CK-12 Trigonometry
Page 23
Now that we know and , we can use the Triangle Sum Theorem to find .
Now, we must convert our angles into headings. See the figures below.
In the first figure, we see that is a heading of east of north. In the second figure, we see that is a heading of west of north. In the third figure, we see that is a heading of east of north.
Answer: The captain’s heading from Island to Island is east of north, from Island to Island is west of north, and from Island to Island is east of north.
Points to Consider
Is there ever a situation where you would need to use the Law of Sines before using the Law of Cosines?
In what situation might you consider using the Law of Cosines instead of Law of Sines if both were applicable?
Why do we only have to use the Law of Cosines one time before we can switch to using the Law of Sines?
Lesson Summary
We have a number of tools to find the missing sides and angles in right and oblique triangles. These tools include: Theorem of Pythagoras
Trigonometric ratios
Law of Cosines
Law of Sines
We can use combinations of the above tools to find all the missing sides and angles in a trying. We call this solving the triangle.
When using the Law of Cosines, we need only to use it once. Then, we can use the Law of Sines, which requires much less computation.
When dealing with the SSS case, find the largest angle first will help us to avoid the Ambiguous case later on.
There are a number of real-world applications that involve using the tools we have learned. We have already explored a few examples in this lesson. We will look at some more situations in the review questions.
Review Questions
Using the information provided, decide which case you are given (SSS, SAS, AAS, ASA, or SSA), and whether you would use the Law of Sines or the Law of Cosines to find the requested side or angle. Make an approximate drawing of each triangle and label the given information. Also, state how many solutions (if any) each triangle would have. If a triangle has no solution or two solutions, explain why.
Given Drawing Case Law Number of Solutions & Explanation
a. find .
b. find .
c. find .
d. find .
e. find .
Using the information in the chart above, solve for the requested side or angle.
Using the information in the chart in question 1 and your answers from question 2, determine what information you are still missing from each triangle.
Find the missing information from question 3, thereby solving each triangle.
The side of a rhombus is and the longer diagonal is . Find the area of the rhombus and the measures of the angles in the rhombus.
Find the area of the pentagon below. Also find the measures of angles and .
In the figure drawn below, angle is . Using the figure below, find the length of the altitude draw to the longest side, the area of the two triangles formed by this altitude, and the measure of all the angles in both triangles.
Refer back to the real-world application at the beginning of the section. Suppose there is a fourth island that tourists can visit. Island is away from Island and the heading from Island to Island is What is the distance from Island to Island ?
What is the angle formed by Island with Islands and ?
What is the angle formed by Island with Island and ?
A golfer is standing on the tee of a golf hole that has a bend to the left. The distance from the tee to the bend is The distance from the bend to the green is How far would the golfer need to hit the ball if he wanted to make it to the green in one shot?
At what angle would he need to hit the ball?
A golfer is standing on the tee, which is from the cup on the green. After he hits his first shot, which is sliced to the right, his ball forms a angle the tee and the cup, and the cup forms a angle with his ball and the tee. What is the degree of his slice?
How far was his first shot?
How far away from the cup is he?
Review Answers
AAS, Law of Sines
SAS, Law of Cosines
SSS, Law of Cosines
SSA, Law of Sines
SSA, Law of Sines
No solution
or
and
and
and
None – there is no solution
and
No solution
or or
The area of the rhombus is . The two larger angles of the rhombus measure and the smaller angles measure .
The area of the pentagon is . Angle is , angle is , and angle is .
The length of the altitude is . The area of triangle The area of triangle RGI is . Angle is , angle TRG is , angle is , angle GRI is , and angle is .
.
He would need to hit the ball .
He would have to hit the ball at a angle.
Supplemental Links
http://demonstrations.wolfram.com/SolvingObliqueTriangles/
Vectors
Learning Objectives
A student will be able to:
Understand directed line segments, equal vectors, and absolute value in relation to vectors.
Perform vector addition.
Perform vector subtraction.
Find the resultant vector of two displacements.
Introduction
Real-World Application: A cruise ship is traveling south at . A westward wind is blowing the ship eastward at . What speed is the ship traveling at and in what direction is it moving?
Not all applications deal with stationary objects. Many applications, such as this one, deal with displacement, velocity, or force.
Displacement is when an object moves a certain distance in a certain direction.
