Given sinθ=1/2 determine the value of sec θ. 0°<θ<90°
Given Choices
2/√3
√3/2
2
1
Answers
Answer:
[tex]sec(\theta)=\frac{2}{\sqrt3}[/tex]
Step-by-step explanation:
Given [tex]sin(\theta)=\frac{1}{2}[/tex] , to find [tex]sec(\theta)[/tex], it will be helpful to visualize a right triangle (triangle with a 90 degree angle) associated with that particular θ. There are a few ways to go about this:
A general solution methodAll of the basic trigonometric functions, applied to an angle) are a ratio of two specific sides of any right triangle that holds that angle.
Remember that the Sine of an angle is defined specifically, the ratio of the opposite side (the side across from the angle in the Sine function), and the hypotenuse (the side across from the right angle). You might remember this through SohCahToa
[tex]sin(\theta)=\frac{opp}{hyp}[/tex]
In our case, since [tex]sin(\theta)=\frac{1}{2}[/tex] , so [tex]\frac{opp}{hyp}=\frac{1}{2}[/tex] . While there are an infinite number of triangles that have that ratio of those sides, they are all "similar" triangles (corresponding angles congruent, and corresponding sides are proportional, yielding common ratios of sides), and for ease, we can consider simply the triangle where the value of the numerator is the length of the opposite side, and the value of the denominator is equal to the hypotenuse. So, [tex]opp=1[/tex], and [tex]hyp=2[/tex].
While we haven't actually talked about θ yet, we can still set up the triangle that has these sides so that we can visualize what the triangle looks like. (see image)
This triangle represents the triangle for the unknown θ in the original sine function. We're tasked with finding the secant of that particular unknown θ.
Working toward Secant
Here, it will be helpful to remember either the reciprocal identities for[tex]sec(\theta)=\frac{1}{cos(\theta)}[/tex], or the definition of the secant function [tex]sec(\theta)=\frac{hyp}{adj}[/tex].
I find that most people remember the reciprocal identities more easily than keeping track of the definitions, so, since secant is related to cosine, it will be important to remember that [tex]cos(\theta)=\frac{adj}{hyp}[/tex]. From there, take the reciprocal of the cosine-value to get the secant-value (which matches the definition of the secant function).
Either way, it comes down to knowing the lengths of the side adjacent to theta, and the hypotenuse. We already know the length of the hypotenuse, so we just need the length of the adjacent side.
Applying the Pythagorean Theorem
Fortunately, because it is a right triangle, the Pythagorean Theorem applies: [tex]a^{2} +b^{2} =c^{2}[/tex] (where c is the length of the hypotenuse, and a & b are the lengths of the legs)
Substituting the known values for the sides we do know...[tex](adj)^{2} +(1)^{2} =(2)^{2}\\(adj)^{2} +1 =4[/tex]
...isolating "adj" by subtraction...
[tex](adj)^2=4-1\\(adj)^2=3[/tex]
...applying the square root property...
[tex]adj=\sqrt{3}[/tex] or [tex]adj=-\sqrt{3}[/tex]
Identifying which Quadrant the triangle is in
Since we were given that [tex]0^o < \theta < 90^o[/tex], our triangle is an acute triangle (as drawn in the diagram), and is in quadrant I (indicating that both legs will be measured with a positive value.
Thus, we discard the negative solution and conclude that [tex]adj=\sqrt{3}[/tex].
Finding the final solution
From there, [tex]cos(\theta)=\frac{adj}{hyp}[/tex] implies [tex]cos(\theta)=\frac{\sqrt3}{2}[/tex], and through the reciprocal relationship (or simply the definition of secant, whichever is easier for you to remember), [tex]sec(\theta)=\frac{2}{\sqrt3}[/tex]
Note: This method did not require knowing what the angle θ was.
Alternative method using the Unit CircleIf you know well the values of special triangles in the unit circle, you may have identified that [tex]sin(\theta)=\frac{1}{2}[/tex] is associated with [tex]\theta=30^o[/tex]. If so, if you also recall that the ordered pair associated with that point on the unit circle is [tex](\frac{\sqrt3}{2} ,\frac{1}{2} )[/tex], and that the [tex]cos(\theta)=x\text{-coordinate on the unit circle}[/tex], then you can quickly identify that [tex]cos(\theta)=\frac{\sqrt3}{2}[/tex].
