# BASIC TRIGONOMETRY PDF

Fundamental of trigonometry. Chapter 3. Fundamentals of Trigonometry. Introduction: The word “trigonometry” is a Greek word. Its mean “measurement. my permission. The PDF version will always be freely available to the public at no cost (go to kaz-news.info). Feel free to contact. Find the value of trig functions given an angle measure. USING DEFINITIONS AND FUNDAMENTAL IDENTITIES OF TRIG FUNCTIONS.

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Trigonometric Formula Sheet. Definition of the Trig Functions. Right Triangle Definition. Assume that: 0 <θ< π. 2 or 0◦ <θ< 90◦ hypotenuse adjacent opposite θ. An Introduction to Trigonometry. kaz-news.infon. I. Basic Concepts. The trigonometric functions are based on the unit circle, that is a circle with radius r=1. Since the. Basic Trigonometry and. Mathematics IMA. Radians. Pythagoras' Theorem. In a right-angled triangle a2 = b2 + c2, where a is the side not involved with the the.

Next, calculate cos x. We know that we could measure it using a ratio, but for simplicity, we will calculate it, since we know it will be equal to the measurement. What is cos x exactly? How does this compare to what you estimated in 20? How does this compare to what you estimated in 21?

How does this compare to what you estimated in 22? As you may have remembered, that line is called a tangent line. Construct ray AC to do this, you can use the segment tool on the left-side toolbar, but hold it down until you can select the ray tool.

Then construct the intersection point of the tangent line and ray AC and label the intersection point E. At this point, your sketch should look like this. Note that you can drag your labels so that they are not covered up by your lines.

The Ancient Greeks were the first to discover trigonometry, and they considered segment DE to be the tangent of angle x. Using similar triangles, find the relationship between sin x , cos x , and tan x. What is the length of the adjacent segment in that triangle? What does that make tan x equal to? What is tan x approximately? What is tan x exactly? How does this compare to what you estimated in 43?

How does this compare to what you estimated in 44? How does this compare to what you estimated in 45? Save your sketch.

We will keep using this as we continue to explore the trigonometric functions. Did they use trigonometry tables? If so, do they remember what they were and how they worked? Introduction: If you asked your parents or grandparents for homework last night, when they solved trigonometric equations, they probably used a table of values, rather than a calculator.

In ancient times, trigonometry tables were created by drawing a large and extremely precise circle, and measuring the lengths of the segments of sine at different angles. It was extremely time-consuming, difficult, and tedious. In your final trigonometry table, if two values disagree slightly because students used slightly different approximations of an angle, average them for the final table.

If two values disagree significantly, investigate whether one student may have an error. You may want to set up a shared Google spreadsheet so that students can put their values into the spreadsheet, which can automatically average the values.

Drag the point D to the right as much as possible to zoom in on your drawing. Your teacher will assign you several whole-number angle measurements. Move point C so that x is as close to each angle measurement as possible. All students should put their measurements up on the board, averages will be calculated, and everyone will record the final trig table on their own paper. Note that because trig tables are so difficult to create, you have to be creative with how you use them to make the most of the values they give you.

Teacher note: For numbers 1c-e and 4c-e, students will likely need help. These will be good opportunities for discussion.

Using your trig table, find the following: Check using calculator: a. Using your trig table, solve for x in the following triangles: a. Try turning the circle sideways to see this better.

Even though our trigonometry table does not include cosine, how can we use this information to find cos x using our trig table? Using your trig table, solve for x in the following triangle: 1. Now that you know how to find sin x and cos x using your trigonometry table, how can you find values for tan x using the trigonometry table? Find the following using your trig table: a. When you have used your trig table and checked using your calculator, how close have your calculations been?

When have they been off, and by how much? What do you attribute this difference to? Is this a lot of error, or just a little bit? I am sure you will be happy to go back to using your calculator after this, but what have you learned from creating and using this trigonometry table? As you continue to use your calculator to solve a trigonometric equation, try to remember what is happening inside of your calculator.

Secant is a trigonometric function that is less commonly used than sine, cosine and tangent. You may or may not have heard of it before. The Ancient Greeks considered sec x to be the segment AE. Note: In geometry, a secant line refers to a line that intersects a circle in two places.

If you extend the segment AE through both sides of the circle, it would be a secant line. Then, draw ray AE, and construct the intersection between the most recent perpendicular line you created and ray AE. Label this intersection point G, and then hide the two perpendicular lines and ray AE. Finally, construct segment AG and FG. Your sketch should look like this. Now consider the segment FG. This is one of the trigonometric functions of the complementary angle. Which trigonometric function is it?

