solve the trig equation by factoring. step by step please
sec^2(x)-sec(x)=2
Answers
\(\sec^2(x)-\sec(x)=2\implies [\sec(x)]^2-\sec(x)=2 \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{let's first just a few make}}{\sec(x)=Z} \\\\\\ Z^2-Z=2\implies Z^2-Z-2=0 \implies (Z+1)(Z-2)=0 \\\\[-0.35em] ~\dotfill\)
\(Z+1=0\implies \sec(x)+1=0\implies \sec(x)=-1\implies \cfrac{1}{\cos(x)}=-1 \\\\\\ \cfrac{1}{-1}=\cos(x)\implies -1=\cos(x)\implies \cos^{-1}(-1)=x\implies \boxed{\pi =x} \\\\[-0.35em] ~\dotfill\\\\ Z-2=0\implies \sec(x)-2=0\implies \sec(x)=2\implies \cfrac{1}{\cos(x)}=2 \\\\\\ \cfrac{1}{2}=\cos(x)\implies \cos^{-1}\left( \cfrac{1}{2} \right)=x\implies \boxed{\cfrac{\pi }{3}~~,~~\cfrac{5\pi }{3}=x}\)
Related Questions
A sailor is 30m above the water in the crow's nest on a sailboat. The sailor encounters an orca surface at an angle of depression of 15 degrees. The crows nest is 20 m horizontally from the bow (front) of the boat. How far in front of the boat is the orca?
Answers
Given :
A sailor is 30 m above the water in the crow's nest on a sailboat.
The sailor encounters an orca surface at an angle of depression of 15 degrees.
The crows nest is 20 m horizontally from the bow (front) of the boat.
To Find :
How far in front of the boat is the orca.
Solution :
Let, distance of boat front from the crow's nest is x.
So,
\(\dfrac{30}{20+x}=tan \ 15^{\circ}\\\\x=\dfrac{30}{tan \ 15^{\circ}}-20\\\\x=111.94-20\\\\x=91.94\ m\)
Hence, this is the required solution.
The figure below is made up of a square with height, h units, and a right triangle with height, h units, and base length, b units. The area of this figure is 80 square units. Write an equation that solves for the height, h, in terms of b. Show all work necessary to justify your answer.
Answers
Answer:
Step-by-step explanation:
Sol'n,
Here,
The length of all sides of sq= height = h
Height of triangle=h
Base length of triangle=b
Now, We know that,
The entire figure is a trapezium,
so, Area of Trap.= 1/2 * h(length of diagonal one + length of diagonal 2)
or, 80 = 1/2 * h* {h +(b+h)}[ since here, the length of second diagonal is sum of the base and length of one side of sq]
or, 160 = h (2h+b)
2h^2 + hb - 160 = 0....(I)
Hence, I is the required eqn....
The Equation for Area of figure is area of Figure, h² + 1/2(h)(b) = 80
What is Area?The measurement that expresses the size of a region on a plane or curved surface is called area. Surface area refers to the area of an open surface or the boundary of a three-dimensional object, whereas the area of a plane region or plane area refers to the area of a form or planar lamina.
Given:
We have the figure consist of one square and one right triangle.
Now, Area of Figure,
= Area of square + area of Triangle
\(\dfrac{= \text{length} \times \text{width +}}{\times \text{base} \times \text{height}} =\)
\(= 16 \times 12 + \dfrac{1}{2} \times 10 \times 20\)
\(= 192+ 100\)
\(=292 \ \text{unit}^2\)
and, if the square with height, h, units and a right triangle with height, h units, and a base length, b units.
Then, area of Figure = h² + 1/2(h)(b) = 80 square units.
