Answers
Step-by-step explanation:
Hi there!
Please see the answer in the picture.
Hope it helps!
1. Approach
One is given a trigonometric equation with and one is asked to prove that it is true. Using the attached image, combined with the knowledge of trigonometry, one can evaluate each trigonometric function. Then one can simplify each ratio to solve. To yield the most accurate result, one has to each of the ratios in a fractional form, rather than simplifying it into a decimal form. Remember the right angle trigonometric ratios, these ratios describe the relationship between the sides and angles in a right triangle. Such ratios are as follows,
[tex]sin(\theta)=\frac{opposite}{hypotenuse}\\\\cos(\theta)=\frac{adjacent}{hypotenuse}\\\\tan(\theta)=\frac{opposite}{adjacent}\\\\csc(\theta)=\frac{hypotenuse}{opposite}\\\\sec(\theta)=\frac{hypotenuse}{adjacent}\\\\cot(\theta)=\frac{adjacent}{opposite}[/tex]
Please note that the terms (opposite) and (adjacent) are relative to the angle uses in the ratio, however the term (hypotenuse) refers to the side opposite the right angle, this side never changes its name. Use these ratios to evaluate the trigonometric functions. Then simplify to prove the identity.
2. Problem (9)
[tex]\frac{sin(60)+cos(30)}{1+sin(30)+cos(60)}=sin(60)[/tex]
As per the attached image, the following statements regarding the value of each ratio can be made:
[tex]sin(60)=\frac{\sqrt{3}}{2}\\\\cos(30)=\frac{\sqrt{3}}{2}\\\\sin(30)=\frac{1}{2}\\\\cos(60)=\frac{1}{2}[/tex]
Substitute,
[tex]\frac{sin(60)+cos(30)}{1+sin(30)+cos(60)}=sin(60)[/tex]
[tex]\frac{\frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{2}}{1+\frac{1}{2}+\frac{1}{2}}=\frac{\sqrt{3}}{2}[/tex]
Simplify,
[tex]\frac{\frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{2}}{1+\frac{1}{2}+\frac{1}{2}}=\frac{\sqrt{3}}{2}[/tex]
[tex]\frac{\frac{2\sqrt{3}}{2}}{1+\frac{1}{2}+\frac{1}{2}}=\frac{\sqrt{3}}{2}[/tex]
[tex]\frac{\frac{2\sqrt{3}}{2}}{1+1}=\frac{\sqrt{3}}{2}[/tex]
[tex]\frac{\sqrt{3}}{1+1}=\frac{\sqrt{3}}{2}[/tex]
[tex]\frac{\sqrt{3}}{2}=\frac{\sqrt{3}}{2}[/tex]
Thus, this equation is true.
2. Problem (10)
Use a similar strategy to evaluate this equation,
[tex]\frac{1-cos(30)}{sin(30)}=\frac{1-cot(60)}{1+cot(60)}[/tex]
Use the attached image to evaluate the ratios.
[tex]cos(30)=\frac{\sqrt{3}}{2}\\\\sin(30)=\frac{1}{2}\\\\cot(60)=\frac{1}{\sqrt{3}}[/tex]
Substitute,
[tex]\frac{1-cos(30)}{sin(30)}=\frac{1-cot(60)}{1+cot(60)}[/tex]
[tex]\frac{1-\frac{\sqrt{3}}{2}}{\frac{1}{2}}=\frac{1-\frac{1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}}[/tex]
Simplify,
[tex]\frac{1-\frac{\sqrt{3}}{2}}{\frac{1}{2}}=\frac{1-\frac{1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}}[/tex]
[tex]\frac{\frac{2-\sqrt{3}}{2}}{\frac{1}{2}}=\frac{1-\frac{1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}}[/tex]
[tex]\frac{\frac{2-\sqrt{3}}{2}}{\frac{1}{2}}=\frac{\frac{\sqrt{3}-1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}}[/tex]
[tex]\frac{\frac{2-\sqrt{3}}{2}}{\frac{1}{2}}=\frac{\frac{\sqrt{3}-1}{\sqrt{3}}}{\frac{\sqrt{3}+1}{\sqrt{3}}}[/tex]
[tex]2-\sqrt{3}=\frac{\frac{\sqrt{3}-1}{\sqrt{3}}}{\frac{\sqrt{3}+1}{\sqrt{3}}}[/tex]
[tex]2-\sqrt{3}=\frac{\sqrt{3}-1}{\sqrt{3}}*\frac{\sqrt{3}}{\sqrt{3}+1}}[/tex]
[tex]2-\sqrt{3}=\frac{\sqrt{3}-1}{\sqrt{3}+1}[/tex]
Rationalize the denominator,
[tex]2-\sqrt{3}=\frac{\sqrt{3}-1}{\sqrt{3}+1}[/tex]
[tex]2-\sqrt{3}=\frac{\sqrt{3}-1}{\sqrt{3}+1}*\frac{\sqrt{3}-1}{\sqrt{3}-1}[/tex]
[tex]2-\sqrt{3}=\frac{(\sqrt{3}-1)^2}{3-1}[/tex]
[tex]2-\sqrt{3}=\frac{3-2\sqrt{3}+1}{2}[/tex]
[tex]2-\sqrt{3}=\frac{4-2\sqrt{3}}{2}[/tex]
[tex]2-\sqrt{3}=2-\sqrt{3}[/tex]
Therefore, this equation is also true.
