Given cosθ = 3/5, find the five other trigonometric function values. 15
Answers
[tex]cos(\theta )=\cfrac{\stackrel{adjacent}{3}}{\underset{hypotenuse}{5}}\qquad \impliedby \textit{let's now find the \underline{opposite side}} \\\\\\ \textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies \sqrt{c^2-a^2}=b \qquad \begin{cases} c=hypotenuse\\ a=adjacent\\ b=opposite\\ \end{cases} \\\\\\ \sqrt{5^2-3^2}=b\implies \sqrt{25-9}=b\implies \sqrt{16}=b\implies 4=b \\\\[-0.35em] ~\dotfill[/tex]
[tex]sin(\theta )=\cfrac{\stackrel{opposite}{4}}{\underset{hypotenuse}{5}}\qquad \qquad tan(\theta )=\cfrac{\stackrel{opposite}{4}}{\underset{adjacent}{3}}\qquad \qquad cot=\cfrac{\stackrel{adjacent}{3}}{\underset{opposite}{4}} \\\\\\ sec(\theta )=\cfrac{\stackrel{hypotenuse}{5}}{\underset{adjacent}{3}}\qquad \qquad csc(\theta )=\cfrac{\stackrel{hypotenuse}{5}}{\underset{opposite}{4}}[/tex]
The values of the five other trigonometric functions value for the same input θ for which cos(θ) = 3/5 are:
sin(θ) = 4/5tan(θ) = 3/4cot(θ) = 4/3sec(θ) = 5/3csc(θ) = 5/4What are the six trigonometric ratios?Trigonometric ratios for a right angled triangle are from the perspective of a particular non-right angle.
In a right angled triangle, two such angles are there which are not right angled(not of 90 degrees).
The slant side is called hypotenuse.
From the considered angle, the side opposite to it is called perpendicular, and the remaining side will be called base.
From that angle (suppose its measure is θ),
[tex]\sin(\theta) = \dfrac{\text{Length of perpendicular}}{\text{Length of Hypotenuse}}\\\cos(\theta) = \dfrac{\text{Length of Base }}{\text{Length of Hypotenuse}}\\\\\tan(\theta) = \dfrac{\text{Length of perpendicular}}{\text{Length of base}}\\\\\cot(\theta) = \dfrac{\text{Length of base}}{\text{Length of perpendicular}}\\\\\sec(\theta) = \dfrac{\text{Length of Hypotenuse}}{\text{Length of base}}\\\\\csc(\theta) = \dfrac{\text{Length of Hypotenuse}}{\text{Length of perpendicular}}\\[/tex]
What is Pythagoras Theorem?If ABC is a triangle with AC as the hypotenuse and angle B with 90 degrees then we have:
[tex]|AC|^2 = |AB|^2 + |BC|^2[/tex]
where |AB| = length of line segment AB. (AB and BC are rest of the two sides of that triangle ABC, AC being the hypotenuse).
We're given that:
cosθ = 3/5 for some angle θ
Since we've got:
[tex]\cos(\theta) = \dfrac{\text{Length of Base }}{\text{Length of Hypotenuse}}[/tex]
Therefore, we have:
[tex]\dfrac{3}{5} = \dfrac{\text{Length of Base }}{\text{Length of Hypotenuse}}[/tex]
Let we consider a right angled triangle in which there is hypotenuse of length 5 units and base (from the perspective of one of its non-right angle) of 3 units (as shown in the image attached below).
(we couldve taken 3x and 5x instead of 3 and 5, for any positive real number value of 'x' as when we would take their ratio, that common factor 'x' would get cancelled out. We can think of 3 and 5 as the special case of 3x and 5x when x = 1)
Then, from that perspective, let the perpendicular be of the length 'p' units, then as per the pythagoras theorem, we get:
[tex]p^2 + 3^2 = 5^2\\p = \sqrt{25 - 9} = \sqrt{16} = 4 \: \rm units[/tex]
(took only the positive root to remove the square term because the value of p denotes length, which is a non-negative quantity).