Since the sine of the complement is called cosine, what do you think FG should be called? Now consider the segment AG. Since the sine of the complement is called cosine, what do you think AG should be called? Big new idea: Until this point, we have been using degrees to measure the angle x, but now we are going to switch and use something called radians. A radian measures an angle by how many radius distances the arc of the angle passes through. Look at your measurements. Delete any measurements of segments and ratios so that all you have left is the measure of x and four trigonometric functions.

Next, we are going to graph the trigonometric functions. You can put it on a website for them, email it to them, or distribute it to them in another convenient way.

To graph sin x , highlight the measurement of x and then the measurement of sin x. Notice that the height of the point that is tracing the graph is the same as the height of the triangle, which is equal to sin x , because what we are graphing is points of the form x, sin x , so the y-value is equal to sin x.

In Display, you can also erase the traces if you want to start over or make a new graph. Use the graph of sin x to answer the following questions: a. What is the domain of sin x? Are there angles beyond what is shown in our graph? Are these acceptable angles for the domain? What is the range of sin x? Where is sin x positive, and where is it negative? Now graph cos x , after deleting the traces of sin x. To graph cos x , highlight the measurement of x and then the measurement of cos x.

What segment on the circle corresponds with the height of the graph in this case? What is the range of cos x? Where is cos x positive, and where is it negative? Erase the traces of sin x and cos x , and graph tan x. What is the range of tan x? Where is tan x positive, and where is it negative? Are there any angles where tan x is undefined? Now erase all previous traces, and graph sec x. What is its domain? Where is sec x positive, and where is it negative?

Is it ever 0? Is it ever undefined? Now erase all previous traces, and graph cot x. Where is cot x positive, and where is it negative? Now erase all previous traces, and graph csc x. Where is csc x positive, and where is it negative?

Are there any pairs of trigonometric functions that are inversely related that is, when one gets bigger, the other one gets smaller? Are there any pairs of trigonometric functions that are directly related that is, they both get bigger together and smaller together? Three special cases of trigonometric functions are when they are equal to 0, equal to 1, or undefined. Is there any relationship among the trigonometric functions as to when that happens to which ones?

You will graph them in pairs and determine how they are related to each other. Teacher note: It will be important to tease out why sin x and cos x and csc x and sec x are horizontal shifts of each other while tan x and cot x also require a reflection 1. Start by graphing sin x. Highlight the measurement of x and then the measurement of sin x.

The next thing we will do is graph cos x in in a different color, leaving the graph of sin x in place. Graph cos x using the same procedure.

Do not delete the traces of sin x. Right-click on the point, and change its color, so that the sin x trace will be a different color from the cos x trace. What is the relationship between the graphs of sin x and cos x? How can you incorporate a horizontal shift into a function? Can you write cos x as a sin x function with a horizontal shift? How does that make sense with what you know about the relationship between sin x and cos x?

What about the words sine and cosine? How are those words related? Does that relate to the function you wrote in number 8? Erase the traces of sin x and cos x , and graph tan x and cot x in different colors.

What is the relationship of tan x to cot x? Can you write cot x as tan x with a horizontal shift? What else needs to happen besides a horizontal shift in this case? How does that make sense with what you know about the relationship between tan x and cot x? What about the words tangent and cotangent? Does that relate to the function you wrote in number 12? Erase the traces of tan x and cot x , and graph sec x and csc x in different colors. What is the relationship of sec x to csc x?

Can you write csc x as sec x with a horizontal shift? How does that make sense with what you know about the relationship between sec x and csc x? What about the words secant and cosecant? Does that relate to the function you wrote in number 16? If these functions are just horizontal shifts of each other, do we really need separate functions, or would it be sufficient to just have sin x , sec x and tan x? When is it helpful to have cos x , cot x , and csc x? Are there ever cases where it seems redundant to have these additional functions?

See if your graphing calculator has a sine regression. Does it also have a cosine regression? Why do you think this would be? Introduction: For several days now, we have been working with a circle, whose radius is 1 unit.

## Product Identities

This circle is often called the unit circle, because it is a circle with a unit radius. On the unit circle, we can find several different right triangles. See what right triangles you can find.

You should be able to find three different right triangles note that there are two right triangles that are congruent, we can just consider one of those. Start with the right triangle ABC. Note that you will have to highlight all 3 points and then highlight the first point again to construct the polygon A-B-C-A, for example.

Consider this sides of this triangle. What trigonometric function represents the length of AB? What is the length of AC? Lesson 4: Students will use the relationships they discovered previously together with their trigonometric table to solve trigonometric equations. Lesson 5: They will investigate the domains and ranges of these functions particularly where they are undefined, 0 and 1 , and see how the graphs relate back to the segments on the unit circle.

