Answer:
[tex]\tan \left(\dfrac{x}{2}\right)=3[/tex]
Step-by-step explanation:
Trigonometric ratios are the ratios of the sides of a right triangle.
[tex]\boxed{\begin{minipage}{9.4 cm}\underline{Trigonometric ratios} \\\\$\sf \sin(\theta)=\dfrac{O}{H}\quad\cos(\theta)=\dfrac{A}{H}\quad\tan(\theta)=\dfrac{O}{A}$\\\\where:\\ \phantom{ww}$\bullet$ $\theta$ is the angle. \\ \phantom{ww}$\bullet$ $\sf O$ is the side opposite the angle. \\\phantom{ww}$\bullet$ $\sf A$ is the side adjacent the angle. \\\phantom{ww}$\bullet$ $\sf H$ is the hypotenuse (the side opposite the right angle). \\\end{minipage}}[/tex]
The sine trigonometric ratio is the ratio of the side opposite the angle to the hypotenuse.
Given sin(x) = 3/5, the side opposite angle x is 3, and the hypotenuse is 5.
As we have two sides of the right triangle, we can calculate the third side (the side adjacent the angle) using Pythagoras Theorem.
[tex]\boxed{\begin{minipage}{9 cm}\underline{Pythagoras Theorem} \\\\$a^2+b^2=c^2$\\\\where:\\ \phantom{ww}$\bullet$ $a$ and $b$ are the legs of the right triangle. \\ \phantom{ww}$\bullet$ $c$ is the hypotenuse (longest side) of the right triangle.\\\end{minipage}}[/tex]
Therefore:
[tex]\implies A^2+3^2=5^2[/tex]
[tex]\implies A^2+9=25[/tex]
[tex]\implies A^2+9-9=25-9[/tex]
[tex]\implies A^2=16[/tex]
[tex]\implies \sqrt{A^2}=\sqrt{16}[/tex]
[tex]\implies A=4[/tex]
Use the cosine trigonometric ratio to find the value of cos(x), remembering that cosine is negative in Quadrant II.
[tex]\implies \cos x=-\dfrac{4}{5}[/tex]
Now we have the values of sin(x) and cos(x) in Quadrant II, we can use the tangent half angle formula to find the value of tan(x/2).
[tex]\begin{aligned}\implies \tan \left(\dfrac{x}{2}\right)&=\dfrac{\sin x}{1+\cos x}\\\\&=\dfrac{\frac{3}{5}}{1-\frac{4}{5}}\\\\&=\dfrac{\frac{3}{5}}{\frac{1}{5}}\\\\&=\dfrac{3}{5} \cdot \frac{5}{1}\\\\&=3\end{aligned}[/tex]
Therefore, the value of tan(x/2) is 3.
Create an equation:
The f(x) is a rational function that has a limit as x approaches 2 but fails to be continuous there because f(2) is undefined.
In golf, scores that are under par for the entire round are shown as negative scores; positive scores are shown for scores that are over par, and 0 is par. Player A was the winner of the 2014 golf tournament. Her scores were -4 ,-4 ,-4 , and -3 . What was her overall score?
the overall score of Player A in the 2014 golf tournament was -15.
In golf, scores that are below par are represented with negative numbers. Therefore, to calculate the overall score, we need to add up all the scores.
Let's call the scores for each round A1, A2, A3, and A4, respectively. We can write the given scores as:
A1 = -4
A2 = -4
A3 = -4
A4 = -3
To find the overall score, we add up all the scores:
Overall score = A1 + A2 + A3 + A4
Overall score = (-4) + (-4) + (-4) + (-3)
Overall score = -15
Therefore, the overall score of Player A in the 2014 golf tournament was -15.
It's worth noting that in golf, the winner is the player who has the lowest score or the most negative score. So, in this case, Player A had the lowest score, making her the winner of the tournament.
In general, the overall score in golf can be calculated by adding up the scores for each hole or round. The player with the lowest score at the end of the tournament is declared the winner. Scores can be compared between different golfers or rounds, regardless of the difficulty of the course or weather conditions, by using the number of strokes above or below par.
