how can you tell how many solutions you'll get for a trigonometric equations? does it depend on the value of theta or the range?
Answers
Answer: Trigonometric equations are, as the name implies, equations that involve trigonometric functions. Similar in many ways to solving polynomial equations or rational equations, only specific values of the variable will be solutions, if there are solutions at all. Often we will solve a trigonometric equation over a specified interval. However, just as often, we will be asked to find all possible solutions, and as trigonometric functions are periodic, solutions are repeated within each period. In other words, trigonometric equations may have an infinite number of solutions. Additionally, like rational equations, the domain of the function must be considered before we assume that any solution is valid. The period of both the sine function and the cosine function is \displaystyle 2\pi2π. In other words, every \displaystyle 2\pi2π units, the y-values repeat. If we need to find all possible solutions, then we must add \displaystyle 2\pi k2πk, where \displaystyle kk is an integer, to the initial solution. Recall the rule that gives the format for stating all possible solutions for a function where the period is \displaystyle 2\pi :2π:
\displaystyle \sin \theta =\sin \left(\theta \pm 2k\pi \right)sinθ=sin(θ±2kπ)
There are similar rules for indicating all possible solutions for the other trigonometric functions. Solving trigonometric equations requires the same techniques as solving algebraic equations. We read the equation from left to right, horizontally, like a sentence. We look for known patterns, factor, find common denominators, and substitute certain expressions with a variable to make solving a more straightforward process. However, with trigonometric equations, we also have the advantage of using the identities we developed in the previous sections.
Related Questions
A researcher needs to ask a question “what impact does a new curriculum have on a student achievement?” Which of the following is the most appropriate method for answering the question?
A) observation
B)textual analysis
C)experiment
D) survey
Answers
Answer:
C) Experiment
Step-by-step explanation:
With a controlled experiment, a researcher can use a student batch that studied the old curriculum as a control group, and the new batch of students as the 'test' group. To measure student achievement, the researcher would need to either:
1. compare test results for the 2 groups
2. do a longitudinal study - follow the 2 groups of students across several years, and track their 'achievements' (university admissions, jobs, income, etc)
With the other 3 options, the results wouldn't necessarily be accurate or relevant.
The _______ of a right triangle are the two sides of a right triangle that intersect to form the right angle
Answers
Answer:
legs
Step-by-step explanation:
Answer:
The Legs of a right triangle are the two sides of a right triangle that intersect to form the right angle
Step-by-step explanation:
Right Angle Triangle-Two legs plus a hypotenuse make up a right triangle. The hypotenuse is the longest side of the right triangle and the side opposite the right angle, and the two legs meet at a 90° angle. Right triangles come in a variety of shapes, including 45°-45° right triangles and 30°-60° right triangles.
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Why do some people say pure mathematics is beautiful, and what are some examples of this beauty?
take the points, no answer.
Answers
Answer:
Because knowledge is beautyyyy
Step-by-step explanation:
Which symbol correctly compares these two fractions? 6/9 ? 2/3. A = B < C >
Answers
Answer:
A
Step-by-step explanation:
You know that you can simplify 6/9 because both the numerator (top number) and denominator (bottom number) can be divided by 3. Therefore, you divide them by 3 to get 2/3. 2/3 cannot be simplified anymore. 2/3 is equal to 2/3. Therefore, the answer is A.
If this has helped please mark as brainliest
Pls help me last question on math I got to finish the packet
Answers
The solutions to the linear function are;
1. The soccer team makes more money per car ($8) than the French club ($5).
2. The graph that demonstrates Arnold's journey is graph 3
3. The graph that demonstrates Francisco's journey is graph 1
4. The graph that demonstrates Celia's journey is graph 2
How do you solve for the linear function?
To determine which group makes the most money per car, we can calculate the slope of the linear function representing the amount earned per car washed for both the French club and the soccer team. The slope of a linear function is the change in the dependent variable (y) divided by the change in the independent variable (x).
Let's calculate the slope for the soccer team. We can choose any two points to calculate the slope. Let's use the points (3, 24) and (6, 48):
Slope (m) = (change in y) / (change in x) = (48 - 24) / (6 - 3) = 24 / 3 = 8
The soccer team earns $8 per car washed.
We can choose any two points to calculate the slope for the French club. Let's use the points (2, 10) and (4, 20):
(20 - 10) / (4 - 2) = 10 / 2 = 5
The French club earns $5 per car washed
The above answer is dependent on the questions below;
The french club and soccer team are washing cars to earn money. The amount earned, (y) dollars for washind (x) cars is a linear function. Which group makes the most money per car? explain
French club
Number of cars (x) Amount earned (y)
2 10
4 20
6 30
8 40
10 50
Soccer team
Number of cars (x) Amount earned (y)
3 24
6 48
9 72
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A city has a population of 400,000 people. Suppose that each year the population grows by 6.25%. What will the population be after 15 years?
Answers
P(t) = Initial Population * (1 + rate)^t
P(t) = 400,00*(1+0.0625)^t
P(t)= 400,00*(1+0.0625)^15
Pt=400
Solve the initial-value problem y' = (8 sin(x))/sin(y), y(0) = pi/2
Answers
Answer:
8cosx-cosy=8.
Step-by-step explanation:
dy/dx=8sinx/siny;
sinydy=8sinxdx;
-cosy= -8cosx+C;
according to the condition y(0)=pi/2:
-cos(pi/2)= -8cos0+C;
-8+C=0; ⇔ C=8, then
8cosx-cosy=8.
The solution to the given initial-value problem y' = (8 sin(x))/sin(y) with the given condition y(0) = π/2 is equal to y = -cos⁻¹(x).
To solve the initial-value problem y' = (8 sin(x))/sin(y) with the initial condition y(0) = π/2, separate variables and then integrate.
Separate variables.
y' = (8 sin(x))/sin(y). To separate variables, multiply both sides by sin(y) and divide by (8 sin(x)):
sin(y) dy = dx
Integrate both sides.
Now, integrate both sides of the equation with respect to their respective variables:
∫sin(y) dy = ∫dx
Integrating the left side:
∫sin(y) dy = -cos(y) + C₁ (where C₁ is the constant of integration)
Integrating the right side:
∫dx = x + C₂ (where C₂ is the constant of integration)
Apply initial condition.
Now, apply the initial condition y(0) = π/2 to find the values of C₁ and C₂:
When x = 0, y = π/2
cos(π/2) + C₁ = 0 (substitute y = π/2 into -cos(y) + C₁)
C₁ = 0 (since cos(π/2) = 0)
So, the equation becomes:
-cos(y) = x + C₂
Solve for y.
Now, isolate y on one side of the equation:
y =-cos⁻¹(x + C₂)
Find C₂ using the initial condition.
Now, use the initial condition y(0) = π/2 to find C₂:
When x = 0, y = π/2
π/2 = -cos⁻¹(0 + C₂)
cos⁻¹(C₂) = -π/2
C₂ = cos(-π/2) = 0
Final solution.
Finally, substitute C₂ = 0 into the equation y = -cos⁻¹(x + C₂) to get the final solution:
y = -cos⁻¹(x)
Therefore, the solution to the initial-value problem y' = (8 sin(x))/sin(y) with y(0) = π/2 is y = -cos⁻¹(x).
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Pierre inherited $226,500 from his uncle and decided to invest the money. He put part of the money in a money market account that earns 3.5% simple
interest. The remaining money was invested in a stock that returned 5% in the first year and a mutual fund that lost 3% in the first year. He invested
$10,000 more in the stock than in the mutual fund, and his net gain for 1 yr was $2740. Determine the amount invested in each account.
Answers
The value of amount invested in each account are,
x = 12125
y = 57187.5
z = 47187.5
Given that;
Pierre inherited $226,500 from his uncle and decided to invest the money.
And, He put part of the money in a money market account that earns 3.5% simple interest.
The remaining money was invested in a stock that returned 5% in the first year and a mutual fund that lost 3% in the first year.
He invested $10,000 more in the stock than in the mutual fund, and his net gain for 1 yr was $2740.
Hence, By given conditions we can formulate;
x + y + z = 226,500
y = z + 10,000
0.035x + 0.06y - 0.03z = 2740
Now, We can solve the expression we get;
x = 12125
y = 57187.5
z = 47187.5
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solve the equation 1/x+3/x=16
Answers
\( \frac{1}{x} + \frac{3}{x} = 16 \\ \)
\( \frac{4}{x} = 16 \\ \)
\(16x = 4\)
\(x = \frac{4}{16} \\ \)
\(x = \frac{1}{4} \\ \)
\( \begin{gathered}\\ \large\implies\sf{ \frac{1}{x} + \frac{3}{x} = 16} \\ \end{gathered}\)
\( \begin{gathered}\\ \large\implies\sf{ \frac{4}{x } = 16 } \\ \end{gathered}\)
\( \begin{gathered}\\ \large\implies\sf{ x = 16 \times 4 } \\ \end{gathered}\)
\( \begin{gathered}\\ \implies \orange{\underline{\boxed{\large\frak \pink{x = 64 }}}} \\ \end{gathered}\)
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The radius of a circle is 17 centimeters. What is the circle's circumference?
Answers
Answer:
The circle's circumference is approximately 106.81 centimeters.
Explanation:
The formula for the circumference of a circle is C = 2πr, where C is the circumference, π is pi (approximately 3.14), and r is the radius of the circle.
So, for a circle with a radius of 17 centimeters, its circumference can be found by:
C = 2πr
C = 2 x 3.14 x 17
C ≈ 106.81 cm
Therefore, the circumference of the circle is approximately 106.81 centimeters.
what is the answer to this problem 9/14= (x+8)/(3x-2)
Answers
Answer:
x=10
Step-by-step explanation:
PLEASE HELP ASAP WILL DO BRAINLIEST!!!!!!! Part A: Three equations that all represent the same function f are shown. Select the equation that includes the value of f(0) as a number that appears in the equation. Part B: Identify f(0) f(x)=2(x+3)^2-8 f(x)=2x^2+12x=10 f(x)=2(x+5)(x+1) -10 -8 -5 -3 -1 0 1 3 5 8 10
Answers
Answer:
A: f(x)=2x^2 +12x +10
B: f(0) = 10
Step-by-step explanation:
Part A:
f(0) is a constant in the equation if the form is such that for x=0, all terms are zero except one of them. Standard form is one such form. This is the equation in standard form:
f(x) = 2x^2 +12x +10
__
Part B:
f(0) = 0 + 0 + 10
f(0) = 10
Given: f(x) = x - 7 and h(x) = 2x + 3
Write the rule for f(h(x))
Answers
Answer:
2x - 4
Step-by-step explanation:
f(x) = x - 7
= 2x + 3 - 7
= 2x - 4
A triangle has a perimeter of 16x + 4. If Side 1 is 3x + 2, Side 2 is 8x + 7, what is the measure of Side 3?
Answers
Answer:
5x - 5
Step-by-step explanation:
To find the missing side, add the two sides together and subtract them from the perimeter.
side 1 + side 2
3x + 2 + 8x + 7
combine like terms
11x + 9
Subtract from the perimeter
16x + 4 - (11x + 9)
rewrite as an addition problem, so you need to change the signs inside the parentheses.
16x + 4 + (-11x -9)
Now combine like terms
5x - 5
What is the slope of the line that passes through the points (-3,-9) and (22,-4)
Answers
Answer: 1/5
Step-by-step explanation:
Help
Select all of the expressions that have the same value as 892 ÷ 8
8,920 ÷ 80
894 ÷ 10
89.2 ÷ 0.08
8.92 ÷ 0.8
.892 ÷ 0.008
Answers
Answer:
8,920 ÷ 80=111.5
892 ÷ 8=111.5
89.2 ÷ 0.08=111.5
.892 ÷ 0.008=111.5
Step-by-step explanation:
what single decimal multiplier would you use to decrease by 2% followed by a 7% decrease?