Lesson 6: Students will investigate the relationship between sine and cosine, secant and cosecant, and tangent and cotangent. They will graph them in pairs and determine how they are related to each other. Wrap-up at the end will emphasize the relationships between these words i.

Lesson 7: Students will discover the Pythagorean identities by examining the unit circle.

Lesson 8: Students will graph them in pairs and investigate their relationships. Lesson 9: Lesson Wrap-up and posttest. Notes to the teacher: It may be instructive to keep this file open on your computer as a reference. The file that students will be working on in the first four lessons will closely resemble this file.

If student flies begin to look significantly different from this file, they have probably made a mistake. The student version of the curriculum contains blank pages so that if you print double-sided, each new lesson starts on a new page. This is to allow you to hand out one lesson at a time for students if desired.

Trigonometry Pre-test Instructions. Answer each question to the best of your ability. What is the purpose of the trigonometric functions? In other words, what types of problems can they help you solve? If so, what is their domain and range? What is the height of the building? Explain how you found your answer. Do not try to prepare students for the pre-test. This is just to get a baseline to help us know what they understand going into this curriculum.

On the left hand toolbar, you can construct segments, circles, points, polygons. On the top, you can construct perpendicular bisectors, intersections, etc. Make sure when they construct their shapes that they not only look right, but also that they can drag and move the applicable points in the appropriate manner.

Sometimes they can make things look right, but they have constructed them incorrectly, and when they drag the point around, you will be able to tell. Students should construct the following: At this time, you probably want to zoom in.

You can do this by selecting the point where the circle intersects the positive x-axis and dragging it to the right. Then, use the segment tool create a segment that goes from the center of the circle out to the edge of the circle.

This will allow you to change the segment any way you want while keeping it length 1. Construct a segment that goes along the x-axis from the center of the circle to the edge of the circle. Construct a perpendicular between the endpoint of your original segment and your segment that goes along the x-axis. Highlight that point and also highlight that segment.

Then highlight the segment that goes along the x-axis and the new perpendicular line. Highlight the perpendicular line, and click hide. Draw a segment where the perpendicular line was, between the point that was on the circumference of the circle and the point that was found using the perpendicular line. Draw a segment that goes along the base of the triangle, just up to where the perpendicular line intersected the x-axis.

Now you have a right triangle with radius 1, and as you change the angle through the first quadrant, the right triangle is always a right triangle, and the hypotenuse is always 1.

Highlight the points of the triangle, beginning with the the one that is in the origin, and continuing in a counter-clockwise fashion. It is important that we all label our points the same to avoid confusion in the future. This is what your sketch should look like at this point. What is sin x? Knowing that the hypotenuse is 1, what does that tell you? For which line segment is the length equal to sin x? What is cos x? For which line segment is the length equal to cos x? What is sin x approximately?

What is cos x approximately? Highlight the three points that make up the angle x, in this order: Since we have enlarged our graph, one unit does not equal one centimeter. To account for that, we need to divide by the length of the hypotenuse. First, highlight the height of the triangle.

## Trigonometry Functions:

This gives us a measurement of the height of the triangle in our coordinate plane. Remember that this height of the triangle is equal to sin x as we saw earlier. Move point C and see how this measurement changes. Write down a pattern that you see. What is sin x exactly? How does this compare to what you estimated in 17? How does this compare to what you estimated in 18? How does this compare to what you estimated in 19?

Is sin x equal to the height of the triangle? Next, calculate cos x. We know that we could measure it using a ratio, but for simplicity, we will calculate it, since we know it will be equal to the measurement. What is cos x exactly? How does this compare to what you estimated in 20?

How does this compare to what you estimated in 21? How does this compare to what you estimated in 22? As you may have remembered, that line is called a tangent line.

Construct ray AC to do this, you can use the segment tool on the left-side toolbar, but hold it down until you can select the ray tool. Then construct the intersection point of the tangent line and ray AC and label the intersection point E. At this point, your sketch should look like this. Note that you can drag your labels so that they are not covered up by your lines. The Ancient Greeks were the first to discover trigonometry, and they considered segment DE to be the tangent of angle x.

Using similar triangles, find the relationship between sin x , cos x , and tan x.

## Magic Hexagon for Trig Identities

What is the length of the adjacent segment in that triangle? What does that make tan x equal to? What is tan x approximately? What is tan x exactly? How does this compare to what you estimated in 43? How does this compare to what you estimated in 44? How does this compare to what you estimated in 45?

Save your sketch. We will keep using this as we continue to explore the trigonometric functions. Go home and ask your parents or grandparents how they did trigonometry.