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An item cost $310 before tax, and the sales tax is$6.20. Find the sales tax rate. Write your answer as a percentage.
Answer:
2%
Step-by-step explanation:
(sales tax/ cost before tax) x 100%
(6.20 ÷ 310) x 100
0.02 x 100%
= 2
so the answer is 2%
Due to random variations in the operation of an automatic coffee machine, not every cup is filled with the same amount of coffee. Assume that the mean amount of coffee dispensed is 7 ounces and the standard deviation is 0.6 ounce. Use the 68-95-99.7 rule to complete the following.
68% of the data values will be within one standard deviation (0.6) of the mean.
95% of the data values will be within 2 standard deviations (1.2) of the mean.
99.7% of the data values will be within 3 standard deviations (1.8) of the mean.
The seats available to a baseball game come in four types: bleacher, box, club, and grandstand. There are 12,600 box seats and 5,400 club seats available. According to the graph, what is the total number of seats available?
The total number of seats available for the baseball game can be calculated by adding the number of box and club seats available whch is found to be 25,200.
What is the game of baseball?The game of baseball can be used to teach various mathematical concepts, such as probability, statistics, graphing, estimation, and basic arithmetic. The game can also be used to introduce concepts such as geometry and trigonometry, as well as to build problem-solving skills.
The total number of seats available can be calculated by adding the number of box and club seats available, which is
12,600 + 5400 = 18,000.
This total can then be multiplied by the respective percentages of the other two categories to get the total number of seats available.
For instance, the total number of bleacher seats available can be calculated by multiplying 12,000 by 18%.
12,000 x 0.18 = 2,160
Similarly, the total number of grandstand seats available can be calculated by multiplying 12,000 by 42%.
12,000 x 0.42 = 5,040
Hence, the total number of seats available for the baseball game is
18,000 + 2,160 + 5,040 = 25,200.
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What is the slope of a line that passes through the points (-2,4) and (-6,12)
Which points in the scatter plot are outliers?
Select each correct answer.
Point A
Point F
Point H
Point K,
Point M
Point K and F
Looking at the data, you can see that most of the data points form close to a line- if you were to draw a line of best fit through the data, you would see that it would be quite close to most points on the graph. This is the trend of the graph, it is linear.
However, you can see thaf K and F do not come close to this trend at all and are therefore outliers since they do not fall close to the line.
A car was purchased for $16,000. Each year since, the resale value has decreased by 26%.
Let t be the number of years since the purchase. Let y be the resale value of the car, in dollars.
Write an exponential function showing the relationship between y and t.
Answer:
y = 16000(0.86)^t
Step-by-step explanation:
y = 16000(0.86)^t
Find a5 and a-n for the geometric sequence with a1=7, r= -3
Answer:
an = 7·(-3)^(n -1)a5 = 567Step-by-step explanation:
You want the 5th term and the general term of an geometric sequence with a1 = 7 and r = -3.
General termThe general term of an arithmetic sequence is ...
an = a1·r^(n-1)
For a1=7 and r=-3, the general term is ...
an = 7·(-3)^(n-1)
Fifth termUsing n=5, the above equation evaluates to ...
a5 = 7·(-3)^(5 -1)
a5 = 7·(-3)^4 = 7·81
a5 = 567
Find the lateral surface area of the prism
The lateral surface area of the triangular prism is 275.5 sq. mm.
What is lateral surface area?"Being to the side" is the definition of the term "lateral." A triangular prism's lateral area is equal to the sum of its side faces' areas (which are 3 rectangles). Specifically, it is the total surface area less the surface areas of the two bases. Also called the lateral surface area (LSA). Due to the two dimensions involved in its calculation, we measure it in square units.
The lateral surface area of the triangular prism is given as:
LSA = (a + b + c)h
Here, the value of a = 7.5, b = 10.5, and c = 11mm, and h = 9.5mm.
Substitute the values:
LSA = (7.5 + 10.5 + 11) (9.5)
LSA = 275.5 sq. mm
Hence, the lateral surface area of the triangular prism is 275.5 sq. mm.
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Suppose that water is pouring into a swimming pool in the shape of a right circular cylinder at a constant rate of 5 cubic feet per minute. If the pool has radius 7 feet and height 8 feet, what is the rate of change of the height of the water in the pool when the depth of the water in the pool is 5 feet?
Since this is a right circular cylinder, the 8-feet height is irrelevant because we should expect the water's height to rise at a steady rate of 0.032481 feet/min.
What is constant rate?When the ratio of the output to the input remains constant at any particular point along the function, the rate of change is said to be constant.
The slope is another name for the steady rate of change.
The height of 8 feet is unimportant because, because this is a right circular cylinder, we should anticipate that the height of the water will increase at a constant rate.
[tex]V=\pi*r^2*h\\\\V=\pi*7^2*h\\\\V=\pi*49h\\\\\frac{dV}{dh} =49\ \pi \\\\\frac{dV}{dt}*\frac{dt}{dh}=49\ \pi \\\\5*\frac{dt}{dh} =49\pi\\\\\frac{dt}{dh} =\frac{49}{\frac{\pi}{5} } \\\\\frac{dt}{dh} =\frac{5}{\frac{49}{\pi} } \\\\\frac{dt}{dh} = 0.032481\ feet/min[/tex]
Since this is a right circular cylinder, the 8-feet height is irrelevant because we should expect the water's height to rise at a steady rate of 0.032481 feet/min.
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At the beginning of the school year, Ms. Lopez asks her students to select 4 books to read
from a list of 20 books.
a. How many different groups of 4 books can be selected?
b. How many different ways can 4 books be selected if the books must be listed in order of
preference?
Answer:
Step-by-step explanation:
A] five as 20 divided by four is five
Ayudaaaa necesito sacar los límites!!!
Debido a restricciones de longitud, los límites de la función aparecen descritos en la explicación de esta pregunta.
¿Cómo determinar los limites de una función?
En este problema debemos determinar los límites de un punto y los límites laterales de la función. El limite de un punto se determina mediante la evaluación directa de la función, mientras los límites laterales de la función se determinan viendo a que valor tiende la función.
A continuación, determinamos los límites de las funciones:
Caso 1:
Subcaso A (x = 1)
f(x) = - 1
Subcaso B (x = 5)
f(x) = 1
Subcaso C (x = 3⁻)
f(x) → - ∞
Subcaso D (x = 3⁺)
f(x) → + ∞
Subcaso E (x = 3)
No existe
Case 2:
Subcaso A (x = 0)
f(x) = 0
Subcaso B (x = - 2)
f(x) = 0.5
Subcaso C (x = 3⁻)
f(x) → - ∞
Subcaso D (x = - 3⁺)
f(x) → + ∞
Subcaso E (x = - 3)
No existe
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The systolic blood pressure of adults in the USA is nearly normally distributed with a mean of 117 and standard deviation of 22 .
Someone qualifies as having Stage 2 high blood pressure if their systolic blood pressure is 160 or higher.
a. Around what percentage of adults in the USA have stage 2 high blood pressure? Give your answer rounded to two decimal places.
b.
b. If you sampled 2000 people, how many would you expect to have BP> 160? Give your answer to the nearest person. Note: I had a bit of an issue encoding rounded answers, so try rounding both up and down if there's an issue!
C. Stage 1 high BP is specified as systolic BP between 140 and 160. What percentage of adults in the US qualify for stage 1? d. Your doctor tells you you are in the 30th percentile for blood pressure among US adults. What is your systolic BP? Round to 2 decimal places.
According to the question we can say that the area to the right of z = 2.00 is 0.0228 or 2.28%.
a. To find the percentage of adults in the USA with stage 2 high blood pressure, we need to find the area under the normal distribution curve to the right of 160.
Using a standard normal distribution table or a statistical software, we can find that the z-score corresponding to 160 systolic blood pressure is:
z = (160 - 117) / 22 = 2.00
The area to the right of z = 2.00 is 0.0228 or 2.28%. Therefore, around 2.28% of adults in the USA have stage 2 high blood pressure.
b. To find how many people out of a sample of 2000 would be expected to have systolic blood pressure greater than 160, we can use the same z-score from part (a) and the standard normal distribution formula:
z = (x - μ) / σ
Rearranging the formula to solve for x, we get:
x = z * σ + μ
Substituting the values, we get:
x = 2.00 * 22 + 117 = 161.4
So the expected number of people with systolic blood pressure greater than 160 out of a sample of 2000 is:
2000 * (1 - P(Z < 2.00)) = 2000 * (1 - 0.9772) = 45.6
Rounding up, we can expect about 46 people to have systolic blood pressure greater than 160 out of a sample of 2000.
c. To find the percentage of adults in the USA who qualify for stage 1 high blood pressure, we need to find the area under the normal distribution curve between 140 and 160.
Using the standard normal distribution formula and z-scores, we get:
z1 = (140 - 117) / 22 = 1.05
z2 = (160 - 117) / 22 = 1.95
Using a standard normal distribution table or a statistical software, we can find the area to the left of z1 and z2, and then subtract the two areas to get the area between z1 and z2:
P(z1 < Z < z2) = P(Z < z2) - P(Z < z1) = 0.9744 - 0.8531 = 0.1213
Therefore, approximately 12.13% of adults in the USA qualify for stage 1 high blood pressure.
d. To find the systolic blood pressure corresponding to the 30th percentile, we need to find the z-score that has an area of 0.30 to its left.
Using a standard normal distribution table or a statistical software, we can find the z-score that corresponds to the area 0.30:
z = -0.52
Using the standard normal distribution formula and the given mean and standard deviation, we can solve for the systolic blood pressure:
x = z * σ + μ = -0.52 * 22 + 117 = 105.16
Therefore, if you are in the 30th percentile for blood pressure among US adults, your systolic blood pressure is approximately 105.16.
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unfortunately, the set up of these problems is very confusing for me because they keep altering and changing per problem
Answer:
2292.03
Step-by-step explanation:
Start with the formula for continuously compounded interest.
Then substitute all given values in the formula.
Finally, solve for the only variable remaining.
[tex] A = Pe^{rt} [/tex]
A = future value = $5000
P = principal (deposited amount) = unknown
r = 6.5% = 0.065
t = time = 12 years
[tex] 5000 = Pe^{0.065 \times 12} [/tex]
[tex] 5000 = Pe^{0.78} [/tex]
[tex]5000 = P \times 2.18147[/tex]
[tex] P = \dfrac{5000}{2.18147} [/tex]
[tex] P = 2292.03 [/tex]
Answer: $2292.03
Answer:
P = $2366.91
(maybe try answering without a comma)
Step-by-step explanation:
The formula for continuous compounding is:
A = Pe^(rt)
Where:
A = final amount
P = principal amount (initial deposit)
e = Euler's number (approximately 2.71828)
r = annual interest rate (as a decimal)
t = time (in years)
We are given:
r = 6.5% = 0.065 (annual interest rate)
t = 12 years
A = $5000 (final amount)
So we can rearrange the formula to solve for P:
P = A / e^(rt)
Substituting the values:
P = 5000 / e^(0.065*12)
P = $2366.91 (rounded to the nearest cent)
Therefore, you would need to deposit $2366.91 in an account with a 6.5% interest rate, compounded continuously, to have $5000 in your account 12 years later.
Hurry!!!! Tyra wrote the equation at the right.
a. Give a possible value for x and y that would make the equation true.
b. If the value of y is 12, what is the value of x?
can you solve this question?
y'=?
The solution of given derivative of y = √(arctan(x)) is:
Y' = 1 / (2√arctan(x)(1 + x²))
What do you mean by Differentiate ?In general, the term "differentiate" means to distinguish or recognize the differences between two or more things or concepts.