Example: A car travels south.
Velocity is when an object travels at a certain speed in a certain direction.
Example: The wind is blowing at from the northeast.
Force is when a push or pull is exerted on an object in a certain direction.
Example: A upward force is required to lift a crate.
The problem above is an example of velocity. We will refer back to this problem later on.
In the examples above, we could not simply use triangles to represent these as we have been in the past few sections. The boat’s engines are working to give the boat a constant speed in the water. The wind is simultaneously working to make the boat go at to the direction that the engines are making the boat go. We need another tool to represent not only direction but also magnitude (length) or force. This is why we need vectors. Vectors capture the interactions of real world velocities, forces and distance changes.
Any application in which direction is specified requires the use of vectors. A vector is any quantity having direction and magnitude. Vectors are very common in science, particularly physics, engineering, electronics, and chemistry in which one must consider an object’s motion (either velocity or acceleration) and the direction of that motion.
In this section, we will look at how and when to use vectors. We will also explore vector addition, subtraction, and the resultant of two displacements. In addition we will look at real-world problems and application involving vectors.
Directed Line Segments, Equal Vectors, and Absolute Value
A vector is represented diagrammatically by a directed line segment or arrow. A directed line segment has both magnitude and direction. Magnitude refers to the length of the directed line segment and is usually based on a scale. The vector quantity represented, such as influence of the wind or water current may be completely invisible.
A wind is blowing from the northwest. If , then the vector would look like this:
An object affected by this wind would travel in a southeast direction at .
A vector is said to be in standard position if its initial point is at the origin. The initial point is where the vector begins
and the terminal point is where it ends. The axes are arbitrary. They just give a place to draw the vector.
If we know the coordinates of a vector’s initial point and terminal point, we can use these coordinates to find the magnitude and direction of the vector.
Magnitude:
Vectors have magnitude. This measures the total distance moved, total velocity, force or acceleration. “Distance” here applies to the magnitude of the vector even though the vector is a measure of velocity, force, or acceleration. In order to find the magnitude of a vector, we use the distance formula. A vector can have a negative magnitude. A force acting on a block pushing it at north can be also written as vector acting on the block from the south with a magnitude of . Such negative magnitudes can be confusing; making a diagram helps. The south can be re-written as north without changing the vector.
Example 1:
If we know the coordinates of the initial point and the terminal point, we can find the magnitude by using the distance formula.
Initial point
Terminal point
If we don’t know the coordinates of the vector, we must use a ruler and the given scale to find the magnitude.
Direction:
If a vector is in standard position, we can use trigonometric ratios such as sine, cosine and tangent to find the direction.
Example 2:
If a vector is in standard position and its terminal point has coordinates of what is the direction?
The horizontal distance is while the vertical distance is . We can use the tangent function since we know the opposite and adjacent sides of our triangle.
The direction of the vector is
If the vector isn’t in standard position and we don’t know the coordinates of the terminal point, we must a protractor to find the direction.
Equal Vectors:
Two vectors are equal if they have the same magnitude and direction. Look at the figures below for a visual understanding of equal vectors.
Vector Addition and Subtraction
As you know from Algebra, . When we think of vector subtraction, we must think about it in terms of adding a negative vector. A negative vector is the same magnitude of the original vector, but its direction is opposite.
In order to subtract two vectors, we can use either the triangle method or the parallelogram method from above. The only difference is that instead of adding vectors and , we will be adding and .
Example using the triangle method:
Vector Addition
The sum of two or more vectors is called the resultant of the vectors. There are two methods we can use to find the resultant: the triangle method and the parallelogram method.
The Triangle Method:
To use the triangle method, we draw the vectors one after another and place the initial point of the second vector at the terminal point of the first vector. Then, we draw the resultant vector from the initial point of the first vector to the terminal point of the second vector. This method is also referred to as the tip-to-tail method.
To find the sum of the resultant vector we would use a ruler and a protractor to find the magnitude and direction.
The resultant vector can be much longer than either , or it can be shorter. Below are some more examples of the tip-to-tail method.
Example 1:
Example 2:
The Parallelogram Method:
Another method we could use is the parallelogram method. To use the parallelogram method, we draw the vectors so that their initial points meet. Then, we draw in lines to form a parallelogram. The resultant is the diagonal from the initial point to the opposite vertex of the parallelogram. It is important to note that we cannot use the parallelogram method to find the sum of a vector and itself.