Did they use trigonometry tables? If so, do they remember what they were and how they worked? The goal of this activity is to create a trigonometry table for sin x. If you asked your parents or grandparents for homework last night, when they solved trigonometric equations, they probably used a table of values, rather than a calculator.

In ancient times, trigonometry tables were created by drawing a large and extremely precise circle, and measuring the lengths of the segments of sine at different angles. It was extremely time-consuming, difficult, and tedious.

Teacher Note: In your final trigonometry table, if two values disagree slightly because students used slightly different approximations of an angle, average them for the final table. If two values disagree significantly, investigate whether one student may have an error. You may want to set up a shared Google spreadsheet so that students can put their values into the spreadsheet, which can automatically average the values.

Drag the point D to the right as much as possible to zoom in on your drawing. Your teacher will assign you several whole-number angle measurements. Move point C so that x is as close to each angle measurement as possible. All students should put their measurements up on the board, averages will be calculated, and everyone will record the final trig table on their own paper. The goal of this activity is to use your trigonometry table to solve different problems involving trigonometry.

Note that because trig tables are so difficult to create, you have to be creative with how you use them to make the most of the values they give you. Teacher note: For numbers 1c-e and 4c-e, students will likely need help. These will be good opportunities for discussion. Using your trig table, find the following: Check using calculator: Using your trig table, solve for x in the following triangles: Side note: We already talked about the name of tangent.

Try turning the circle sideways to see this better.

Even though our trigonometry table does not include cosine, how can we use this information to find cos x using our trig table? Using your trig table, solve for x in the following triangle: Now that you know how to find sin x and cos x using your trigonometry table, how can you find values for tan x using the trigonometry table? Find the following using your trig table: When you have used your trig table and checked using your calculator, how close have your calculations been? When have they been off, and by how much?

What do you attribute this difference to? Is this a lot of error, or just a little bit? I am sure you will be happy to go back to using your calculator after this, but what have you learned from creating and using this trigonometry table?

As you continue to use your calculator to solve a trigonometric equation, try to remember what is happening inside of your calculator. Secant is a trigonometric function that is less commonly used than sine, cosine and tangent. You may or may not have heard of it before. The Ancient Greeks considered sec x to be the segment AE. In geometry, a secant line refers to a line that intersects a circle in two places.

If you extend the segment AE through both sides of the circle, it would be a secant line. Then, draw ray AE, and construct the intersection between the most recent perpendicular line you created and ray AE. Label this intersection point G, and then hide the two perpendicular lines and ray AE. Finally, construct segment AG and FG. Your sketch should look like this.

Now consider the segment FG. This is one of the trigonometric functions of the complementary angle. Which trigonometric function is it? Since the sine of the complement is called cosine, what do you think FG should be called? Now consider the segment AG.

Since the sine of the complement is called cosine, what do you think AG should be called? Big new idea: Until this point, we have been using degrees to measure the angle x, but now we are going to switch and use something called radians. A radian measures an angle by how many radius distances the arc of the angle passes through. Look at your measurements.

Delete any measurements of segments and ratios so that all you have left is the measure of x and four trigonometric functions. Next, we are going to graph the trigonometric functions. You can put it on a website for them, email it to them, or distribute it to them in another convenient way. To graph sin x , highlight the measurement of x and then the measurement of sin x. Notice that the height of the point that is tracing the graph is the same as the height of the triangle, which is equal to sin x , because what we are graphing is points of the form x, sin x , so the y-value is equal to sin x.

In Display, you can also erase the traces if you want to start over or make a new graph. Use the graph of sin x to answer the following questions: What is the domain of sin x? Are there angles beyond what is shown in our graph? Are these acceptable angles for the domain? What is the range of sin x?

Where is sin x positive, and where is it negative? Now graph cos x , after deleting the traces of sin x. To graph cos x , highlight the measurement of x and then the measurement of cos x. What segment on the circle corresponds with the height of the graph in this case?

What is the range of cos x?

## Downloadable Trig Table PDF

Where is cos x positive, and where is it negative? Erase the traces of sin x and cos x , and graph tan x.

What is the range of tan x? Where is tan x positive, and where is it negative? Are there any angles where tan x is undefined? Now erase all previous traces, and graph sec x. What is its domain? Where is sec x positive, and where is it negative?On the unit circle, we can find several different right triangles. Lesson 5: Theorem A.

What is the length of AC? What about the words secant and cosecant? Try taking the principal Pythagorean identity, and dividing through by sin2 x. What is the height of the building?

Students may need help, especially getting started on this. Note that you will have to highlight all 3 points and then highlight the first point again to construct the polygon A-B-C-A, for example. Then, draw ray AE, and construct the intersection between the most recent perpendicular line you created and ray AE.

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