In mathematics, differentiation refers to the process of finding the rate at which a function changes with respect to one of its variables. It is a fundamental concept in calculus and involves calculating the derivative of a function. The derivative gives us the slope of a tangent line to the curve of the function at a specific point
To differentiate Y = √arctan(x), we need to use the chain rule and the formula for differentiating the arctan function, which is:
d/dx arctan(x) = 1 / (1 + x²)
Using the chain rule, we have:
Y = √arctan(x)
Y = [tex](arctan(x))^(1/2)[/tex]
Y' =[tex](1/2)(arctan(x))^(-1/2) * d/dx (arctan(x))[/tex]
Y' = [tex](1/2)(arctan(x))^(-1/2) * (1 / (1 + x^{2} ))[/tex]
Putting it all together, we have:
Y' = (1 / 2√arctan(x)) * (1 / (1 + x²))
Simplifying the expression, we get:
Y' = 1 / (2√arctan(x)(1 + x²))
Therefore, the derivative of Y = √arctan(x) is:
Y' = 1 / (2√arctan(x)(1 + x²))
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Find areas of the trapezoids.
(I'm giving 100 points to whoever answers.)
Answer:
a) Area of STAR = 48 square units
b) Area of SKCO = 42 square units
Step-by-step explanation:
The formula for the area of a trapezoid is half the sum of the bases multiplied by the height:
[tex]\boxed{\sf Area=\dfrac{a+b}{2} \cdot h}[/tex]
The bases of a trapezoid are the parallel sides.
The height of a trapezoid is the perpendicular distance between the two bases.
a) Trapezoid STARThe bases are parallel sides SR and TA.
The height is the perpendicular distance between SR and TA.
Therefore:
a = SR = 4 unitsb = TA = 8 unitsh = 8 unitsSubstitute these values into the formula and solve for area:
[tex]\begin{aligned}\sf \implies Area\;STAR&=\dfrac{4+8}{2} \cdot 8\\\\&=\dfrac{12}{2} \cdot 8\\\\&=6\cdot 8\\\\&=48\;\sf square\;units\end{aligned}[/tex]
b) Trapezoid SKCOThe bases are parallel sides SK and OC.
The height is the perpendicular distance between SK and OC.
Therefore:
a = SK = 4 unitsb = OC = 10 unitsh = 6 unitsSubstitute these values into the formula and solve for area:
[tex]\begin{aligned}\sf \implies Area\;SKCO&=\dfrac{4+10}{2} \cdot 6\\\\&=\dfrac{14}{2} \cdot 6\\\\&=7 \cdot 6\\\\&=42\;\sf square\;units\end{aligned}[/tex]
let f(x)=2x^2+16x+33.
by completing the square write f(x)=a(x-h)^2+k
Step-by-step explanation:
use the formula for completing the square
A ship is 115 miles from one radio transmitter and 140 miles from another transmitter. If the angle between the signals is 132°, how far apart are the transmitters? Round to the nearest tenth.
The transmitters are approximately 115.1 miles apart.
How to find how far apart are the transmittersThis is a trigonometry problem that can be solved using the Law of Cosines.
We can use the law of cosines to solve this problem. Let's call the distance between the transmitters "d". Then we have:
d^2 = 115^2 + 140^2 - 2(115)(140)cos(132°)
d^2 = 13212.4
d ≈ 115.1 miles (rounded to the nearest tenth)
Therefore, the transmitters are approximately 115.1 miles apart
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Armando kicks a football into the air. The function (x)--5x^2 + 39x + 0.24 models the height of the football from the ground, in feet, with respect to the time x in seconds. Use a graph or table to estimate
the time for the ball to return to the ground after being kicked
The graph of the quadratic function is on the image at the end, there we can see that the ball takes 7.8 seconds to return to the ground.