To find the sum of the resultant vector, we would again use a ruler and a protractor to find the magnitude and direction.
If you look closely, you’ll notice that the parallelogram method is really a version of the triangle or tip-to-tail method. If you look at the top portion of the figure above, you can see that one side of our parallelogram is really vector translated.
Resultant of Two Displacements
We can use vectors to find direction, velocity, and force of moving objects. In this section we will look at a few applications where we will use resultants of vectors to find speed, direction, and other quantities. A displacement is a distance considered as a vector. If one is away from a point, then any point at a radius of from that point satisfies the condition. If one is to the east of north, then only one point satisfies this.
We will now refer back to the application at the beginning of the section.
A boat’s engines are capable of moving it at . Its compass says that it is moving south. The ocean current at that spot happens to be east. What is the true speed and direction of the boat’s path? In order to find the direction and the speed the boat is traveling, we must find the resultant of the two vectors representing south and east. Since these two vectors form a right angle, we can use the Theorem of Pythagoras and trigonometric ratios to find the magnitude and direction of the resultant vector.
First, we will find the speed.
The ship is traveling at a speed of
To find the direction, we will use tangent, since we know the opposite and adjacent sides of our triangle.
The ship’s direction is
Real-World Application:
A hot air balloon is rising at a rate of , while a wind is blowing at a rate of . Find speed at which the balloon is traveling as well as the angle it makes with the horizontal.
First, we will find the speed at which our balloon is rising.
Since we have a right triangle, we can use the Theorem of Pythagoras to find calculate the magnitude of the resultant.
The balloon is traveling at rate of per second.
To find the angle the balloon makes with the horizontal, we will find the angle A in the triangle and then we will subtract it from
We will use the tangent function to find angle .
The balloon forms an angle of with the horizontal.
Here are some other things to consider using the above problem:
a. How far from the lift off point is the balloon in ? Assume constant rise and constant wind speed. (Total displacement)
After two hours, the balloon will be from the lift off point ( in two hours).
b. How far must the support crew travel on the ground to get under the balloon? (Horizontal displacement)
After two hours, the horizontal displacement will be ( in two hours).
c. If the balloon stops rising after and floats for another , how far did it travel total? How far away does the crew have to go to be under the balloon when it lands?
After two hours, the balloon will have risen vertically ( in two hours). After an additional two hours of floating in the wind, the balloon will have traveled horizontally ( times in two hours). We must recalculate our resultant vector using Pythagorean Theorem.
The balloon has traveled from its lift off point. The crew will have to travel (horizontal displacement) to be under the balloon when it lands.
Points to Consider
Is it possible to find the magnitude and direction of resultants without using a protractor and ruler and without using right triangles?
How can we use the Law of Cosines and the Law of Sines to help us find magnitude and direction of resultants?
Lesson Summary
Vectors are used in situations where we have moving objects or force being applied to objects. These situations deal with displacement, velocity, and force.
Vectors have both magnitude and direction. Equal vectors have the same magnitude and direction. We always calculate the absolute value of the magnitude using coordinates of the initial and end points. Length is not a magnitude. The magnitude of a vector is commonly associated with a positive value. East is the same vector as West. Both have a magnitude of .
If we know the coordinates of the initi
al and terminal points of a vector, we can use the Theorem of Pythagoras and trigonometric ratios to calculate magnitude and direction.
When adding vectors, we can use either the triangle method (tip-to-tail) or the parallelogram method (tail to tail).
When subtracting two vectors, we add the negative if the vector being subtracted. A negative vector has the same magnitude but the opposite direction.
Review Questions
Vectors and are perpendicular. Make a diagram of each addition or subtraction. Find the magnitude and direction (with respect to and ) of their resultant if:
Use and to find the magnitude and direction of each resultant. Make a diagram of each addition or subtraction.
Operation Diagram Resultant
a.
b.
c.
d.
e.
f.
Does ? Explain your answer.
A plane is traveling north at a speed of while an easterly wind is blowing the plane west at . What is the direction and the speed of the plane?
Two workers are pulling on ropes attached to a tree stump. One worker is pulling the stump east with of forces while the second working is pulling the stump north with of force. Find the magnitude and direction of the resultant force on the tree stump.