trig unit 4

You want to buy a plot of land next to your house. The lot measures 48 yards by 100 yards by 120 yards. The price of the land is $2,000 per acre (1 acre = 4840 square yards). How much will this addition cost?

$967.81

Find u*v when θ is the angle between u and v. ||u|| = 7 ||v|| = 4 θ=π/3

14

Let u = 5i - j and w = -8i + 4j. Find 3u - 2w and then write the component form of the answer

3u - 2w = 31i - 11j<31, -11>

Find the projection of u unto v. Then write u as the sum of the two orthogonal vectors, one which is proj,u u={-4,-1} v={-3,-2}

<- 4, -1>

Find the projection of u unto v. Then write u as the sum of the two orthogonal vectors, one which is proj,u u={12,16} v={-6,-4}

<12,16>

Let u = <3,6> and v = <-2,-5>. Find u - 2v

<7,16>

Find the projection of u unto v. Then write u as the sum of the two orthogonal vectors, one which is proj,u u={8,4} v={2,-4}

<8,4>

A race car with a gross weight of 2200 pounds is parked on a ramp at a slope of 18°. Assuming that the only force required to overcome is the force of gravity, find the force required to keep the vehicle from rolling down the ramp

A force of about 680 pounds is required to keep the car from rolling down the ramp

Which of the following careers uses trigonometry?

All of these use trigonometry

If asked to solve an oblique triangle and you are given only the three angles, it would be best to use the Law of Cosines

False

In physics, the standard metric unit for work is the Jules Verne. One Jules Verne is equivalent to one Newton of force causing a displacement of one meter

False

In trying to solve an oblique triangle, if you are given AAS, ASA, or SSA, use the Law of Cosines. If given SSS or SAS, use the Law of Sines

False

The Law of Sines can be used to solve any right triangle

False

There are many applications in which two vectors are added to produce a resultant vector, but it never works in reverse. Vectors cannot be decomposed into the sum of two vector components

False

Three of the following tricks are used to solve trigonometry equations, but one is not. Can you find the one that is NOT used?

Sometimes we turn single angles into double and half angles to solve trigonometry equations.

Given an oblique triangle where <A=100 degrees, side a=5 and side b=12. Find <B, if possible

This is an ambiguous case with no solution

If two vectors u and v are orthogonal then their dot product = 0. Therefore, given u = <-1, 7> and v = <7, 1>, vectors u and v are orthogonal

True

If you are given a vector in linear combination form, v = 14i + 2j, then the component form is v = <14, 2>

True

If ||v||=1, v is a unit vector: Moreover, ||v||=0 if, and only if, v is the zero vector 0

True

In solving a trig equation, the primary goal is to isolate the trig function on one side of the equation

True

Is it possible to use the Law of Sines AND the Law of Cosines in the same problem?

True

The two basic vector operations are scalar multiplication and vector addition. When dealing with vectors, numbers are usually referred to as scalars

True

The vectors <4,2> and <-4,-2> are opposite directions and are orthogonal to the vector u = <2,- 4>. That's because the dot product of the vectors equals zero

True

We now have three different formulas for finding the area of a triangle

True

When using the Law of Sines in the Ambiguous Case (SSA), there are three distinct possibilities including: there is a single-solution; there is no solution; and there are two solutions

True

w=f*PQ is the dot product form of work w and w=||proj pqF|| ||PQ|| is the projection of work, w

True

Find the work done in moving a particle from P to Q if the magnitude and direction of the force are given by v. P(4,9) Q(-1,5) v = <3,4>

Work = -31

Find the work done in moving a particle from P to Q if the magnitude and direction of the force are given by v. P(3,5) Q(2,8) v = <4,10>

Work = 26

A mover exerts a horizontal force of 40 pounds on a couch as it is pushed up a ramp that is 12 feet long and inclined at an angle of 30 degrees above the horizontal. Find the work done in pushing the couch and state whether you used the projection form or dot product form for work. Round to two decimal places

Work = 415.69 foot-pounds. Projection form was used for work

A young child pulls a wagon by exerting a force of 6 pounds on a handle that makes a 12 degree angle with the horizontal. Find the work done by the child if the wagon is pulled 15 feet across the sidewalk

Work = 88.03 foot-pounds

Let u be the vector represented by the line segment from P(0,4) to Q(-3,- 4) and v be the vector represented by the line segment from R(3,3) to S(0,-5). Are the vectors equivalent? If so, why?