Thus, we have:
From the perspective of the angle θ:
Length of the base = 3 unitsLength of the perpendicular = 4 unitsLength of the hypotenuse = 5 units.Thus, using these values, and the definition of the trigonometric ratios, we get:
sin(θ) = 4/5tan(θ) = 3/4cot(θ) = 4/3sec(θ) = 5/3csc(θ) = 5/4Thus, the values of the five other trigonometric functions value for the same input θ for which cos(θ) = 3/5 are:
sin(θ) = 4/5tan(θ) = 3/4cot(θ) = 4/3sec(θ) = 5/3csc(θ) = 5/4Learn more about Pythagoras theorem here:
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Related Questions
(15 points )
Find the x in the problem
(x−1)^2=9
Show steps if you can :)
Answers
Answer:
x=4 or x= -2
Step-by-step explanation:
(x-1)(x-1)
x(x-1)-1(x-1)
x^2-x-x+1
x^2-2x+1=9
x^2-2x+1-9=0
x^2-2x-8=0
p= -8
s= -2
1------ -8
2------ -4
x^2+2x-4x-8=0
x(x+2)-4(x+2)
(x+2)(x-4)=0
x+2=0
x-4=0
x= -2
or
x=4
4 qt =
С
Show me the steps please
Answers
Answer:
cups =15.99994. according to the holy bible.
Step-by-step explanation:
What is the factorization of the following trinomial? 15x2 - 44x - 36
(3 x - 4) (5 x + 9)
(5 x - 9)(3 x +4)
(5 x -18) (3 x + 2)
(5 x + 18 ) (3 x - 2 )
Answers
Evaluate f(2) for the function f(x)=5(x-3)+17
Answers
Hello!
Plug in the value of x:
f(x)=5(x-3)+17
f of x is 5 times the difference of x and 3 plus 17
plug in 2:
f of 2 is 5 times the difference of 2 and 3 plus 17
f(2)=5(2-3)+17
f(2)=5(-1)+17
f(2)=-5+17
f(2)=12
[tex]\huge\boxed{\mathfrak{Answer:{\boxed{\rm{f(2)=12\star\star}}}}}[/tex]
Hope everything's clear.
Let me know if you have any questions!
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an architect is designibg a tapered office tower where rge ground floor us the largwst amd the floor space is reduced by 6.3% permifloor if the ground gloor has an arra pf 11,400,0ft2 ehat is tje area of the floor to the nearest tenth of a ft2?
Answers
The area of the ground floor after it has been reduced to the nearest tenth is 10,681,800 square foot.
What is the area of the ground floor?In order to determine the area of the ground floor after it has been reduced, the area of the floor has to reduced by 6.3%.
Area that was reduced = 6.3% x 11,400,000 = 718,200
New area = 11,400,000 - 718,200 = 10,681,800 square foot
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What is the perimeter of the polygon shown below?
A. 24 m
B. 45 m
C. 51 m
D. 57 m
Answers
Answer:
c no
jskdha
shdjszdj
see
What is the area of the following circle?
Answers
Answer:
34.54
Step-by-step explanation:
you do 2 * 3.14 * 5.
to find the area 2pie r the 3.14 comes from the directions, the 2 comes from the circle because the full length of the circle is 10 but you divide by 2 to get 5 and the 5 also comes from the circle and that is the radius.
I understand I need help on this.
Answers
Answer:
A. 168 units³
Step-by-step explanation:
volume = area of bas × height
volume = (6)(7)/2 × 8
volume = 168 units³
PLEASE HELP ME I CANT GET IT RIGHT
Answers
2,7, 12, ...
Find the 46th term.
Answers
Given is an A.P. in which -
First term,a = 2Common difference = Second term - First term[tex]\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad = a_2-a_1[/tex]
[tex]\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad = 7-2 [/tex]
[tex]\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad = 5[/tex]
Solution:-nth term of an A.P ,is given by-
[tex]\green{ \underline { \boxed{ \sf{a_n =a+(n-1)d }}}}[/tex]
where
[tex]a_n = nth \;term[/tex]a = first termn = number of termd = common differencePutting n = 46
[tex]\begin{gathered}\\\implies \sf a_{46}= 2 + (46-1)\times 5 \\\end{gathered} [/tex]
[tex]\begin{gathered}\\\implies \sf a_{46}= 2 + 45\times 5 \\\end{gathered} [/tex]
[tex]\begin{gathered}\\\implies \sf a_{46}= 2 + 225 \\\end{gathered} [/tex]
[tex]\begin{gathered}\\\implies \sf a_{46}= 227 \\\end{gathered} [/tex]
[tex]\longrightarrow[/tex]Therefore, the 46th term is 227
Hence, the [tex]46th[/tex] term is [tex]227[/tex].
What is the sequence?
A sequence is a list of objects, typically numbers, in which order matters, repetition is allowed, and the same elements can appear multiple times at different positions in the sequence.
Here given sequence is
[tex]2,7, 12,...[/tex]
Here,
[tex]a=2[/tex]
[tex]d=a_2-a_1\\d=7-2\\d=5[/tex]
The general formula of the sequence is
[tex]a_n=a+(n-1)d[/tex]
where,
[tex]a_n=[/tex] nth term
[tex]a[/tex] is the first term
[tex]d[/tex] is the common difference
[tex]n[/tex] is the number of terms
Now substituting the values in general formula is
[tex]a_{46}=2+(46-1)5\\\\a_{46}=2+(45)5\\\\a_{46}=2+225\\\\a_{46}=227\\[/tex]
Hence, the [tex]46th[/tex] term is [tex]227[/tex].
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The area of a circle is 144 in 2. What is the circumference, in inches? Express you answer in terms of pi.
Answers
Answer:
24π inches
Step-by-step explanation:
The area of a circle is denoted by: , where r is the radius. Here, we know that the area is A = , so we can plug this in to find r:
A = [tex]\pi r^{2}[/tex]
[tex]144\pi = \pi r^{2}[/tex]
[tex]r^{2}[/tex] = 144
r = 12
The circumference is denoted by: , where r is the radius. We've found the radius so just plug 12 in for r:
Thus, the circumference is 24π inches.
Hope this helps!
Correct Question :
The area of a circle is 144π in². What is the circumference, in inches? Express your answer in terms of pi.
[tex] \: [/tex]
Step-by-step explanation:
Before finding the circumference of the circle, we have to find the radius of it.
We already know that,
[tex] \longrightarrow \qquad{ \underline {\boxed{ \pmb{ \mathfrak{ \pi r{}^{2} = \: \: Area_{(circle)}}}}}} \: \: \bigstar \\ \\ [/tex]
So, using this formula we'll find the radius and now we'll substitute the given values :
[tex] \longrightarrow \qquad{ \sf{ \pi \times r{}^{2} = \: 144 \pi }} \\ \\[/tex]
Cancelling π from both sides we get :
[tex]\longrightarrow \qquad{ \sf{ \cancel\pi \times r{}^{2} = \: 144 \cancel\pi }} \\ \\[/tex]
[tex]\longrightarrow \qquad{ \sf{ r{}^{2} = \: 144 }} \\ \\ [/tex]
[tex]\longrightarrow \qquad{ \sf{ r{}^{} = \: \sqrt{144} }} \\ \\ [/tex]
[tex]\longrightarrow \qquad{ \sf{ \pmb{ r{}^{} = \: 12 }}} \\ \\ [/tex]
So,
The radius of the circle is 12 inches.[tex] \\ [/tex]
Now we'll find the circumference of the circle :
[tex] \\{ \longrightarrow \qquad{ \underline {\boxed{ \pmb{ \mathfrak{ \: \: Circumference_{(circle)} = 2 \pi r}}}}}} \: \: \bigstar \\ \\ [/tex]
Now, we'll substitute the required values in the formula :
[tex] \\ { \longrightarrow \qquad{ \sf{ \: \: Circumference_{(circle)} = 2 \times \pi \times 12}}} \: \: \\ \\ [/tex]
[tex]{ \longrightarrow \qquad{ \sf{ \: \: Circumference_{(circle)} = 2 \times 12 \pi }}} \: \: \\ \\ [/tex]
[tex]{ \longrightarrow \qquad{ \sf{ \pmb{ \: \: Circumference_{(circle)} = 24 \pi }}}} \: \: \\ \\ [/tex]
Therefore,
The circumference of the circle is 24π inches 1. Which best describes your ability to solve quadratic equations?
Answers
Answer:
you just have to do it yourself
Please help me!!!!!!!
Answers
Ay, help?
What is the measure of angle P?
Enter your answer as a decimal in the box. Round only your final answer to the nearest hundredth.
m∠P=
°
Answers
Answer:
P=67.38
Step-by-step explanation:
All three sides are given. You can use any one of the the three major functions on your calculator to g
Adjacent = PQ = 5
Hypotenuse = PR = 13
Cos(P) = Adjacent / Hypotenuse
Cos(P) = 5/13
Cos(P) = .3846
Now you need to use the inverse cos to get P
P = cos-1(.3846)
P =67.38
=====================================
Now we'll try the tangent
Tan(P) = opposite / adjacent
Tan(P) = 12/5
Tan(P) =2.4
P = Tan-1(2.4)
P = 67.38
Both functions give the same thing
[tex]\\ \rm\Rrightarrow cosP=\dfrac{Base}{Hypotenuse}[/tex]
[tex]\\ \rm\Rrightarrow cosP=\dfrac{5}{13}[/tex]
[tex]\\ \rm\Rrightarrow cosP=0.38[/tex]
[tex]\\ \rm\Rrightarrow P=cos^{-1}(0.38)[/tex]
[tex]\\ \rm\Rrightarrow P=67.67°[/tex]
HELP PLS I CAnt DO THIS
Answers
Using the two given solutions and the general absolute value function we will get:
|x + 9| = 27
Which absolute value function has that solution set?The two solutions are:
x = 0 and x = -18.