This method still ends the same: recalling the reciprocal relationship between cosine and secant, giving [tex]sec(\theta)=\frac{2}{\sqrt3}[/tex].
Related Questions
Click an item in the list or group of pictures at the bottom of the problem and, holding the button down, drag it into the correct position in the answer box. Release your mouse button when the item is place. If you change your mind, drag the item to the trashcan. Click the trashcan to clear all your answers. Indicate the equation of the given line in standard form. The line that contains the point Q( 1, -2) and is parallel to the line whose equation is y - 4 = 2/3 (x - 3)
Answers
The value of x, y, and z are 15/2, (15√3)/2, and (15√3)/2 after applying the trigonometric ratios.
What is trigonometry?Trigonometry is a branch of mathematics that deals with the relationship between sides and angles of a right-angle triangle.
The question is incomplete.
The complete question is in the picture, please refer to the attached picture.
To find the value of x:
In right-angle triangle x-z-15:
sin30 = x/15
x = 15sin30 = 15/2
To find the value of y:
sin60 = y/x = y/(15/2)
sin60 = 2y/15
√3/2 = 2y/15
y = (15√3)/2
To find the value of z:
cos30 = z/15
√3/2 = z/15
z = (15√3)/2
Thus, the value of x, y, and z are 15/2, (15√3)/2, and (15√3)/2 after applying the trigonometric ratios.
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Solve the following triangle. Round side measure to the nearest tenth and angle measure to the nearest degree:
Answers
Answer: [tex]AB=4.7, BC=8.8, \angle C=28^{\circ}[/tex]
Step-by-step explanation:
As angles in a triangle add to 180 degrees,
[tex]\angle C=180^{\circ}-90^{\circ}-62^{\circ}=\boxed{28^{\circ}}[/tex]
We know that:
[tex]\sin 62^{\circ}=\frac{BC}{10}\\\\BC=10\sin 62^{\circ} \approx \boxed{8.8}[/tex]
Similarly,
[tex]\cos 62^{\circ}=\frac{AB}{10}\\\\AB=10\cos 62^{\circ} \approx \boxed{4.7}[/tex]
Which algebraic expression is a trinomial?
O x³ + x²-√x
O 2x³-x²
○ 4x³ + x²-1/x
O x-x+√6
Answers
Answer:
D. x^6 - x + √6
Step-by-step explanation:
Last option:
⇒ x^6 - x + √6
Cannot be simplified further and has three terms.
Therefore is a trinomial.
The algebraic expression that is a trinomial is 2x³-x².
What is a polynomial?An expression that consists of variables, constants, and exponents that are combined using mathematical operations like addition, subtraction, multiplication, and division is referred to as a polynomial.
A trinomial is a polynomial that has three terms.
In the given options, the only expression that has three terms is "2x³-x²". It has the terms 2x³, -x², and 0x (which is often omitted).
The other options have different numbers of terms or they have terms with fractional or radical exponents, which do not fit the definition of a trinomial.
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2. Mike is looking to purchase a Ford Explorer, which has a sticker price of $32,415. He's able to negotiate $3,000
off sticker price, and the tax rate is 6.5%. License and title fees are $395. If he trades in his minivan for $9,000
(deducted from the negotiated price before taxes), the monthly payment will be $500.95 for 60 months. He is
also offered a lease with $2,900 down and payments of $429 for 36 months.
a) How much will Mike pay total if he chooses the lease option?
b) How much more will he pay in the first three years if he chooses to buy instead?
c) How much interest will he pay if he buys?
d) Which option would you choose? Explain.
3. Maya bought a new Ford Focus for $17,400. She made a down payment of $4,000 and made monthly payments
of $396 for three years. Find the APR.
Answers
Answer:
.wfhytrsssssssssd
Step-by-step explanation:
dddddddddddddddddd
look at the picture
Answers
The box plots below show attendance at a local movie theater and high school basketball games:
Two box plots are shown. The top one is labeled Movies. Minimum at 130, Q1 at 162, median at 185, Q3 at 195, maximum at 290. The bottom box plot is labeled Basketball games. Minimum at 85, Q1 at 170, median at 200, Q3 at 225, maximum at 230.