How long will take the ball to return to the ground?We know that the height of the ball is modeled by the quadratic function:
h(x) = -5x² + 39x + 0.24
Where x represents the time in seconds.
To do this, we need to use a graph of the quadratic function, when we have the graph, we need to identify the zeros of the quadratic (when the height is zero, the ball is in the ground).
On the image at the end you can see the graph, there you can see that we have a zero at x = 7.8, it means that the ball takes 7.8 seconds to return to the ground.
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I’m terrible with this stuff please help
As, a1/a2 = b1/b2 = c1/c2 = 2 .Thus, the given system of equations forms the identical lines.
Explain about the solution of system of equations:A list of numbers x, y, z, etc. that simultaneously make all of the equations true is the solution of a system of equations. A system of equations' solution set is the whole of all possible answers. Finding all answers using formulas containing a certain number of parameters is known as solving the system.
For the given system of equations:
6x - y - 2 = 0
here, coefficients a1 = 6, b1 = -1 and constant c1 = -2
Now,
3x - 1/2y - 1 = 0
here, coefficients a2 = 3, b2 = -1/2 and constant c2 = -1
taking the ratios:
a1/a2 = 6/3 = 2
b1/b2 = -1/(-1/2) = 2
c1/c2 = -2/(-1) = 2
As, a1/a2 = b1/b2 = c1/c2 = 2 .Thus, the given system of equations forms the identical lines.
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A line passes through the point (-8,7) and has a slope of 3/2.
Write an equation in slope-intercept form for this line.
The equation in slope-intercept form for the line that passes through the point (-8,7) and has a slope of 3/2 is y = (3/2)x + 19.
Insurance helps you transfer the
A. Risk
B. Money
C. Insurance
D. Health
from your bank account to the insurance company.
Generally, insurance helps the policyholder to transfer the A. Risk from their bank account to the insurance company.
What is insurance risk?Insurance risk refers to the exposure of a policyholder to financial loss and consequent inability to meet their liabilities.
Insurance risks are transferable to insurance companies who pool resources together from many policyholders to meet and spread the risks.
Insurance does not transfer money, insurance, or health to the insurance company, but it transfers risk.
Thus, the correct option is Option A.
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In a marriage ceremony of Pemba's daughter, he has to make arrangement for accommodation of 150 persons. For this purpose he plans to build a conical tent in such a way that each person has 4 sq.m. of the space on ground and 20 cu.m. of air to breadth. What should be the height of the tent? Find it.
Answer:
This is the answer of that question
3
Select the correct answer.
Which phase of the business cycle would be marked by an increase in productivity while employment and profits also rise?
Answer:
The phase of the business cycle that would be marked by an increase in productivity while employment and profits also rise is known as the Expansion phase.
the equation of a circle centered at the origin x^2 + y^2 = 64. what is the radius of the circle?
Answer:
r = 8 units
Step-by-step explanation:
The equation of the circle centered at the origin is x² + y² = r²
x² + y² = 64
x² + y² = 8²
r = 8 units
Answer:
8
Step-by-step explanation:
An equation of a circle is depicted by the notation [tex](x-h)^2+(y-k)^2=r^2[/tex], where
- [tex](h,k)[/tex] is the center point of the circle
- [tex]r[/tex] is the radius (the span of units from the origin to the border of the circle)
In this unique problem, the center is the origin (0,0), so [tex]h[/tex] and [tex]k[/tex] respectively equal 0.
Since the equation of the circle is [tex]x^2+y^2=64[/tex] , we must square root 64 since it's in it's [tex]r^2[/tex] form to obtain the value of [tex]r[/tex], which is the radius.
[tex]\sqrt{64}[/tex]=±8
Despite mathematically, -8 or 8 will work in this case, you cannot have a negative radius; therefore, the radius of the circle is 8.
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Junior Saans obtained a $3,275 loan at 7.5% for 24 months. His monthly payment on the loan is $147.37. After 8 payments, the balance on the loan is $2,237.27. If he pays off the loan when the next payment is duc, what is the final payment? Step I Find the previous balance. Step 2 Find the interest for the 9th month. Step 3. Find the final payment.