Answers
9514 1404 393
Answer:
0.9114
Step-by-step explanation:
The multiplier for a series of changes is the product of the multipliers for each of the changes:
(1 -2%)(1 -7%) = (0.98)(0.93) = 0.9114
6 * 10x^{-3}/8*10x^{-6}
Answers
The simplified form of the given expression, 6 × 10⁻³ / 8 × 10⁻⁶, is 3/4 × 10³
Simplifying an expressionFrom the question, we are to simplify the given expression
The given expression is
6 × 10⁻³ / 8 × 10⁻⁶
Simplifying the expression
6 × 10⁻³ / 8 × 10⁻⁶
This can be written as
6/8 × 10⁻³/10⁻⁶
Reduce the fraction
3/4 × 10⁻³/10⁻⁶
Apply the division law of indices
3/4 × 10⁻³ ⁻ ⁽⁻⁶⁾
3/4 × 10⁻³ ⁺ ⁶
3/4 × 10³
Hence, the value is 3/4 × 10³
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Two angles are complementary. The larger angle is 50∘ more than the smaller angle. Find the measure of the larger angle.
Answers
The measure of the smaller complementary angles is 40°
What are complementary anglesWhen two angles add up to 90°, we say they complement each other in terms of mathematics. Hence in geometry, complementary angles are defined as two angles whose sum is 90 degrees.
The larger angle = 50°
we shall represent the smaller angle with the letter x so that;
x + 50° = 90°
subtract 50° from both sides of the equation
x + 50° - 50° = 90° - 50°
x = 40°
Therefore, the smaller complementary angle has a measure of 40°.
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If 7 ballpoint pens cost $9.66, how much will 12 pens cost? (100 points)
Answers
Answer:
16.56
Step-by-step explanation:
9.66/7
=1.38 each
1.38x12=16.56
Answer:
16.56
Step-by-step explanation:
Which fractions are equivalent to 1012?
Answers
\(50/60\)
That should be your answer
Answer:
Hello, There! I'm here to help!
QuestionWhich fractions are equivalent to 10/12?
AnswerAccording to The Question I'm guessing you mean 10/12
If so The Following Fractions are Equilvalent to 10/12
1. 20/24
2. 30/36
3. 40/48
5. 50/60
Step-by-step explanation:
To get Eqilvalent You Can Multiply Number to the fractions to find out what's Equilvalent
Therefore, I hope this helps you!
If the diameter of the circle is 14 units, calculate its area: then calculate its circumference
Answers
Answer:
Area: 153.86
Circumference: 43.96
Which inequality is represented by this graph?
Answers
Answer:
\( x \ge -53 \)
Step-by-step explanation:
A solid dot means \( \le \) or \( \ge \).
The solid dot is on -53.
The shading is to the right, so it is greater than or equal to.
Answer: \( x \ge -53 \)
Answer: the last one
Step-by-step explanation: well it is clearly an grater than solusand and it is colored in not doted so its grater than or equal to
I NEES HELP STATISTICS
Answers
- This year at least one of their clients will have a starting salary of at least $40,000.
There is not enough information to determine whether this statement is true or false. The mean starting salary from last year does not provide information on this year's salaries.
- Last year some of their clients had a starting salary of exactly $40,000.
This statement is true. Since the mean includes all starting salaries, it is possible for some clients to have had a starting salary of exactly $40,000.
- Last year fewer than half of their clients had a starting salary of $40,000 or less.
This statement is false. Since the mean starting salary is $40,000, it is impossible for fewer than half of the clients to have had a starting salary of $40,000 or less.
- Last year some of their clients had a starting salary of more than $36,000.
This statement is true. Since the mean starting salary is $40,000, it is possible for some clients to have had a starting salary of more than $36,000.
- Last year at least one of their clients had a starting salary of at least $40,000.
This statement is true. Since the mean starting salary is $40,000, it is possible for at least one client to have had a starting salary of at least $40,000.