Related Questions
Find the equation of the linear function represented by the table below in slope-intercept form.
Need Help with this last question plz
Answers
Answer:
y = -3x-4
Step-by-step explanation:
First we find the slope
m = (y2-y1)/(x2-x1)
= (-16 - -7)/(4-1)
= (-16+7) /3
= -9/3
= -3
The slope intercept form is
y = mx+b where m is the slope and b is the y intercept
y = -3x+b
Using a point
-10 = -3(2)+b
-10 = -6+b
-10+6 = b
-4 = b
y = -3x-4
Determine the period
Answers
Answer:
10
Step-by-step explanation:
acellus
The period of the given graph is 10.
Use the concept of a period of function defined as:
One full wave is set up in a medium for the length of time it takes a particle of that medium to complete one vibration or one wave period.
In the given wave,
X-axis represents time
The Y- axis represents the amplitude of the curve
As we can see,
The first peak of this graph is located at x = 1 and the second peak is at,
x = 13
Since the period is the time interval between two consecutive peaks in a graph.
In this case, the time interval is 10.
Hence,
The period of the given graph is 10.
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Find the surface area of the composite figure
Answers
Answer: SA = 644 cm²
Step-by-step explanation:
STEP ONE: Find the surface area of the smaller rectangle (green)
There are in total 5 sides because one of the sides is not shown on the surface level.
Upper base = 3 × 3 = 9 cm²
4 sides = 3 × 8 × 4 = 96 cm²
Total = 9 + 96 = 105 cm²
STEP TWO: Find the surface area of the larger rectangle (orange)
There are in total 5 whole sides with one side covered by the base of the smaller rectangle (green) which should later be subtracted.
Front and back = 7 × 10 × 2 = 140 cm²
Left and right = 7 × 12 × 2 = 168 cm²
Lower base = 10 × 12 = 120 cm²
Upper base = 10 × 12 - 3 × 3 = 111 cm²
Total = 140 + 168 + 120 + 111 = 539 cm²
STEP THREE: Find the total surface area
SA = Green + Orange = 105 + 539 = 644 cm²
Hope this helps!! :)
Please let me know if you have any questions
The sequences below are either arithmetic sequences or geometric sequences. For each sequence, determine whether it is arithmetic or geometric, and write
the formula for the nth term e, of that sequence
Sequence
Type
term formula
(a) 13, 16, 19
Arithmetic
Geometric
(6) 2, 10, 50
Arithmetic
Geometrie
Sum
o
el
Answers
Answer:
a) Arithmetic yₙ = 3*(n - 1) + 13
b) Geometric yₙ = 2*5⁽ⁿ⁻¹⁾
Step-by-step explanation:
The formula for the nth term e, of that sequence is; yₙ = 3*(n - 1) + 13 and yₙ = 2*5⁽ⁿ⁻¹⁾
What is an arithmetic sequence?An arithmetic sequence is a sequence of integers with its adjacent terms differing with one common difference.
If the initial term of a sequence is 'a' and the common difference is of 'd', then we have the arithmetic sequence as:
a, a + d, a + 2d, ... , a + (n+1)d, ...
Its nth term is
T_n = a + (n-1)d
(for all positive integer values of n)
(a) 13, 16, 19
Here, yₙ = 3*(n - 1) + 13
This is an Arithmetic.
b) 2, 10, 50
Here, yₙ = 2*5⁽ⁿ⁻¹⁾
This is Geometric .
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Help help help help help help help help
Answers
The simple interest on a sum of money in 2 years is Rs 36 less than the compound interest compounded annually. If the rate of interest is 12 % p.a., find the sum.
Answers
======================================================
Explanation:
Compute the simple interest on a deposit of x
The per annum (p.a.) interest rate is r = 0.12 from the 12% value mentioned.