Remember that the absolute value equation is something like:
|x - a| = b
If we want to have these solutions, then:
|0 - a| = b
|-18 -a| = b
From the first one, we have:
|-a| = b
Replacing that on the second equation we get:
|-18 -a| = |-a|
Notice that if we take a = -9, then we have:
|-18 + 9| = |+9|
|-9| = |+9|
This is true, so a = -9.
Then we find the value of b:
|18 + 9| = b = 27
Then the absolute value equation is:
|x + 9| = 27
If you want to learn more about absolute value equations, you can read.
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An hour before show time, only 228 people are seated for a musical performance. according to ticket sales, 94% of the people have yet to arrive. how many tickets were sold for the musical performance? explain your thinking.
Answers
Answer:
3800 tickets sold
Step-by-step explanation:
We know the number of people seated, and we know the percentage of total ticket sales that represents. So, we can find the total number of tickets sold.
seated + not-arrived = total sold
228 + 0.94 × total sold = total sold
228 = 0.06 × total sold . . . . . . . . . . . . subtract 0.94 × total sold
228/0.06 = total sold = 3800
There were 3800 tickets sold for the performance.
1. Find the length of the arc shown in red:
2. Find the area of the circle:
Answers
Answer:
Step-by-step explanation:
If b=4 units and h=5 units, the area of the triangle is _____
units2.
Answers
Answer:
Step-by-step explanation:
Area = 1/2 b h
= 1/2 * 4 * 5
= 10 unit^2.
1.3 1.5 2.2 2.2 2.8
What is the mean median mode range
Answers
Step-by-step explanation:
mean = sum of all data points / number of data points
(1.3+1.5+2.2+2.2+2.8)/5 = 10/5 = 2
median is the number where half of the other data points are smaller or equal, and the other half are larger or equal.
median = 2.2
mode is the data point occurring most often = 2.2
range is the difference between largest and smallest data point : 2.8 - 1.3 = 1.5
Solve each pair of simultaneous equations.
A)4x+y=9
2x -y =3
B) x-y = 1
10x+y=21
C) 6x - 2y =4
3x + 2y =-4
D) 2x - 3y = -6
-2 +4y =8
E) x-5y = -32
2x + 5y =26
Answers
Answer:
I solved the first question. Hope this helped you and can solve other by yourself.
Given the following geometric sequence, find the 12th term: {2, -4, 8, ...}.
-4096
2048
4096
-2048
Answers
A geometric sequence has a common ratio.
The formula for the nth term is
[tex]\sf{a_n =ar^{n-1} }[/tex]
where,
an = nth term of the sequence,a = first term of the sequence and r = common ratioGiven -
A geometric sequence 2, -4, 8, ...To find -
the 12th term of the given geometric sequenceSolution -
A.T.Q, a = 2 and r = -2
[tex]\rightarrow\sf{a_{12} =2(-2)^{12-1} }[/tex]
[tex]\rightarrow\sf{a_{12 }=2(-2)^{11} }[/tex]
[tex]\rightarrow\sf{a_{12} =2×(-2048) }[/tex]
[tex]\rightarrow\bf{a_{12 }=-4096 }[/tex]
Hence, the 12th term is -4096.
Liliana is making a vase with a circular base. She wants the area of the base to be between 135 cm2 and 155 cm2. Which circle could represent the base of the vase? Use 3. 14 for Pi.
Answers
The area of the vase lies between the area [tex]135\ cm^2[/tex] and [tex]155\ cm^2[/tex] will be [tex]153.86\ cm^2[/tex].
What will be the area?As we know that the area of the circle is given by the formula
[tex]A=\pi r^2[/tex]
So now from the question, we will find the radius for both the given areas.
Radius for the first area
[tex]r=\dfrac{A}{\pi} =\dfrac{135}{\pi} =6.56\ cm[/tex]
Radius for the second area
[tex]r=\dfrac{A}{\pi} =\dfrac{155}{\pi} =7.03\ cm[/tex]
Now we can see the range of the radius lies
[tex]6.56\leq r\leq 7.03[/tex]
The radius must be r=7 cm lie between two quantities.
Then the area at r= 7cm will be
[tex]A=\pi r^2[/tex]
[tex]A=\pi\times 7^2=153.86\ cm^2[/tex]
Thus the area for the vase will be [tex]A=153.86\ cm^2[/tex] ar radius r= 7 cm
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PLEASE HELP ME I CANT GET IT RIGHT
Answers
Answer:
Area=½b×h
Area =½×20×8=80m²
=80m²hope it helps!