Yes, they are equivalent because ||u|| = 73−−√73 and ||v|| = 73−−√73 and the slopes are the same, 8383

Solve the missing parts of the triangle, and round your answers to the nearest hundredth

a = 73.00b = 22.10A = 35 degrees

Three of the following four professionals use trigonometry on the job. Which professional does not use trigonometry in a direct way?

dentist

There are five possible orientations of two vectors. If the cos θ = 1 then ______. (fill in the blank)

the vectors are the same direction

Find the component form of the sum of u and v with direction angles θu and θv ||u|| =4 and θu=30 degrees and ||v|| =4 and θv=60 degrees

u + v = 5.5i + 5.5j

Let u = <3, 3> and w = <-5, 1>. Find (a) u + w, (b) 3u, and (c) 2u + w

u + w = <-2,4>3u = <9,9>2u +w = <1,7>

Let u = <-3,6> and w = <-2,-7>. Find (a) u + w, (b) 2w, and (c) 4u + w

u + w = <-5,-1>2w = <-4,-14>4u +w = <-14,17>

Find the component form of u = 2i - 7j and v = -i - 3j and also the dot product

u = <2, -7> and v = <-1, -3> Dot product = 19

Find the component form and magnitude of vector v that has initial point (2,5) and terminal point (5,9)

v = <3,4> ||v|| = 5

A softball pitcher was recorded throwing a pitch at a velocity of 82 miles per hour. If he threw the pitch at an angle of 25 degrees below the horizontal, find the vertical and horizontal components of the velocity

v = <72.32, -34.65>

Find the component form and magnitude of the vector v

v={4,-5} ||v||= √41

Solve: cos(x+π/3)-cos(x-π/3)=-√3 on the interval [0,2π)

x=π/2

Solve: cos x(cox x + 1) = 0 in the interval (0, 2π)

x=π/2, 3π/2, π

Solve: 2cscθ=secθcscθ on the interval [0,2π)

x=π/3, 5π/3

Solve: sin(x+π/4)-sin(x-π/4)=1 on the interval [0,2π)

x=π/4, 7π/4

Use the dot product to find the magnitude of w = <6, 5>

||w||= √61

Find the angle θ between the vectors u = 6i + 8j and v = -4j. (negative 4j)

Θ = 143.13 degrees

Solve sin θ - 2 = cos θ - 2 in the interval (0, 2π)

θ=π/4, 5π/4

Whether the navigation is for a boat, an airplane, or other forms of transportation, the Law of Cosines can be used to calculate the navigation components, but never the Law of Sines

False

Two markers A and B on the same side of a river are 44 feet apart. A third marker is located across the river at point C. A surveyor determines that the measure of Angle CAB = 64 degrees and the measure of angle ABC = 56 degrees. Find the distance between points A and C and the distance across the river

AC = 42.12 feet and the distance across the river = 37.86 feet

Which of the following careers uses trigonometry?

All of these use trigonometry.

Use the Law of Cosines to find the measure of Angle A to the nearest degree. The three sides of the triangle are: a = 51, b = 39, and c = 43

Angle A = 77 degrees

A triangular garden plot has sides of lengths 115 feet, 76 feet, and 92 feet. Find the measure of the largest angle

Angle C = 85.8 degrees

Find the area of a garden plot that has side lengths of 51.6 ft., 63.6 ft., and 86.4 ft

Area = 1630 square feet

A vector is in standard position if its terminal point is at the origin

False

An oblique triangle is a triangle with one 90-degree angle

False

Any vector in two dimensions can be thought of as having an influence in four different directions

False

Heron's Formula is used to find a missing side of an oblique triangle

False

There is always at least one solution when solving a trig equation. The answer "no solution" does not apply in trig equations

False

Trigonometry is a new field of mathematics. In 1940, Hipparchus, a Greek mathematician, wrote 12 books on the table of chords and was named the founder of trigonometry. Hipparchus was the first mathematician to study triangular geometry

False

Vectors are often denoted in italic uppercase letters such as A, B, and C

False

The angles of elevation to an airplane from points A and B on level ground are 52 ̊ and 70 ̊. The points A and B are 2.6 miles apart, and the airplane is west of both points. Find the altitude of the plane