We do so over t = 2 years
i = p*r*t
i = x*0.12*2
i = 0.24x
Let A = 0.24x
Do the same for the compound interest. We'll have n = 1 to reflect annual compounding.
i = p*(1+r/n)^(n*t) - p
i = x*(1+0.12/1)^(1*2) - x
i = 1.2544x - x
i = 0.2544x
Let B = 0.2544x
We're told that "The simple interest on a sum of money in 2 years is Rs 36 less than the compound interest compounded annually"
This means the interest earned in the simple interest (value A) is Rs 36 less than the amount earned with quantity B
So,
A = B-36
0.24x = 0.2544x - 36
0.24x - 0.2544x = -36
-0.0144x = -36
x = -36/(-0.0144)
x = 2500
The amount invested or deposited is Rs 2500
Find $24.75 as a fraction of $70.
Give your answer in its lowest terms.
Answers
Answer:
99/280
Step-by-step explanation:
24.75/70
Multiply the top and bottom by 100
2475/7000
Divide the top and bottom by 25
99/280
Answer: $[tex]\frac{9.9}{28}[/tex]
Step-by-step explanation:
[tex]\frac{24.75}{70}[/tex] is the fraction.
First for convenience, let us convert the decimal into a whole number by multiplying both numerator and denominator by 100. This is not necessary and the final answer will be the same but doing this step will make it easier to divide.
{[tex]\frac{24.75 * 100}{70 * 100}[/tex]} = [tex]\frac{2475}{7000}[/tex]
Now, since it is not in its lowest form we need to simplify it.
For ease, we can find the HCF of 2475 and 7000 which is 25 and divide both numerator and denominator with it.
After simplifying, the final answer is [tex]\frac{99}{280}[/tex].
However we need to put back the decimal places.
Final answer is $[tex]\frac{9.9}{28}[/tex]
Answers plssssssssssssssssss
Answers
Answer:
3/2
Step-by-step explanation:
if u divide two fractions, it is same as multiplying with its reciprocal
Write down at least five number pairs to solve the equation vw = 50
Answers
Answer:
1, 50
2, 25
5, 10
10, 5
25, 2
Answer:
Step-by-step explanation:
V W
1 x 50 = 50
1.5 x 33 1/3 = 50
2 x 25 = 50
2.5 x 20 = 50
3 x 16 2/3 = 50
3.5 x 14 2/7 = 50
4 x 12 1/2 = 50
4.5 x 11 1/9 = 50
5 x 10 = 50
5.5 x 9 = 50
6 x 8 1/3 = 50
6.5 x 7 2/3 = 50
7 x 7 1/7 = 50
7.5 x 6 2/3 = 50
8 x 6 1/4 = 50
8.5 x 5 8/9 = 50
9 x 5 5/9 = 50
9.5 x 5 1/4 = 50
10 x 5 = 50
10.5 x 4 3/4 = 50
11 x 4 5/9 = 50
11.5 x 4 1/3 = 50
12 x 4 1/6 = 50
12.5 x 4 = 50
13 x 3 6/7 = 50
13.5 x 3 5/7 = 50
14 x 3 4/7 = 50
How long would it take the car to travel a distance of 2 meters at this average speed?
Answers
Answer:
1 hour
Step-by-step explanation:
2 ÷2=1 speed divide by distance
What is the average rate of change for the function f(x) = –4x2 + 1 over the interval –3/2 ≤ x ≤ 0?
Answers
Answer:
6
Step-by-step explanation:
To find the average rate of change, we must calculate the slope over the interval -3/2 to 0. We can use the formula below.
f(b) - f(a) / b - a
f(0) - f(-3/2) / 0 - -3/2
1 - -8 / 3/2
9 / 3/2
18/3
6
Brainliest, please!
Which statement is true regarding the graphed functions
f(4)=g(4)
F(4)=g(2)
F(2)=g(-2)
F(-2)=g(-2)
Answers
When you are on the other side of the = you are opposite.
Evaluate C=5/9(F−32) for F = 77 degrees.
Answers
Answer:
b:25
Step-by-step explanation:
77 degrees Fahrenheit is 25 degrees Celsius. So, the correct answer for the given equation is the second option.
An equation is a combination of two or more expressions separated by an equal sign(=), indicating that the expressions are equal to each other. The expressions, though, are a group of numbers, variables, mathematical operations, etcetera. An equation can be represented on the Cartesian plane.
The relation between Celsius and Fahrenheit is given by the following equation:
[tex]C =\dfrac{5}{9}(F-32)[/tex],
Substitute F = 77 to convert 77 degrees Fahrenheit to Celsius.
