theta = 1[/tex]

[tex] 1-1 +sin^{2}\theta=\sqrt{3} sin \theta cos \theta = 1[/tex]

[tex] Sin\theta(sin\theta -\sqrt{3}cos\theta)[/tex]

Either

[tex] sin\theta=\sqrt{3} cos \theta [/tex]

[tex] \sqrt{sin\theta}{cos \theta}=\sqrt{3} [/tex]

[tex] Tan \theta =\sqrt{3}[/tex]

[tex] Tan \theta =Tan 60,Tan(180+60),Tan (360+60)[/tex]

[tex] \theta =60°,240°,420°[/tex]

in terms of π is

[tex] \theta =⅓π,\frac{4}{3}π,[/tex]

[tex] Sin\theta=1[/tex]

[tex]Sin\theta=Sin0° ,Sin180°[/tex]

In terms of π is

[tex]\theta=0π,π[/tex]

[tex]\theta=0π,⅓π,π,\frac{4}{3}π,[/tex]

Related Questions

Dose anyone know how to solve this?

Answers

Answer:

Domain = [-5, -2) U (1, 3)

Range = (-3, 4]

Step-by-step explanation:

Domain is the x values

Domain = [-5, -2) U (1, 3)

Range is the y values

Range = (-3, 4]

I need help I’ll give brainlist answer.

Answers

Answer:

All beside D

Step-by-step explanation:

4x + 8y = 20 and -4x + 2y = -30

Answers

Step-by-step explanation:

Given :-

4x + 8y = 20 -4x + 2y = -30

Adding the equations :-

[tex]: \implies[/tex] 10y = -10

[tex]: \implies[/tex] y = -1

Put it (i) :-

[tex]: \implies[/tex] 4x - 8 = 20

[tex]: \implies[/tex] 4x = 20 + 8

[tex]: \implies[/tex] 4x = 28

[tex]: \implies[/tex] x = 28/4

[tex]: \implies[/tex] x = 7

What is the area of a circle with a radius of 3? Give your answer in terms of π.

Answers

Answer:

9π

Step-by-step explanation:

the area of a circle =

π×r² = π×3² = 9π

radius=r=3

Area

πr²π(3)²9π units ²

Mr. Carr is hiring a plumber to repair his sink. The cost of Plumber T is shown in the table.

A table titled Plumber T Costs is shown. It has three columns and two rows. Row one is labeled Hours, h. Row two is labeled Cost, c. One hour, one hundred twenty-eight dollars. Two hours, two hundred fifteen dollars. Three hours, three hundred two dollars.

Part A. What is the initial cost and hourly rate of Plumber T?

Part B. Write an equation to represent the function shown in the table.

Part C. Plumber W charges $95 per hour and an additional $32 for the parts needed. What is the difference in cost between Plumber W and Plumber T if the repair Mr. Carr needs takes 5 hours?

Enter the correct answers in the boxes.

A. Initial cost: $

Rate per hour: $

B. Equation:

C. Difference in cost: $

Answers

The hourly rate of plumber T is $87 per hour and have an initial cost of $41.

Linear equation

Linear equation is in the form:

y = mx + b

where y, x are variables, m is the rate of change and b is the initial amount.

From the table, using points (1, 128) and (2, 215):

[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1} (x-x_1)\\\\y-128=\frac{215-128}{2-1}(x-1)\\ \\y= 87x+41[/tex]

The hourly rate of plumber T is $87 per hour and have an initial cost of $41.

Find out more on Linear equation at: https://brainly.com/question/14323743

Answer:

Initial cost: $ 41

Rate per hour: $87

Equation: y=87x+41

Step-by-step explanation:

Sorry I do not know the difference in cost.

Item 3
A rectangle is 29 feet long and 7 feet long.

What is the area of this rectangle?

203 ft²

143 ft²

107 ft²

72 ft²

Answers

Answer: 203

Step-by-step explanation:

to find the area, you have to multiply the length of the sides together. 29x7 = 203

Find the volume of the given cylinder

Answers

Answer:

The answer is 32 I just know it

if you know how to do this pls help thankyou!

Answers

Answer : 49

I hope you understand sorry for the ugly writing

f , what is the truncation error for S3?

0.016
0.031
0.219
0.234

Answers

Answer: it’s B, 0.031

Step-by-step explanation: edge 2021

Answer by 11:59 today!!!

Answers

Answer:

x = 30

Step-by-step explanation:

These angles are all supplementary (they all add up to 180) so:

x + 3x + 2x = 180

6x = 180

x = 30

If Jayna's shuffler is working properly and randomizing the playlist after each song, what is the probability of playing a slow song?

ANSWER IS 33.33%

Answers

Answer:

1 song is a slow song, 2 songs are not

Step-by-step explanation:

that would give you 1/3 probability = 33.3333333333333%

Gulliver tossed a coin 100 times and got 43 heads. What percent of the tosses where tails?

Answers

Answer:

53 %

Step-by-step explanation:

you have the percentage of 43% for heads and you subtract 43 from one hundred and you get 57%

Answer: 57%

Dont ask how but it is 57%

Im just like dat

What is the next number in the pattern below? 6, 13, 21, 30, ____

Answers

Answer:

Step-by-step explanation:

6+7=13

13+8=21

21+9=30

30+10=40

I NEED HELPPPPPP ¶×¶ ¶×¶

Find the RATIO and the EXACT VALUE of the given Sec A.

Answers

Answer:

Step-by-step explanation:

take angle A as reference angle

using sec rule

sec A=hypotenuse/adjacent

sec A=13/5

sec A=2.6

A=68 degree

for ratio,

sec A=13/5

the answer is 13/5

expiation:

5. How many 1/2 cups can be filled with 500 milliliters of liquid?

Answers

Answer:

1/2 = 120ml

Step-by-step explanation:

500/120 = 4.16666666667 round to whole number equal 4 time

can y’all help me i’m taking a test

Answers

Answer:

Over time, the function is DECREASING. Cho ran at a faster rate during segment K.

Step-by-step explanation:

Decreasing during segment K

A box of Pop-tarts costs $3.00 for 8. How much per pop-tart?

Answers

it’s $0.375 per pop tart

What does this mean and what is the answer?

-3(-7)

Answers

Answer:

-3 * -7 = 21

Step-by-step explanation:

-3(-7) = -3 * -7

Multiply

Remember: negative * negative = positive

So -3*-7 = 21

If my answer is incorrect, pls correct me!

If you like my answer and explanation, mark me as brainliest!

-Chetan K

Cora charges $20 for each lawn she mows. However, she has to pay her father $12
each day she uses his lawn mower. How much does she earn if she mows 3 lawns in
one day?

Answers

Answer:

$48

Step-by-step explanation:

Cora earns $20 for each lawn

She gives her father $12 EACH DAY for his lawn mower

In a day, she mows 3 lawns which is worth= 20*3= $60

but since it is in a day, she gives her father $12 for the day from the $60

Therefore, her final earning is, $60-$12= $48

convert 8m 32 cm into centimeters

Answers

Answer:

832 cm one meter us equal to 100 cm so 8 multiply 100 plus 32 is equal to 832

help will give brainly

Answers

Answer:

Is it 72?

Step-by-step explanation:

The Volume is 12x12x12= 1,728

2x-y=10 y=-4x+2 substitution method

Answers

Answer:

the solution is (2, -6)

Step-by-step explanation:

Substitute the second equation into the first, replacing y in the first:

2x - (-4x+2) = 10

Simplifying, we get:

2x + 4x - 2 = 10, or:

6x = 12, which yields x = 2.

Substituting 2 for x in the second equation yields y = -4(2) + 2 = 0, or y = -6

Then the solution is (2, -6).

Help me find answers ASAP please

Answers

The answer to the question is 40 because it is

How do I find the angles

Answers

Answer:

if there smaller than 90 degrees then it's acute if the angle is bigger then 90 degrees it's obtuse

Step-by-step explanation:

How do I solve this?

Answers

Answer:

Step-by-step explanation:

C = √a^2+b^2

√7^2+24^2

√49+576

√625

C=25

The ratio of the measurement of an angle to its complement is 1: 2. Find the measurement of the angle and it’s complement.

Answers

Answer:

1.x=90 degrees

2.smaller angles =30

3.larger angles =(30×2)=60

Step-by-step explanation:

Complement angle=90 degrees

90/3 (because need to separate into 3 parts) =30 degrees

1part =30 degrees

1:2

(1×30):(2×30)

30:60

what does the following expression mean?
7!

A. 7+6+5+4+3+2+1
B. 7·6·5·4·3·2·1

Answers

Answer: B

Step-by-step explanation:

A factorial is multiplying the integer and all of the integers below it together.

The product of a whole number 'n' with every whole number until 1 is called the factorial. The correct option is B.

What is a factorial?

The product of a whole number 'n' with every whole number until 1 is called the factorial. The factorial of 4 is, for example, 43221, which equals 24.

The expression 7! means 7 factorial which is expanded as,

7! = 7 × 6 × 5 × 4 × 3 × 2 × 1

Hence, the correct option is B.

Learn more about Factorial:

https://brainly.com/question/16674303

#SPJ2

Which of these best describes the circumference?
A The square root of the area of a circle
B. The ratio of the radius of a circle to its diameter
C. The radius of a circle times 3.14
D. The diameter of a circle times 3.14

Answers

Answer:

Step-by-step explanation:

Since the formula of circumference is 2πr

Solve the problem and enter your solution for the variable below. Do not include units in your answer. The scale below shows the weight, in grams, of two identical blocks. How many grams does one of the blocks weigh?

Answers

Answer:

1) 100+10+10 = 120

2) 120 : 2 = 60

1 block — 60

The transportation department predicts that about 800,000 cars will be registered in a certain state by year 2012. Their license plates are made using and two letters followed by and three digits from 0 to 9. Will there be enough possible arrangements?

Answers

Answer:

There won't be enough arrangement

Step-by-step explanation:

Given

[tex]Letters = 2[/tex]

[tex]Digits= 3[/tex]

[tex]Cars = 800000[/tex]

Required

Determine if the possible arrangement will be enough

The 2 letters can be selected from any of the 26 alphabets without restriction.

So, the selection of letters is:

[tex]Letters = 26 * 26[/tex]

The 3 digits can be selected from any of the 10 digits without restriction.

So, the selection of digits is:

[tex]Digits = 10*10*10[/tex]

The total arrangement is:

[tex]Total = Letters * Digits[/tex]

[tex]Total = 26*26*10*10*10[/tex]

[tex]Total = 676000[/tex]

By comparison:

[tex]Total < Cars[/tex]

i.e.

[tex]676000 < 800000[/tex]

Hence, the arrangement will not be enough

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Solve the trigonometric equation over the interval [0, 2π) [tex] \large{ {cos}^{2} \theta + \sqrt{3} sin \theta cos \theta = 1}[/tex]Show your work, thanks!​

Answers

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Solve the trigonometric equation over the interval [0, 2π)
[tex] \large{ {cos}^{2} \theta + \sqrt{3} sin \theta cos \theta = 1}[/tex]
Show your work, thanks!