The altitude of the plane is approx. 6.2 miles

Find the direction angle of vector v = 5i - 5j

The angle θ = 315 degrees

Find the area of an oblique triangle where angle C = 25 degrees, side a = 12, and side b = 9

The area = 23

Find the area of an oblique triangle where angle B = 130 degrees, side a = 62 and side c = 20

The area = 475

Use the Law of Cosines to find the measure of Angle C of an oblique triangle where a = 1.42, b = 0.75, and c = 1.25. Round Angle C to two decimal places

The area = 61.50

The distance from the moon to the sun is about 4,580,000 miles. When the earth and moon are at their closest, the angle shown in the image is 87.19 degrees. Approximately, what is the closest distance between the earth and the moon?

The closest approximate distance between the earth and the moon is 225,000 miles

A pilot is flying over a straight highway and determines the angle of depression to two towers 2.4 miles apart to be 67 degrees and 59 degrees. Find the air distance the plane must travel until touching down on the near end of the highway

The distance is 2.54 miles

A small ship travels from a lighthouse whose light only shines east on a bearing of N70 ̊E, a distance of 24 miles. It then makes a turn to the right and travels for 15 miles parallel to the lighthouse and finds itself directly in the beam of the lighthouse's light. How far from the lighthouse is the ship?

The distance is 23.6 miles

A Jeep Cherokee with a gross weight of 4800 pounds is parked on a slope of 18°. Assuming that the only force required to overcome is the force of gravity, find the force required to keep the vehicle from rolling down the hill. Find the force perpendicular to the hill

The force needed to keep the Jeep from rolling is 1483 pounds. A force of 4565 pounds is perpendicular to the hill

Because of a tornado, a tree was found leaning 8 degrees from the vertical. At a point 50 meters from the tree, the angle of elevation to the top of the tree is 30 degrees. Find the height of the tree

The height of the tree is 31.7 feet

A force of 160 pounds is required to pull a refrigerator up a moving truck ramp inclined at 23 degrees from the horizontal. What is the weight of the refrigerator?

The refrigerator weighs 409.5 lbs

Find the bolt spacing, or the angle in degrees, between each hole in a bolt circle whose diameter is 6.250 inches

There are 36 degrees between the center of one hole and the center of the next hole.

Find the bolt spacing, or the angle in degrees, between each hole in a bolt circle whose diameter is 8.250 inches

There are 60 degrees between the center of one hole and the center of the next hole

A machinist is a person who works with machines to create or modify parts for other machines. They fit and assemble parts and make repairs. Machinists must apply their knowledge of mechanics, mathematics, materials, layout, and machining procedures to create a specific part. They must also study sketches or blueprints to be able to manufacture or repair the part

True

A unit vector is a vector with a magnitude of 1 and has the same direction as the original vector, but its length is equal to one

True

Angle between two vectors if θ is the angle between two nonzero vectors u and v, then cos θ= u*v/||u|| ||v||

True

Sometimes when solving a trig equation it is necessary to square both sides of the equation. With this process, you may get some extraneous solutions that you must check in the original equation

True

A mover exerts a horizontal force of 50 pounds on a couch as it is pushed up a ramp that is 10 feet long and inclined at an angle of 30 degrees above the horizontal. (a) Find the work done in this situation and (b) find the work done if the mover changes the angle to 35 degrees. Which angle created less work?

a. Work = 433 foot-poundsb. Work = 409 foot-pounds. The 35-degree angle is less work

Three of the following four professionals use trigonometry on the job. Which professional does not use trigonometry in a direct way?

accountant

Use the Law of Cosines and the Law of Sines to solve all the missing parts of the given oblique triangle

b = 10.53 Angle A = 78 deg. Angle C = 62 deg

Given an oblique triangle with Angle B = 62 degrees, side a = 11, and side c = 7, find the length of side b using the Law of Cosines. Round your answer to the nearest tenth

b = 9.9

The pitcher's mound on a baseball field is 50 feet from home plate, and the distance between the bases is 70.71 feet. The measure of Angle ABC = 45 degrees. How far is it from the pitcher's mound to the third base?

b or the distance from the pitcher's mound to third base is 50 feet