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The schizophrenia scale on a widely used personality scale is standardized to have a mean of 50 for a national group of normal adults. You administer the scale to a random sample of 20 students from a large university and obtain the following results:
52 58 54 60 46 56 54 55 62 50
52 66 64 54 62 60 58 56 66 65
Answers
The terms used are mean, standard deviation, and z-score. Here's a step-by-step explanation:
1. Calculate the sample mean:
(52+58+54+60+46+56+54+55+62+50+52+66+64+54+62+60+58+56+66+65) / 20 = 1146 / 20 = 57.3
2. Calculate the sample standard deviation:
a. Find the squared difference of each score from the sample mean:
[(52-57.3)^2 + (58-57.3)^2 + ... + (66-57.3)^2 + (65-57.3)^2] / 19 = 984.7 / 19 = 51.826
b. Take the square root of the result: √51.826 = 7.2 (rounded to one decimal place)
3. Calculate the z-score for each student:
a. Subtract the national mean (50) from the sample mean (57.3) and divide the result by the sample standard deviation (7.2): (57.3-50) / 7.2 = 1.0139 (rounded to four decimal places)
The z-score for this sample of 20 students is 1.0139. This indicates that, on average, the students in this sample scored about 1.0139 standard deviations above the national mean of 50 for normal adults on the schizophrenia scale.
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The original price for a piece of machinery is $150,000. If the piece of machinerydecreases in value by 22.5% each year, which graph models the value of the piece of machinery after x years?
Answers
The correct graph that models this exponential decay would show a decreasing trend as x (the number of years) increases. we need to consider the information given regarding its decrease in value.
The piece of machinery decreases in value by 22.5% each year. This means that its value after one year is \(100\% - 22.5\% = 77.5\%\) of its original value. In other words, its value is 0.775 times its original value.
To model the value of the machinery after x years, we need to consider the exponential decay function. The general form of an exponential decay function is:
y = \(a(1 - r)^x\)
Where:
- y represents the value after x years.
- a represents the initial value.
- r represents the decay rate (expressed as a decimal).
In this case, the initial value (a) is $150,000 and the decay rate (r) is 22.5% or 0.225.
Therefore, the equation representing the value of the piece of machinery after x years would be:
y = \(\$150,000 * (1 - 0.225)^x\)
Simplifying the equation further, we have:
y = \(\$150,000 * (0.775)^x\)
From this equation, we can see that the value of the machinery decreases exponentially over time. The correct graph that models this exponential decay would show a decreasing trend as x (the number of years) increases.
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Consider the region in the first quadrant bounded by the x-axis and the graph of the function f(x) = 4 - x². Find the volume of the solid of revolution formed by revolving the region about the x-axis.
Answers
The volume of solid of revolution which is formed by revolving the region about the x-axis as per given conditions is equal to 14π cubic units.
To find the volume of the solid of revolution formed by revolving the region bounded by the x-axis,
and the graph of the function f(x) = 4 - x² about the x-axis,
Use the method of cylindrical shells.
The volume of the solid of revolution can be calculated by integrating the area of each cylindrical shell.
The height of each cylindrical shell will be given by the function f(x) = 4 - x²,
and the radius will be x (the distance from the x-axis).
The differential thickness of each shell will be dx.
The volume of each cylindrical shell can be expressed as,
dV = 2πx × f(x) × dx
To find the total volume,
Integrate this expression over the range of x-values that define the region in the first quadrant.
Since the graph of f(x) = 4 - x² is symmetric about the y-axis,
Integrate from 0 to the x-value where f(x) = 0.
To find this x-value, set 4 - x² = 0 and solve for x,
4 - x² = 0
⇒x² = 4
⇒x = ±2
Since considering the region in the first quadrant, take the positive value, x = 2.
The integral for the volume becomes,
V = ∫₀² 2πx × (4 - x²) dx
Simplifying the integrand,
V = 2π ∫₀² (8x - x³) dx
Evaluating the integral,
⇒V = 2π [4x²/2 - x⁴/4] |[0 to 2]
⇒V = 2π [2x² - x⁴/4] |[0 to 2]
⇒V = 2π [(2(2)² - (2)⁴/4) - (2(0)² - (0)⁴/4)]
⇒V = 2π [(8 - 4/4) - (0 - 0/4)]
⇒V = 2π (8 - 1)
⇒V = 14π
Therefore, the volume of the solid of revolution formed by revolving the region bounded by the x-axis
and the graph of the function f(x) = 4 - x² about the x-axis is 14π cubic units.
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Pamela and her friend Nicole are each baking apple pies and tarts for a bake sale, using the same recipes. Pamela baked 5 apple pies and 7 apple tarts, using a total of 68 apples. Nicole made 5 apple pies and 9 apple tarts, which used 76 apples. How many apples does each dessert require?
Answers
Answer:
Each apple pie requires 8 apples, and each apple tart requires 4 apples.