Which of the following best describes how to measure the spread of the data?
- The IQR is a better measure of spread for movies than it is for basketball games.
- The standard deviation is a better measure of spread for movies than it is for - basketball games.
- The IQR is the best measurement of spread for games and movies.
- The standard deviation is the best measurement of spread for games and movies.
Answers
The standard deviation is a better measure of spread for movies than it is for basketball games.
What is box- plot?A box plot is a graphical rendition of statistical data based on the minimum, first quartile, median, third quartile, and maximum.
As, Movies is more centric to mean and not skewed on a particular side whereas basketball games is not centric to mean but is skewed right then left.
Now, Standard deviation just measures the deviation or spread with respect to mean of collected data.
So, In the movies the data is more collected. Thus standard deviation is best suited for movie data.
Now, interquartile range. IQR relies on if the data is skewed.
So, in the basketball data the standard deviation will not give the information.
Hence, the standard deviation is a better measure of spread for movies than it is for basketball games.
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A tree is making a shadow 22 feet long. The angle of elevation from the tip of
the shadow to the top of the tree is 50°.
feet. Round your answer to the nearest
The height of the tree is about
foot.
Answers
Answer:
26.2 feet
Step-by-step explanation:
tan(50°) = h/22
h = 22 × tan(50°) = 26.2 feet
patulong po,thankyou
Answers
1.8 x 1.2 = 2.16
2.16 x 12.5 = 27
Because we just multiplied all three numbers by each other, we have identified our missing volume. The volume of the rectangular box is 27 cubic meters.
Your final answer: A.
What algebraic expression is a product with a factor of 5
Answers
Answer:
y=5x
Step-by-step explanation:
A product means multiplication, and you need to multiply x by 5 in order to make the product have a factor of 5.
Math Topic: Sum, Difference, and Product of two functions
Help with this!
Answers
The expression are; [tex](h - g) (x) = 3x^{3} -x^{2}[/tex], [tex](h + g) (x) = 3x^{3} +x^{2}[/tex] , [tex](h.g) (-1) = -3[/tex].
What is a function?The function is a type of relation, or rule, that maps one input to specific single output.
Given function;
g(x) = [tex]x^{2}[/tex]
h(x) = [tex]3x^{3}[/tex]
The expression;
[tex](h - g) (x) = 3x^{3} -x^{2}[/tex]
[tex](h + g) (x) = 3x^{3} +x^{2}[/tex]
[tex](h.g) (-1) = x^{2} .3x^{3} \\\\(h.g) (-1) = 3x^5\\\\(h.g) (-1) = 3(-1)^5\\\\(h.g) (-1) = -3[/tex]
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What is 3/4 divided by 1/2
Answers
Fraction: 3/2
Decimal: 1.5
Answer:
1 1/2
Step-by-step explanation:
Convert both fractions into decimals
⇒ 3/4 (0.75) and 1/2 (0.50)
Divide the decimals:
⇒0.75 ÷ 0.50
Find the quotient
⇒ 1.5
Convert the quotient to a fraction
⇒ 1 1/2
Terms:
Operations with fractions
the height of a plant in centimeters is modeled as a function of time, in days. consider this graph of the function enter the average rate of change for the height of the plant measured in centimeters per day between day 0 and day 20
Answers
The average rate of change for the height of the plant measured in centimeters per day between day 0 and day 20 is 1.32 cm per day.
What is average height?The average height is the ratio of change in height of the plant and the time taken taken for that change.
Given is the height of a plant in centimeters is modeled as a function of time, in days.
For day 0, the height is 18cm and the height for day 20 is 42 cm, then the average height change is
height per day = 42 - 18/ 20 -0
height per day = 1.2 cm/day
Thus, the average height of plant is 1.2 cm/day.
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If the sides of a square measure 9sqrt3 units, then find the length of the fishbowl.
Answers
The length of the diagonal of the square is 22.05 units
How to find the diagonal of a square?A square have all its sides equal to each other.