[tex]C =\dfrac{5}{9}(77-32)[/tex]
[tex]=\dfrac{5}{9}(77-32)\\=\dfrac{5}{9}\times45\\= 25[/tex]
Thus, the second option is correct.
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What is the degree of -9x^5+3x^2+3-7x+10x^3
Answers
Answer:
5
Step-by-step explanation:
The degree is the highest power of x
The highest power of x is 5 so the degree is 5
I NEED THE ANSWER ASAP PLEASE HELP!!
The price of a stock went from $32 to $28. If p is the percent decrease in the value of the stock, which proportion can be used to compute p? A. B. C. D.
Answers
Explanation:none
Answer: C
Step-by-step explanation:
Will Mark Brainnlest Please help me
Answers
Answer:
l = 2, m = - 1, n = - 6
Step-by-step explanation:
A scalar matrix has its diagonal elements equal and all other elements zero, so
2l - 4 = 0 ( add 4 to both sides )
2l = 4 ( divide both sides by 2 )
l = 2
---------------------------------------
3l + n = 0
3(2) + n = 0
6 + n = 0 ( subtract 6 from both sides )
n = - 6
--------------------------------------
3m - n = 3
3m - (- 6) = 3
3m + 6 = 3 ( subtract 6 from both sides )
3m = - 3 ( divide both sides by m )
m = - 1
Solve x^3 = 1 over 8.
1 over 2
±1 over 2
1 over 4
±1 over 4
Answers
Answer:
x = 1/2
Step-by-step explanation:
x^3 = 1/8
Take the cube root of each side
x^3 ^ (1/3) = (1/8)^ (1/3)
Rewriting 8 as 2^3
x^3 ^ (1/3) = (1/2^3)^ (1/3)
x = 1/2
x³ = 1/8
8 can be written as 2³
x³ = 1/2³
(x)³ = (1/2)³
When exponents are same bases are taken and exponents are eliminated
x = 1/2
Hence, the required answer is 1/2Another learning aid provides a step-by-step walk-through showing the solution process for a problem similar to the one you are working on. You would click Continue to advance through each step in the solution process to show intermediate calculations at key points. While you are attempting to solve the given exercise, this learning aid can be referenced multiple times and does not load a new version of the exercise when it is accessed. Choose the described learning aid below.
Answers
Choose the described learning aid below.
a. Animation
b. view an example
c. video
d. help me solve this
e. Textbook
The answer to this question is option b. View an example helps to give a step by step walk through showing the solution to a similar problem.
The learning aid that is referenced in this question is view an example because it is a learning aid that helps the learner to solve a problem just by looking at the solution of another problem.
The solution that is being viewed has to be quite similar with the problem. Such that the learner would be able to follow through with finding a solution to his problem by following the steps in the example.
here is a similar question https://brainly.com/question/24454973?referrer=searchResults
What is the answer? Don’t give me 4.
Answers
Answer:
what?
Step-by-step explanation:
........................................................................
The product of twice a number and four is the same as the difference of seven times the number and 5/3
Answers
Answer:
2N÷4=7N-5/3
Hope this helps!
I need some help with math
#1 rewrite this in standard form, then state the center of the circle as an ordered pair and identify the radius. Show your work
y^(2)-14y+x^(2)-4x+37=0
#2 Write the equation of the circle with center (-2,15) and radius 3.
Answers
Answer:
[tex](y-7)^2+(x-2)^2=16[/tex]
and
[tex](x+2)^2+(y-15)^2 = 9[/tex]
Step-by-step explanation:
The standard equation of a circle is [tex](x-h)^2+(y-k)^2=r^2[/tex] where the coordinate (h,k) is the center of the circle.
Second Problem:
We can start with the second problem which uses this info very easily.(h,k) in this problem is (-2,15) simply plug these into the equation. [tex](x--2)^2+(y-15)^2=r^2[/tex] .We can also add the radius 3 and square it so it becomes 9. The equation.This simplifies to [tex](x+2)^2+(y-15)^2 = 9[/tex].First Problem:
The first problem takes a different approach it is not in standard form. But we can convert it to standard form by completing the square.[tex]y^2-14y+x^2-4x+37=0[/tex] first subtract 37 from both sides so the equation is now [tex]y^2-14y+x^2-4x=-37[/tex].[tex]y^2-14y+x^2-4x+37=0[/tex] by adding [tex](-\frac{b}{2a} )^2[/tex] to both the x and y portions of this equation you can complete the squares. [tex](-\frac{b}{2a})^2=(-\frac{-14}{2(1)})^2[/tex] and [tex](-\frac{-4}{2(1)})^2[/tex] which equals 49 and 4.Add 49 and 4 to both sides and the equation is now:[tex]y^2-14y+49+x^2-4x+4=-37+49+4[/tex] You can simplify the y and x portions of the equations into the perfect squares or factored form [tex](y-7)^2[/tex] and [tex](x-2)^2[/tex].Finally put the whole thing together. [tex](y-7)^2+(x-2)^2=16[/tex].I hope this helps!