Step-by-step explanation:
We see that both Pamela and Nicole bake the same amount of apple pies, but different amounts of apple tarts. Because of this, we can subtract the two to try to figure out the amount of apples for each apple tart. We subtract 68 from 76, giving us 8. Nicole baked 9 apple tarts, while Pamela baked 7, and 9-7=2. So we can bake two apple tarts with 8 apples, so one apple tart requires 4 apples (we divide by 2). Now that we know the amount of apples per each apple tart, we multiply 7 apple tarts that Pamela made by 4 apples, giving us 28. We subtract that from the total amount of apples Pamela used, which was 68, giving us 40. From this we can deduct that 5 apple pies need 40 apples, and we divide by 5, giving us 1 apple pie requires 8 apples.
8. Gianna is making bows for her cheerleading squad. She has 4 yards of ribbon to make the
bows. If each bow requires 1/5 a yard of ribbon, will she have enough ribbon to make 18 bows?
Answers
Using the concept of proportion, Gianna will have enough yards to make 18 bows
Will she have enough ribbon to make 18 bowsTo solve this problem, we need to apply the concept of proportions.
In order to make 1 bow, she needs 1/5 yard of ribbon
Representing this in an equation form will be;
1 bow = 1/5 yard
18 bow = x yard
cross multiply both sides and solve for x
x * 1 = 18 * (1/5)
x = 3.6 yards
From the above, there will be enough yards to make 18 bows.
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juan organizes the stamps in his collection by country and by the decade in which they were issued. the prices he paid for them at a stamp shop were: brazil and france, $6$ cents each, peru $4$ cents each, and spain $5$ cents each. (brazil and peru are south american countries and france and spain are in europe.)in dollars and cents, how much did his south american stamps issued before the $70\text{'s}$ cost him?
Answers
The cost of Juan's South American stamps issued before the '70s is $10.
To determine the cost of Juan's South American stamps issued before the '70s, we need to consider the prices for stamps from Brazil and Peru, which are both South American countries. According to the information provided, stamps from Brazil and Peru cost 6 cents each.
Since the specific quantity of stamps is not given, we assume that Juan has an equal number of stamps from Brazil and Peru. Therefore, the cost of his South American stamps from these countries can be calculated as follows:
Cost of stamps from Brazil = 6 cents each
Cost of stamps from Peru = 6 cents each
Total cost of South American stamps = (Cost of stamps from Brazil + Cost of stamps from Peru) × Quantity
As the quantity is unknown, we cannot calculate the exact cost. However, the given answer options provide the closest approximation.
Looking at the answer options, the only option that falls within the range of the given prices is $10. Therefore, the cost of Juan's South American stamps issued before the '70s is $10.
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At the beginning of the week, a puppy weighed 514 pounds. During the week, the puppy gains 8 ounces.
How much does the puppy weigh, in pounds, at the end of the week?
Enter your answer as a mixed number in simplest form by filling in the boxes
Answers
♣ explanation ♣
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Add 8 ounces onto 5 to get 13, and then, add the unit fraction 1/4 onto it and you get 13 1/4 as your non-simplified mixed number answer.
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But wait! The answer must be simplified into the mixed number, "5 3/4" by dividing the current answer by 8 1/2, or any other number that involves similarities to the fractions' characteristics.
Consider the following rational expression: 4y + 16 y+ 4 Step 2 of 2: Find the restricted values of y, if any, for the given rational expression Answer How to enter your answer (opens in new window) 2
Answers
The given rational expression is 4y + 16 y + 4. To find the restricted values of y, we need to identify any values of y that would make the expression undefined.
In this case, the expression is in the form of a sum, so we don't have any denominators that could lead to division by zero. Therefore, there are no restricted values of y for this rational expression.
The expression 4y + 16 y + 4 is defined for all real numbers. We can evaluate it for any value of y without encountering any restrictions.
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*look at the image*
whoever answers first will be marked brainliest
Answers
*looks at the image woooow
He expression 1 ÷ (4 × −4 × 4 × −4 × 4) is equivalent to (14
× −14
× 14
× −14 ×
14
)
Answers
The expression 1 ÷ (4 × -4 × 4 × -4 × 4) is not equivalent to (14 × -14 × 14 × -14 × 14). The simplified value of the given expression is 1/1024, whereas the value of the second expression is 537,824.