Answer:
Step 1: Find the previous balance.
After 8 payments, the remaining balance on the loan is $2,237.27. Therefore, the previous balance would be the balance before the 8th payment.
Total number of payments = 24
Number of payments already made = 8
Number of payments remaining = 24 - 8 = 16
Using the given monthly payment, we can find the balance before the 8th payment:
PV = PMT x [(1 - (1 + r/n)^-n*t)/(r/n)]
where PV is the present value or previous balance, PMT is the monthly payment, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the time period in years.
Plugging in the given values, we get:
PV = $147.37 x [(1 - (1 + 0.075/12)^(-12*2))/(0.075/12)]
PV = $3,090.60
Therefore, the previous balance was $3,090.60.
Step 2: Find the interest for the 9th month.
We know that the balance at the end of the 8th month was $2,237.27. We can use this and the given interest rate to find the interest for the 9th month.
Interest = Balance x (Annual interest rate/12)
Interest = $2,237.27 x (0.075/12)
Interest = $13.99
Step 3: Find the final payment.
To find the final payment, we need to add the interest for the 9th month to the monthly payment and subtract it from the remaining balance.
Final payment = Remaining balance + Interest for 9th month - Monthly payment
Final payment = $2,237.27 + $13.99 - $147.37
Final payment = $2,103.89
Therefore, the final payment is $2,103.89.
Hope This Helps!
A boat heading out to sea starts out at Point A, at a horizontal distance of 1189 feet from a lighthouse/the shore. From that point, the boat’s crew measures the angle of elevation to the lighthouse’s beacon-light from that point to be 10∘ .At some later time, the crew measures the angle of elevation from point B to be 3 Find the distance from point A to point B. Round your answer to the nearest tenth of a foot if necessary.
The boat is 1189 feet away from the lighthouse at position A, and the angle between the boat and the beacon light is 10°. The distance between point A and point B is 2811.4 feet.
What's the difference between points A and B?The boat is 1189 feet away from the lighthouse at position A, and the angle between the boat and the beacon light is 10°.
As a result, if h is the lighthouse's height, we have:
tan 10 = h/1189
h = 1189.tan 10
h = 209.7 feet
The elevation angle at position B is 3°. The following equation provides the distance (d) from the lighthouse:
tan 3 = h/d
d = h/tan 3 = 4000.4 feet
Therefore, the distance between points A and B is 400.4 - 1189
= 2811.4 feet
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Considering only the values of α and β for which cos(α−β)cosαcosβ is defined, which of the following expressions is equivalent to cos(α−β)cosαcosβ?
Select the correct answer below:
tanα−tanβ
1+tanαtanβ
cotαcotβ−1
cotα+cotβ
According to the given condition, the correct expressions is:
cotαcotβ−1
What is trigonometric equations?Trigonometric equations are mathematical equations that involve trigonometric functions, such as sine, cosine, tangent, cotangent, secant, or cosecant, and their variables. Trigonometric equations typically involve finding the values of the unknowns that satisfy the given equation, subject to certain restrictions or conditions on the domain of the trigonometric functions.
According to the given information:
Using trigonometric identities, we can simplify the expression cos(α−β)cosαcosβ:
cos(α−β)cosαcosβ = cos(α−β) * (cosα * cosβ)
Next, we can use the identity cos(α−β) = cosαcosβ + sinαsinβ to substitute into the expression:
cos(α−β)cosαcosβ = (cosαcosβ + sinαsinβ) * (cosα * cosβ)
Now, we can distribute and simplify:
cos(α−β)cosαcosβ = cosα² * cosβ² + sinαsinβ * cosα * cosβ
Finally, using the identity cotα = 1/tanα, we can rewrite the expression as:
cos(α−β)cosαcosβ = cotαcotβ - 1
So, the equivalent expression is cotαcotβ - 1.
Therefore, according to the given condition. The correct answer is:
cotαcotβ−1
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