Therefore, using Pythagoras theorem,
c² = a² + b²
c² = (9√3)² + (9√3)²
c² = 81 × 3 + 81 × 3
c² = 243 + 243
c² = 486
square root both sides
c = √486
c = 22.045407685
c = 22.05 units
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-
Find the area of a rectangle with a length of 15 inches and a width of 9 inches. The area of the rectangle is
inches. Enter only the number.
The solution is
square
Answers
135inches2
Step-by-step explanation:
by multiplying both length and width you can get the sum
The price p in dollars that a store sets for an items is given by the equation p=c + 1/10c where c dollars is the stores cost for the items if the store sets a price of $55 for an ideal what is the cost for the item
Answers
The cost of the item whose price $ 55 is $ 50.
What is Cost Price?The cost price is the price paid by an individual to buy an item or any other commodity. Or this can also be taken as the total amount of money spent by the manufacturer to assemble a product or deliver a service.
Here, Price equation in terms of cost
P = C + C/10
P = (10C+C)/10
P = 11C/10
C = 10P /11
For P = $ 55, the cost will be
C = 10 X 55 / 11
C = 10 X 5
C = $ 50
Thus, the cost of the item whose price $ 55 is $ 50.
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La diferencia de las medidas de dos ángulos suplementarios es 20% determina el complemento del menor de dichos ángulos
Answers
Smaller angle = 50
Larger angle = 130
Let smaller angle be x degree
=> larger angle= [tex]3x - 20[/tex]
Since these angles are supplementary , So
[tex]= > x + 3x - 20 = 180\\= > 4x = 200\\= > x = 50\\= > 3x - 20\\ = > 150 - 20 = 130[/tex]
So smaller angle would be 50 deg & larger angle would be 130 deg.
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RATS is a parallelogram. M
Answers
Answer:
Answer:
x = 12
Step-by-step explanation:
m∠S + m∠T = 180
8x + 7x = 180
15x = 180
x = 12
does this help?
Solve 8x + c = k for x.
A. x=8(k-c)
OB. x=
_k+c
8
O C. x = 8(k+c)
O D. x=
k-c
8
Answers
Answer: [tex]x=\frac{c-k}{8}[/tex]
Step-by-step explanation:
[tex]1) \text { } 8x+c=k \text{ (given)}\\\\2) \text { }8x=c-k \text{ (subtract } k \text{ from both sides)}\\\\3) \text { } \boxed{x=\frac{c-k}{8}} \text{ (divide both sides by 8)}[/tex]
a die is tossed twice and the number of dots facing up in each toss is counted and noted in the order of occurrence. (a) find the sample space. (b) find the set a corresponding to the event “number of dots in first toss is not less than number of dots in second toss.” (c) find the set b corresponding to the event “number of dots in first toss is 6.” (d) does a imply b or does b imply a? (e) find and describe this event in words. (f) let c correspond to the event “number of dots in dice differs by 2.”
Answers
A die is tossed twice and a number of dots facing up in each toss are counted and noted in the order of occurrence.
a) The sample space is given as:
Sample Space (S)= {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5),(2,6),(3,1),(3,2),(3,3)(3,4),(3,5),(3,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(5,1)(5,2)(5,3),(5,4),(5,5),(5,6),(6,1),(6,2),(6,3),(6,4),c),(6,6)}.
What is the sample space?A sample space is a collection of a set of possible outcomes of a random experiment. The sample space is represented using the symbol S.
b) the set corresponding to the event number of dots in the first toss is not less than the number of dots in the second toss.
{(6,6),(6,5)(6,3)(6,2)(6,1)(6,4),(5,1)(5,2)(5,3),(5,4),(5,5),(4,1),(4,2),(4,3),(4,4),(3,1),(3,2), (2,1), (2,2),(1,1)}
c) find the set b corresponding to the event number of dots in the first toss is 6.
{(6,1),(6,2),(6,3),(6,4),c),(6,6)}.
e) a does not imply b or does b imply a.
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The table below shows the participation in the town soccer league during its first five seasons.
Town Recreational Soccer League Participation
Number of participants
88
95
100
108
115
Year
1
2
23
3
4
fo
5
Which is the equation of the trend line that can be generated by using the data from years 1 and 5?