In a trapezoid the sum of the bases is 28 and the area is 84.
What is the altitude or height of the trapezoid?
Answers
Answer: If you sketch this out, you should be able to convince yourself that if you drew a line parallel to the bases and halfway between them, and a vertical at the end of that line, there would be an extra triangle on the longer base that would just fit into the space at the end of the shorter base, if you cut and pasted it.
You should also be able to convince yourself by what you know about similarity that the length of that parallel halfway line is just halfway between the lengths of the bases (you can add them and divide by two).
So your trapezoid (trapezium, we call ’em this side of the pond) has the same area as a rectangle with an altitude equal to the trapezoid’s and a width equal to the sum of those bases divided by two. And since you know about rectangles, you’re home and dry. I suggest you do the sketch, fill in the numbers, and then you’ve completed a model piece of homework that should earn full marks and the teacher’s approval.
Step-by-step explanation:
When three or more numbers are added, the sum is the same regardless of the grouping. Which property is this Associative, Commutative, or Distributive property
Answers
Answer:
Associative property
Step-by-step explanation:
have fun
HELPPP ILL GIVE BRAINLIST ALSO Interpret each plotted point in the graph.
Answers
(1,1.5) traveling 1.5 inches in 1 second
(3,4.5) traveling 4.5 inches in 3 seconds
Harold deposited $70 in an account earning 5% interest compounded annually. To the nearest cent, how much interest will he earn in 2 years? Use the formula B=p(1+r)t, where B is the balance (final amount), p is the principal (starting amount), r is the interest rate expressed as a decimal, and t is the time in years.
Answers
Answer:
10.50
Step-by-step explanation:Harold deposited $70 in an account earning 5% interest compounded annually. To the nearest cent, how much interest will he earn in 2 years? Use the formula B=p(1+r)t, where B is the balance (final amount), p is the principal (starting amount), r is the interest rate expressed as a decimal, and t is the time in years.
15 of 15 Find the size of the final unknown interior angle in a polygon whose other interior angles are: 162°, 115°, 125°, 148° and 105°.
answer ?
Answers
Answer: 147 Degrees
Step-by-step explanation
The size of the final unknown interior angle in a polygon is 147 degrees.
Given that,
The other interior angles are 162°, 115°, 120°, 148° and 85°.
We assume that there is an equation (2n - 4) 90.
Here n be 7.
Based on the above information, the calculation is as follows:
= (2n - 4) 90
= ((2) (7) - 4) 90
= 10 (90)
= 900
Now the size of the unknown interior angle is
= 900-(162+125+148+105+98+115)
= 900 - 753
= 147°
Therefore we can conclude that the size of the final unknown interior angle in a polygon is 147 degrees.
What is the slope of the line that passes through the points(-5,-9) and (-5,-13)? Write your answer in simplest form.
Answers
Answer:
slope is undefined
ex. a straight vertical line like x = 2 would have an undefined slope
Step-by-step explanation:
formula is
y2-y1 / x2-x1
-13 + 9 / -5 + 5
-4 / 0
slope is undefined
straight vertical line like x = 2 would have an undefined slope
undefined slope is the slope of any vertical line that goes up or down. There is no horizontal movement and hence the denominator is zero while calculating the slope. Thus the slope of the line is undefined.
cuemathcomgeometryundefinedslope
mathbootcampscom
the slope of a line passing through two points with the same x-coordinate, like (-5, -9) and (-5, -13), is undefined because the line is vertical and doesn't have a defined slope.
Slope (m) = (change in y) / (change in x)
In this case, you have two points: (-5, -9) and (-5, -13). The x-coordinates are the same (-5) for both points, which means there is no change in the x-coordinate. Therefore, the denominator in the slope formula becomes zero:
Slope (m) = (change in y) / (0)
When the denominator is zero, division is undefined in mathematics. This corresponds to a vertical line on a graph. A vertical line doesn't have a well-defined slope because it doesn't have a constant rate of change in the x-direction since the x-values don't change. Instead, the vertical line runs parallel to the y-axis.
So, the slope of a line passing through two points with the same x-coordinate, like (-5, -9) and (-5, -13), is undefined because the line is vertical and doesn't have a defined slope.
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I think of a number. I multiply it by 5. I then subtract 19. Finally, I double the resu The final number is 22. What number did think of?
Answers
Answer:
8.2 is your answer.
Step-by-step explanation:
first you would add 22 and 19 together to get 41
then you divide by 5 to get your answer 8.2
22+19=41÷5=8.2