Therefore, the statements that must be true based on the given information are:
- Last year some of their clients had a starting salary of exactly $40,000.
- Last year some of their clients had a starting salary of more than $36,000.
- Last year at least one of their clients had a starting salary of at least $40,000.
What is the surface area of the box formed by the pattern below?
Answers
Answer:
h
Step-by-step explanation:
Use the quadratic formula to solve the equation below. –19x = -4x2 + 30 OA. x = -6, x = -1.25 OB. x = -6, x = -1.25 OC. x = 3, x = –2.5 OD. x = 6, x = 1.25ill send a picture of the question
Answers
In order to solve this quadratic equation, first let's put it in the standard form:
\(\begin{gathered} y=ax^2+bx+c \\ \\ -19x=-4x^2+30 \\ 4x^2-19x-30=0 \\ \\ a=4,b=-19,c=-30 \end{gathered}\)Then, let's use the quadratic formula to find the zeros:
\(\begin{gathered} x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ x_1=\frac{19+\sqrt[]{19^2+480}}{8}=\frac{19+29}{8}=\frac{48}{8}=6 \\ x_2=\frac{19-29}{8}=\frac{-10}{8}=-1.25 \end{gathered}\)So the zeros are x = 6 and x = -1.25, therefore the correct option is A.
please hurry
Solve for t: 5t - 4 = 11
Answers
Answer:
t=3
Step-by-step explanation:
5t-4=|11
+4 |+4
5t= | 15
÷5 | ÷5
t = 3
The perimeter of a rectangle is 50 inches. If the width of the rectangle is 11 inches, what is the length
Answers
Answer:
The length would be 14 inches
Step-by-step explanation:
width = 11 x 2 = 22
So from that we can subtract 22 from 50 is is 28 and 28 divided by 2 is 14.
Mr. Plaggenier divided Camp Greenfield into 5 squads for an athletic scrimmages. Each squad competes against each of the other 4 squads 2 times. The scrimmage lasts 2 hours. If only one pair of teams competes at a time and all of the competitions take the same amount of time, how long is each competitions?
A)6 min.
B) 8 min.
C) 10 min.
D) 12 min.
Answers
Answer: (a)
Step-by-step explanation:
Given
There are 5 squads for an athletic scrimmages
If each team played 2 matches
Total no of matches will be
\(\Rightarrow \dfrac{5(5-1)}{2}\times 2=20\ \text{matches}\)
So, 20 matches are played in 2 hours
Each match takes
\(\Rightarrow \dfrac{120}{20}=6\ \text{minute}\)
Option (a) is correct.
You make $29/hour, work 40 hours/week, 50 weeks a year. You save 12% of your
GROSS Your employer matches your savings up to 6%. Your spouse ALSO
contributes $150/month to your retirement fund.
Your average return over 50 years of work is 6.7%. How much will you have
accumulated once you retire in 50 years?
Answers
Gross income per hour = $29/hour
Gross income per week = $29/hour * 40 hours/week = $1160/week
Gross income per year = $1160/week * 50 weeks/year = $58,000/year
Next, let's calculate the amount you save per year:
Savings per year = Gross income per year * 12% = $58,000 * 12% = $6,960
Now, let's calculate the employer matching contribution:
Employer matching contribution = Savings per year * 6% = $6,960 * 6% = $417.60
Next, let's add the spouse's contribution to the savings per year:
Savings per year = Savings per year + Spouse's contribution = $6,960 + $150/month * 12 months/year = $8,760
Finally, let's use this information to calculate the amount you will have accumulated in 50 years:
Amount accumulated in 50 years = Savings per year * (1 + 6.7%)^50 + Employer matching contribution * (1 + 6.7%)^50
Amount accumulated in 50 years = $8,760 * (1 + 0.067)^50 + $417.60 * (1 + 0.067)^50
This is the approximate amount you will have accumulated in 50 years of work, taking into account your gross income, savings, employer matching contributions, spouse's contributions, and average return on investment.