Oy-6.75x-688.25
Oy-6.75x-1.75
O y-6.75x+81.25
Oy-6.75x+148.75
Answers
Using a calculator, the equation of the trend line that can be generated by using the data from years 1 and 5 is:
y = 6.75x + 81.25.
How to find the equation of linear regression using a calculator?To find the equation, we need to insert the points (x,y) in the calculator.
In this problem, the points, in the format (x,y), are given as follows:
(1,88), (2,95), (3,100), (4,108), (5,115).
Hence, using a calculator, the regression line is:
y = 6.75x + 81.25.
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Given the function f ( x ) = x 2 − 6 x + 10 , what is the average rate of change of the function over the interval [ 4 , 8 ] ?
Answers
The rate of change of the function over the interval [ 4, 8 ] will be 6.
What is the rate of change of the function?
The rate of change function represents the rate at which one quantity changes in relation to another. Simply put, the rate of change is the difference between the amount of change in one item and the comparable amount of change in another.
Given function:-
F(x) = x² - 6x + 10The formula for calculating the rate of change will be given as:-
[tex]ROC = \dfrac{F(x_2)-F(x_1)}{x_2-x_1}[/tex]
We have x₁ = 4 and x₂ = 8 given in the question
F(x₂) = x²₂ - 6x₂ + 10 = (8)² - ( 6 x 8 ) + 10 =
F(x₂) = 64 - 48 + 10
F(x₂) = 26
F(x₁) = x²₁ - 6x₁ + 10 = (4)² - ( 6 x 4 ) + 10 =
F(x₁) = 16 - 24 + 10
F(x₁) = 2
[tex]ROC = \dfrac{F(x_2)-F(x_1)}{x_2-x_1}[/tex]
[tex]ROC = \dfrac{26 -2}{8-4}=\dfrac{24}{4}=6[/tex]
Therefore the rate of change of the function over the interval [ 4, 8 ] will be 6.
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ƒ(x) = x² - x - 1
What is the average rate of change of f over
the interval -1 ≤ x ≤ 1?
Answers
Answer: -1
Step-by-step explanation:
[tex]f(-1)=(-1)^2 - (-1)-1=1\\\\f(1)=1^{2}-1-1=-1\\\\\therefore \frac{f(-1)-f(1)}{-1-1}=\frac{1-(-1)}{-1-1}=\frac{2}{-2}=\boxed{-1}[/tex]
Shauna solved the polynomial equations given in the table.
Determine whether each polynomial is correct. Select Correct or Incorrect for each equation.
Equation
-
(5a - 8) + (2a² + 2a − 1) = 2a² + 7a - 9
(6² +6b-4) (3b +6²) = 26² +36-4
(9c-4c²) - (2c² + c) = -6c² + 10c
-
Correct
Incorrect
Answers
Answer:
(5a - 8) + (2a² + 2a − 1) = 2a² + 7a - 9 - correct.
Step-by-step explanation:
(5a - 8) + (2a² + 2a − 1) = 2a² + 7a - 9 - correct;
(6² +6b-4) (3b +6²) = 26² +36-4 - incorrect (correct is: 18b²+...36²);
(9c-4c²) - (2c² + c) = -6c² + 10c - incorrect (correct is: -6c² + 8c).
Find the x-intercept of 2x+2y=2
Answers
Answer:
x-intercept: (1,0)
y-intercept: (0,1)
Step-by-step explanation:
Given: Find the x and y intercepts of 2x + 2y = 2
X-interceptTo solve for a x - intercept we have to substitue 0 in for y and solve for [tex]x[/tex]
Lets do it:
2x + 2(0) = 2
2x + 0 = 2
2x = 2
x = 1
Therefore, this means that when y = 0, x = 1 or (1,0)
Y-intercept
To solve for a y - intercept we have to substitue 0 in for x and solve for [tex]y[/tex]
Lets do it:
2(0) + 2y = 2
0 + 2y = 2
2y = 2
y = 1
Therefore, this means that when x = 0, y = 1 or (0,1)
PLEASE HELP WITH THIS PROBLEM!
Answers
Answer: Options 1 and 2
Step-by-step explanation:
To model exponential decay, the base of the exponential has to be less than 1.
Model exponential decay equation are
y = 0.34(0.98)^x
y = 8 (0.28)^x
Model exponential decay equation
Model exponential decay equation has the base value which is less than 1.Exponential decay is the process of reducing amount by consistent percentage rate over a time period. It can be expressed by the equation y=a(1-b)x where y is the final amount, a original amount, b decay factor, and x is the amount of time that has passed.Option C and D have the base value greater than 1. So it can't be model exponential decay equation.
Hence, the model exponential decay equation are
y = 0.34(0.98)^x
y = 8 (0.28)^x
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Expand each expression A ( x+1)^2 B(x-3)^2. C ( x+4)^2 D (x-1/2)^2
Answers
The expanded expressions are shown below:
(x + 1)² = x² + 2 · x + 1(x - 3)² = x² - 6 · x + 9(x + 4)² = x² + 8 · x + 16(x - 1/2)² = x² - x + 1/43 · (x - 5)² = 3 · (x² - 10 · x + 25) = 3 · x² - 30 · x + 75(1/2) · (x - 2)² = (1/2) · (x² - 4 · x + 4) = (1/2) · x² - 2 · x + 2How to expand perfect square trinomials
Perfect square trinomials are polynomials of grade 2 with the form (a + b)² = a² + 2 · a · b + b². In this case, we have to expand six perfect square trinomials:
a) (x + 1)² = x² + 2 · x + 1
b) (x - 3)² = x² - 6 · x + 9
c) (x + 4)² = x² + 8 · x + 16
d) (x - 1/2)² = x² - x + 1/4
e) 3 · (x - 5)² = 3 · (x² - 10 · x + 25) = 3 · x² - 30 · x + 75
f) (1/2) · (x - 2)² = (1/2) · (x² - 4 · x + 4) = (1/2) · x² - 2 · x + 2
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Create a real-life representation of the following multiplication problem. Explain what the result represents in the problem.
24(1/3) = 8
Answers
The real-life problem will be:- The total number of chocolates is 24 and there are 3 boxes what will be the number of chocolates filled in each box. The number of chocolates in each box will be 8.
What is an expression?Expression in maths is defined as the collection of the numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
The given expression is:-
[tex]24\times \dfrac{1}{3}[/tex] = 8
The real-life problem will be given the as:-The total number of chocolates is 24 and there are 3 boxes what will be the number of chocolates filled in each box. The number of chocolates in each box will be 8.
Hence the number of chocolates in each box will be 8.
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A cell phone towerbcast a shadow that is 40 feet long . An 10- foot- tall stop sign located near the tower casts a shadow that is 8 feet long. How tall is the cell phone tower?
Answers
Using proportions, considering the relation between the height and the shadow, it is found that the cell phone tower is 50 feet.
What is a proportion?A proportion is a fraction of a total amount, and the measures are related using a rule of three.
When the shadow is of 8 feet, the height is of 10 feet. What is the height when the shadow is of 40 feet? The rule of three is:
8 feet - 10 feet
40 feet - h feet
Applying cross multiplication:
8h = 10 x 40
Simplifying by 8:
h = 10 x 5 = 50 feet.
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A dairy farmer wants to make 45% protein supplement and a standard 15% protein ration to make 1800 pounds of high grade 35% protein ration how many pounds of each should be used.
Answers
1200 pounds of protein supplement and 600 pounds of standard protein is used.
Let a be the protein supplement and b be the standard protein.
Given that protein supplement and standard protein make 1800 pounds of high grade protein.
i.e., a+b = 1800 ------- (1)
45% a + 15% b = 35% 1800
0.45a + 0.15b = 1800 (3.5)
By (1), a = 1800 - b
Substituting, we get
0.45 (1800-b) + 0.15b = 630
810 - 0.45 b + 0.15b = 630
- 0.3 b = - 180
b = -180/-0.3
b = 600
Substituting b = 600 in (1),
we get
a = 1200.
Hence, 1200 pounds of protein supplement and 600 pounds of standard protein is used.
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Eli read 1 book in 5 months If he read constantly how many books did he read?
Answers
Answer:
(life span - age when he started reading)/5months =number of books he read
Step-by-step explanation:
(life span - age when he started reading)/5months =number of books he read