To evaluate the given expression, we can simplify the factors in the denominator first:
4 × -4 = -16
-16 × 4 = -64
-64 × -4 = 256
256 × 4 = 1024
Now we can substitute these values into the original expression:
1 ÷ (1024) = 1/1024
We can simplify the expression on the right-hand side by factoring out 14 and -14:
14 × -14 × 14 × -14 × 14 = (14 × -14) × (14 × -14) × 14
= (-196) × (-196) × 14
= 38416 × 14
= 537,824
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Hi i don’t know how to answer #4 and #5. I’m in college calculus 1.
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Hence,
\(\begin{gathered} \tan ^2\alpha+1=\sec ^2\alpha \\ \text{Substitute tan}\alpha=-3\text{ into the eqaution above},\text{ we have } \\ (-3)^2+1=\sec ^2\alpha \\ 9+1=\sec ^2\alpha \\ 10=\sec ^2\alpha \\ \sec ^2\alpha=10 \\ \sec \alpha=\sqrt[]{10} \\ \text{but sec}\alpha=\frac{1}{\cos \alpha} \\ \text{hence, }\frac{1}{\cos\alpha}=\sqrt[]{10} \\ \cos \alpha=\frac{1}{\sqrt[]{10}} \end{gathered}\)\(\begin{gathered} \text{But tan}\alpha=\frac{\sin \alpha}{\cos \alpha} \\ \sin \alpha=\tan \alpha\cos \alpha \\ \sin \alpha=-3\text{ x }\frac{1}{\sqrt[]{10}} \\ \sin \alpha=-\frac{3}{\sqrt[]{10}} \\ \\ Hence,\text{ }\sin \alpha=-\frac{3}{\sqrt[]{10}} \end{gathered}\)Which of the following is equivalent to (16^3/2)^1/2
Answers
Answer:
8
Step-by-step explanation:
(16^3/2)^1/2
We know a^b^c = a^(b*c)
16 ^(3/2*1/2)
16 ^3/4
Rewriting 16 as 2^4
2^4^3/4
2^(4*3/4)
2^3
8
1. How can you find the decimal forms of 3/12 and 2/9 .
Answers
Answer:
3/12 as a decimal is 0.25
2/9 as a decimal is 0.2 reapeted
Answer:
Answer written below
Step-by-step explanation:
Divide 3 by 12 and 2 by 9 using long division solve like normal long division if your divisor is bigger than your dividend add a zero on top and add a zero to the dividend.
Decimal form of 3/12:0.25
Decimal form of 2/9:0.22223
you use the parallelogram-shaped sponge to create the T-shirt design. The area of the design is 66 square inches. How many times do you use the sponge to create the design?
Answers
We use the parallelogram-shaped sponge to create the T-shirt design. The area of the design is 66 square inches. 22 times we will use the sponge to create the design.
Parallelogram:
The term "parallelogram" comes from the Greek word "parallelogram on" which means "surrounded by parallel lines". A parallelogram is a quadrilateral bounded by parallel lines. A shape with opposite sides parallel and equal. Parallelograms fall into three main types: squares, rectangles, and rhombuses, each with its own characteristics.
According to the Question:
Area of the parallelogram-shaped sponge = base × height
= 3 inches × 1 inch
= 3 square inches
And,
Area of design = 66 square inches
Now,
Number of times to use the sponge to create the design
= Area of design ÷ Area of parallelogram-shaped sponge
= 66 square inches ÷ 3 square inches
= 22 Times
Therefore,
The sponge can be used 22 times to create the design.
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5. Select Yes or No to indicate whether each ordered pair is a point of intersection
between the line x - y = 6 and the circle y² - 26 = -x².
Ordered Pair
(1,-5)
(1,5)
(5,-1)
Answers
To determine if each ordered pair is a point of intersection between the line x - y = 6 and the circle y² - 26 = -x², we need to substitute the values of x and y in both equations and see if they are true for both.
Select Yes or No to indicate whether each ordered pair is a point of intersectionFor the ordered pair (1, -5):
x - y = 6 becomes 1 - (-5) = 6, which is true.
y² - 26 = -x² becomes (-5)² - 26 = -(1)², which is false.
Therefore, (1, -5) is not a point of intersection.
For the ordered pair (1, 5):
x - y = 6 becomes 1 - 5 = -4, which is false.
y² - 26 = -x² becomes (5)² - 26 = -(1)², which is true.