the equation of the line that goes through the points (5,5) and is parallel to the line going through the points (-5,5) and (6,1) can be written in the form y=mx+b where m is and b is
Answers
Answer:
y = -4/11x + 75/11
Step-by-step explanation:
y2 - y1 / x2 - x1 1 - 5 / 6 - (-5) -4/11
y = -4/11x + b
5 = -4/11(5) + b
5 = -20/11 + b
75/11 = b
The basic Parabola y = x2 is transformed to make y = (x – 5)2 +3. The
vertical translation is
a. Up
b. Down
c Left
d. Right
e. 3 Units
f. 5 Units
Answers
Answer:
a and f
Step-by-step explanation:
The transformation is 3 units up, which is concluded from the fact that there is a positive 3 meaning up. This also represents the y-intercept.
Hope this makes sense :)
A sample is generated from a population of 20 items. Each of the 20 items are given a label, and a 20-sided die is rolled 3 times to determine which 3 items are in the sample. This is an example of a __________.
A. systematic sample
B. random sample
C. convenience sample
D. self-selecting sample
Answers
Answer:
Option B, random sample
it should be the answer
The given information is an example of a random sample. The correct option is B.
What is a random sample?In statistics, a simple random sample is a subset of people chosen at random from a larger group, all of whom were chosen with the same probability. It is a method of choosing a sample at random.
It is given that A sample is generated from a population of 20 items. Each of the 20 items is given a label, and a 20-sided die is rolled 3 times to determine which 3 items are in the sample.
The above information is an example of a random sample. Follow the given definition above.
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which of the following is a function
please help!!
Answers
Find the nth term of each of the sequences.
(a) 16, 19, 22, 25, 28, ...
(b) 1,3,9,27,81,...
Answers
Answer:
a) 16, 19, 22, 25, 28, 31, 34, 37, 40
b) 1, 3, 9, 27, 81, 243, 729, 2187
Explanation:a) Add 3 on every number.
b) Multiply every number by 3.
*NEED AN ANSWER ASAP* My book didn’t explain how to solve these ones.
Answers
Answer:
1-2. AC = 13.6
BC = 8.9
3-4. A triangle such as the given value isn't possible
Mrs. barker works 40 hours per week regularly, at a rate of $36.80 per hour. when she works overtime, her rate is time and a half of her regular hourly rate. what is Mrs. Barker hourly overtime rate
Answers
Answer:
$55.20 is Mrs. Barker hourly overtime rate.
Step-by-step explanation:
The term "time and a half" in your question means that Mrs. Baker's overtime rate is 1.5 times her usual hourly rate.
So, ----> $36.80 times 1.5 = $55.20
using Euclids division algorithm, find the largest number that divides 475 and 871 leaving 7 as the remainder in each case
Answers
Answer:36
Step-by-step explanation:
gcd(475-7,871-7)=gcd(468,864)=36
[tex]\begin{array}{c|c|c|c|c|c}&0&1&1&5&2\\468&864&468&396&72&\boxed{36}\\468&396&72&36&0\\\end{array}[/tex]
help me match these??????? please and thank you!!!!
Answers
Answer:
[tex]\sqrt{12}=2\sqrt{3}[/tex]
[tex]\sqrt{18} = 3\sqrt{2}[/tex]
[tex]\sqrt{24}=2\sqrt{6}[/tex]
[tex]\sqrt{72}=6\sqrt{2}[/tex]
Step-by-step explanation:
Hi there!
[tex]\sqrt{12}[/tex]
To simplify a square root, check if the radicand has any perfect square factors. For example, 12 can be rewritten as 4*3:
[tex]=\sqrt{4*3}\\=\sqrt{4}*\sqrt{3}[/tex]
4 is a perfect square. Its square root is 2:
[tex]=2*\sqrt{3}\\=2\sqrt{3}[/tex]
[tex]\sqrt{18} \\= \sqrt{9*2} \\= 3\sqrt{2}[/tex]
[tex]\sqrt{24} \\=\sqrt{4*6}\\=2\sqrt{6}[/tex]
[tex]\sqrt{72} \\=\sqrt{36*2} \\=6\sqrt{2}[/tex]
I hope this helps!
