1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Alona [7]
3 years ago
12

Find all solutions of each equation on the interval 0 ≤ x < 2π.

Mathematics
2 answers:
Korvikt [17]3 years ago
8 0

Answer:

x = 0 or x = \pi.

Step-by-step explanation:

How are tangents and secants related to sines and cosines?

\displaystyle \tan{x} = \frac{\sin{x}}{\cos{x}}.

\displaystyle \sec{x} = \frac{1}{\cos{x}}.

Sticking to either cosine or sine might help simplify the calculation. By the Pythagorean Theorem, \sin^{2}{x} = 1 - \cos^{2}{x}. Therefore, for the square of tangents,

\displaystyle \tan^{2}{x} = \frac{\sin^{2}{x}}{\cos^{2}{x}} = \frac{1 - \cos^{2}{x}}{\cos^{2}{x}}.

This equation will thus become:

\displaystyle \frac{1 - \cos^{2}{x}}{\cos^{2}{x}} \cdot \frac{1}{\cos^{2}{x}} + \frac{2}{\cos^{2}{x}} - \frac{1 - \cos^{2}{x}}{\cos^{2}{x}} = 2.

To simplify the calculations, replace all \cos^{2}{x} with another variable. For example, let u = \cos^{2}{x}. Keep in mind that 0 \le \cos^{2}{x} \le 1 \implies 0 \le u \le 1.

\displaystyle \frac{1 - u}{u^{2}} + \frac{2}{u} - \frac{1 - u}{u} = 2.

\displaystyle \frac{(1 - u) + u - u \cdot (1- u)}{u^{2}} = 2.

Solve this equation for u:

\displaystyle \frac{u^{2} + 1}{u^{2}} = 2.

u^{2} + 1 = 2 u^{2}.

u^{2} = 1.

Given that 0 \le u \le 1, u = 1 is the only possible solution.

\cos^{2}{x} = 1,

x = k \pi, where k\in \mathbb{Z} (i.e., k is an integer.)

Given that 0 \le x < 2\pi,

0 \le k.

k = 0 or k = 1. Accordingly,

x = 0 or x = \pi.

earnstyle [38]3 years ago
6 0

Answer:

Step-by-step explanation:

You might be interested in
Solve for b.<br><br> Help pls thank u !!!!
Nadusha1986 [10]

Step-by-step explanation:

3b - 3.76 = 1.34

3b = 3.76+ 1.34

3b = 5.1

b= 5.1/3

= 1.7

5 0
3 years ago
PART A. A fruit juice recipe uses three ingredients. Two of the ingredients use a ratio of 2 ounces of pineapple juice to 5 ounc
Rashid [163]

Answer:

(B) The equation of line is 5 X = 2 Y.

Step-by-step explanation:

2 ounces of pineapple juice

5 ounces of fruit punch

Other is 4 ounces pineapple juice and 10 ounces fruit punch

(A) Graph is shown below:

(B) Let X is the pineapple juice and Y is the fruit punch

(2, 5)  and (4, 10)

So, the equation of line is

Y - 5 = \frac{10-5}{4-2}\times (x -2)\\\\Y-5=2.5 (X-2)\\\\Y - 5 = 2.5 X - 5 \\\\2.5 X = Y\\\\ 5 X = 2 Y

8 0
3 years ago
Which statement is true?
Rom4ik [11]

Answer:

  (b)  Both vertical and horizontal reflection

Step-by-step explanation:

The figure will be a horizontal reflection of itself about any vertical line through two of the smaller 6-pointed stars.

The figure will be a vertical reflection of itself about any horizontal line through two of the smaller 6-pointed stars.

  the pattern has both vertical and horizontal reflection

__

<em>Additional comment</em>

A pattern will have horizontal reflection if there exists a vertical line about which the pattern can be reflected to itself. That is, there exists one (or more) vertical lines of symmetry.

Similarly, the pattern will have vertical reflection if there is a horizontal line about which the pattern can be reflected to itself. Such a line is a horizontal line of symmetry.

3 0
3 years ago
mr. morris has a class of 14 students. he can spend $24 on each student to buy math supplies for the year. he first buys all the
Natali [406]
Hello!

First of all let's divide and find how much each calculator cost.

95.2/14=$6.80

Now we will subtract from 24.

24-6.8= 17.2

Mr. Morris has $17.2 left to spend on each student.

I hope this helps!
4 0
3 years ago
Read 2 more answers
ANSWER AND I WILL DO YOU A FAVOR!!!!
Sonja [21]
T=21+4.5X

21 is the original temp.

4.5 is the constant rate I got when I subtracted 75-21=54 then divided 54/12min=4.5.

75 =21 + 4.5(12)
3 0
3 years ago
Other questions:
  • What is the percentage of sugar in the syrup made of 10 kg of water and 5 kg of sugar?
    6·1 answer
  • The odds that you will pick a spade from a standard deck of 52 cards is 4:39. ToF
    10·2 answers
  • 15PTS PLEASE HELP ASAP!<br> (dont write random answers pls)<br> look at the pic attached:
    6·1 answer
  • Calculate the slope of a line that goes through the points: (5, 7) &amp; (2, 7). Leave your solution as a simplified (improper)
    8·1 answer
  • {(2,0), (4, -3), (6, 9), (-2,7), (4,9)}<br> What is the range?<br> um...? idk
    11·2 answers
  • 20POINTS Which exponential functions have been simplified correctly? Check all that apply. f (x) = 5 RootIndex 3 StartRoot 16 En
    7·2 answers
  • Solve each system by substitution. y = -4x - 18 y = -3x - 13​
    5·1 answer
  • Rewrite each of the following expressions without using absolute value. |c| if c&lt;0
    14·1 answer
  • When h is one-half and j is one-third, g is 4. if g varies jointly with h and j, what is the value of g when h is 2 and j is 3?
    10·1 answer
  • 3 positive integers have a mean of 5 and range of 10, what are the integers
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!