Therefore, (1, 5) is a point of intersection.
For the ordered pair (5, -1):
x - y = 6 becomes 5 - (-1) = 6, which is true.
y² - 26 = -x² becomes (-1)² - 26 = -(5)², which is false.
Therefore, (5, -1) is not a point of intersection.
So the answer is:
(1,-5) - No
(1,5) - Yes
(5,-1) - No
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Discover the value of each of the shapes. The total weight is 48
Answers
Helpppppppppppppppppppppp
Answers
17) Evaluate D(4)
Please help
Answers
Answer:
D(4) = 6
Step-by-step explanation:
Using the blue graph, find the y value when x =4
the y value is 6 when x=4
D(4) = 6
Work out the size of an exterior and an interior angle of a regular 12-sided polygon. Interior = ____ Exterior = ____
Answers
\((n-2) \times 180\)
\((12-2)\times 180\)
\(10 \times 180\)
\(1800\)
\(1800\div 12\)
\(=150\)
\(180-150\)
\(=30\)
The interior angle is 150°, and the exterior angle is 30°.
Triangular prism problem
Help, I’m trying to solve this question but I’m stuck on what the length is for the right angle in pink
Answers
The total surface area of the triangular prism is: 111 ft²
What is the surface area of the triangular prism?To find the total surface area of the given triangular prism, we will find the area of all the surfaces and add it up.
Formula for the area of a rectangle is:
Area = Length * Width
Area of a triangle is:
Area = ¹/₂ * base * height
Thus:
Total Surface area = 2(¹/₂ * 3 * 7) + (7 * 5) + (8 * 5) + (3 * 5)
Total Surface area = 21 + 35 + 40 + 15
Total Surface area = 111 ft²
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Determine if (3, 2) is a solution for the system of equations.
y = -x + 5
y = 2x - 6
yes
no
Answers
Answer:
Nope
Step-by-step explanation:
2= -3+5 (works)
2= 2(3)-6 (doesn't work)
Vertical stretch by a factor of 4
Translation 5 units right
Reflection over the x-axis
Answers
The given transformations are vertical stretch by a factor of 4, translation 5 units right, and reflection over the x-axis. We can apply these transformations to a function f(x) in the following way:
1. Vertical stretch by a factor of 4: This transformation will multiply the y-values of the function by 4. The new function will be g(x) = 4f(x).
2. Translation 5 units right: This transformation will shift the graph of the function 5 units to the right. The new function will be h(x) = g(x - 5) = 4f(x - 5).
3. Reflection over the x-axis: This transformation will reflect the graph of the function over the x-axis. The new function will be k(x) = -h(x) = -4f(x - 5).
Therefore, the final transformed function will be k(x) = -4f(x - 5).
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Now change matrix B to a 3 x 3 matrix and enter these values for B:
B =
1.2 1.4 3.1
2.2 1.1 5.6
3.7 4.2 6.7
Then select A • B to calculate the product:
77 39 −33
1.2 1.4 3.1
2.2 1.1 5.6
3.7 4.2 6.7
=
c11 c12 c13
c11 =
c12 =
c13 =
Answers
Answer:
Step-by-step explanation:
56.1,12.1,23.6
Sharon filled the bathtub with 38 gallons of water. How many quarts of water did she put in the bathtub?
Answers
Answer:
152 US quarts
Step-by-step explanation:
4 x 38 = 152
quarts = 4 of something
Answer:
152
Step-by-step explanation:
38 x 4 = 152
given the points p(2, –6) and r(8, 3), what is the component form of ? ⟨6, -3⟩ ⟨10, -3⟩ ⟨6, 9⟩ ⟨10, 9⟩
Answers
Answer:
the answer is (6,9)
Step-by-step explanation:
The vector r - p will be in component form,
(8-2, 3-(-6)) = (6,9)
8.6 divided by 1,000 =
find the quotient. divide decimals by power of ten.
Answers
substitute the value of (p) you just found, 285.7, into one of the equations in the system
Answers
Answer: 214.28
Step-by-step explanation:
tonya pays $300.each month to rent a office were she earns $25 per hour tutoring students. which equation repersants tonyas profit,y, for working x hours?
Answers
Answer:
y=25x + 300
Step-by-step explanation: