Trig Cheat Sheet
tan(θ). © Paul Dawkins - https://tutorial.math.lamar.edu. Page 3. Trig Cheat Sheet. For any ordered pair on the unit circle (x y) : cos(θ) = x and sin(θ) = y.
INVERSE TRIGONOMETRIC FUNCTIONS
sin (sin -1 0.5) = 0.5 sin (sin -1 1.5) ≠ 1.5. (not • The definition undifferentiated to sine and cosine
USEFUL TRIGONOMETRIC IDENTITIES
Unit circle properties cos(π - x) = -cos(x) sin(π - x) = sin(x) tan(π - x) = -tan(x) cos(π + x) = -cos(x) sin(π + x) = -sin(x) tan(π + x) = tan(x).
Angles and Radians of a Unit Circle
Circle. Courtesy of Randal Holt. Original Source Unknown. http://www Positive: sin cos
OER Math 1060 – Trigonometry
sin sin. 2. 2 in cos. 2 π θ π α β α β π α β π π α β α θ β. ⎛. ⎞. = -. │. │ ... circle with a radius of 8 inches. Find.
How To
The three trigonometric functions taught most often in high school are: sine (sin) cosine (cos) and tangent (tan). In the unit circle sine is the measure of ...
degree radian sin cos tan cot sec csc 0 30 45 60 90 120 135 150
Page 1. Table of Trigonometric Functions degree radian sin cos tan cot sec csc. 0. 0. 0. 1. 0 undefined. 1 undefined. 30 π. 6. 1. 2. 3. 2. 3. 3. 3. 2 3. 3.
Untitled
Are the following points on the unit circle? Show your work. 21. 36. (23. 29. 20 Find sin 2x
6.1 ans
circle shown below. Find the other two trigonometric functions of Ø of sin(0) cos(0)
Trig Cheat Sheet ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )x y
opposite tan adjacent ? = adjacent cot opposite ? = Unit circle definition. For this definition ? is any angle. sin. 1 y y ? = = 1 csc y ? = cos.
Trig Cheat Sheet ( ) ( ) ( ) ( ) ( ) ( ) ( )x y
tan adjacent ? = adjacent cot opposite ? = Unit circle definition. For this definition ? is any angle. sin. 1 y y ? = = 1 csc y ? = cos.
Double-Angle Power-Reducing
https://www.alamo.edu/contentassets/35e1aad11a064ee2ae161ba2ae3b2559/analytic/math2412-double-angle-power-reducing-half-angle-identities.pdf
Angles and Radians of a Unit Circle
tan 45 1 Circle. Courtesy of Randal Holt. Original Source Unknown. http://www.MathematicsHelpCentral.com ... Positive: sin cos
INVERSE TRIGONOMETRIC FUNCTIONS
sin -1 or arcsin is the inverse of the restricted sine function y = sin x
Trigonometry
The six trigonometric functions can be used to find the ratio of the side lengths. The six functions are sine (sin) cosine (cos)
degree radian sin cos tan cot sec csc 0 30 45 60 90 120 135 150
Table of Trigonometric Functions degree radian sin cos tan cot sec csc. 0. 0. 0. 1. 0 undefined. 1 undefined. 30 ?. 6. 1. 2. 3. 2. 3. 3. 3. 2 3. 3. 2. 45 ?.
10.4 Trigonometric Identities
and sin(-?) = -sin(?). The remaining four circular functions can be expressed in terms of cos(?) we have sin(?) = tan(?) cos(?) so we get sin(?) = (2).
Tangent Cotangent
and Cosecant - The Quotient Rule
Untitled
sin. COS tan. Express as the function of an acute angle. 32. cos 330°. Cos 30° Are the following points on the unit circle? Show your work.
Trigonometry
An Overview of
Important Topics
2Contents
Trigonometry - An Overview of Important Topics ....................................................................................... 4
UNDERSTAND HOW ANGLES ARE MEASURED ............................................................................................. 6
Degrees ..................................................................................................................................................... 7
Radians ...................................................................................................................................................... 7
Unit Circle .................................................................................................................................................. 9
Practice Problems ............................................................................................................................... 10
Solutions.............................................................................................................................................. 11
TRIGONOMETRIC FUNCTIONS .................................................................................................................... 12
Definitions of trig ratios and functions ................................................................................................... 12
Khan Academy video 2 ........................................................................................................................ 14
Find the value of trig functions given an angle measure ........................................................................ 15
Find a missing side length given an angle measure ................................................................................ 19
Khan Academy video 3 ........................................................................................................................ 19
Find an angle measure using trig functions ............................................................................................ 20
Practice Problems ............................................................................................................................... 21
Solutions.............................................................................................................................................. 24
USING DEFINITIONS AND FUNDAMENTAL IDENTITIES OF TRIG FUNCTIONS ............................................. 26
Fundamental Identities ........................................................................................................................... 26
Khan Academy video 4 ........................................................................................................................ 28
Sum and Difference Formulas ................................................................................................................. 29
Khan Academy video 5 ........................................................................................................................ 31
Double and Half Angle Formulas ............................................................................................................ 32
Khan Academy video 6 ........................................................................................................................ 34
Product to Sum Formulas ....................................................................................................................... 35
Sum to Product Formulas ....................................................................................................................... 36
Law of Sines and Cosines ........................................................................................................................ 37
Practice Problems ............................................................................................................................... 39
Solutions.............................................................................................................................................. 42
UNDERSTAND KEY FEATURES OF GRAPHS OF TRIG FUNCTIONS ................................................................ 43
3Key features of the sine and cosine function.......................................................................................... 46
Khan Academy video 7 ........................................................................................................................ 51
Key features of the tangent function ...................................................................................................... 53
Khan Academy video 8 ........................................................................................................................ 56
Graphing Trigonometric Functions using Technology ............................................................................ 57
Practice Problems ............................................................................................................................... 60
Solutions.............................................................................................................................................. 62
Rev. 05.06.2016-4
4Trigonometry Ȃ An Overview of Important Topics
So I hear you're going to take a Calculus course? Good idea to brush up on yourTrigonometry!!
Trigonometry is a branch of mathematics that focuses on relationships between the sides and angles of triangles. The word trigonometry comes from the Latin derivative of Greek words for triangle (trigonon) and measure (metron). Trigonometry (Trig) is an intricate piece of other branches of mathematics such as, Geometry, Algebra, and Calculus. In this tutorial we will go over the following topics.Understand how angles are measured
o Degrees o Radians o Unit circle o Practice Solutions
Use trig functions to find information about right triangles o Definition of trig ratios and functions o Find the value of trig functions given an angle measure o Find a missing side length given an angle measure o Find an angle measure using trig functions o Practice Solutions
Use definitions and fundamental Identities of trig functions o Fundamental Identities o Sum and Difference Formulas o Double and Half Angle Formulas o Product to Sum Formulas o Sum to Product Formulas o Law of Sines and Cosines o Practice Solutions
5 Understand key features of graphs of trig functions o Graph of the sine function o Graph of the cosine function o Key features of the sine and cosine function o Graph of the tangent function o Key features of the tangent function o Practice Solutions
Back to Table of Contents.
6UNDERSTAND HOW ANGLES ARE MEASURED
Since Trigonometry focuses on relationships of sides and angles of a triangle, let's Angles are formed by an initial side and a terminal side. An initial side is said to be in standard position when it's ǀertedž is located at the origin and the ray goes along the positive x axis. An angle is measured by the amount of rotation from the initial side to the terminal side. A positive angle is made by a rotation in the counterclockwise direction and a negative angle is made by a rotation in the clockwise direction.Angles can be measured two ways:
1. Degrees
2. Radians
7Degrees
A circle is comprised of 360°, which is called one revolution Degrees are used primarily to describe the size of an angle. The real mathematician is the radian, since most computations are done in radians.Radians
1 reǀolution measured in radians is 2ʋ, where ʋ is the constant approdžimately
3.14.How can we convert between the two you ask?
Easy, since 360Σ с 2ʋ radians (1 revolution)Then, 180Σ с ʋ radians
So that means that 1° = గ
ଵ଼ radians 8And ଵ଼
గ degrees = 1 radianExample 1
Convert 60° into radians
60 ڄ
ଵ଼ = 60 ڄ ଷ radianExample 2
Convert (-45°) into radians
-45 ڄ ସ radianExample 3
Convert ଷగ
Example 4
Convert െగ
ଷ radian into degrees Before we move on to the next section, let's take a look at the Unit Circle. 9Unit Circle
The Unit Circle is a circle that is centered at the origin and always has a radius of1. The unit circle will be helpful to us later when we define the trigonometric
ratios. You may remember from Algebra 2 that the equation of the Unit Circle is Need more help? Click below for a Khan Academy videoKhan Academy video 1
10Practice Problems
11Solutions
Back to Table of Contents.
12TRIGONOMETRIC FUNCTIONS
Definitions of trig ratios and functions
In Trigonometry there are six trigonometric ratios that relate the angle measures of a right triangle to the length of its sides. (Remember a right triangle contains a90° angle)
A right triangle can be formed from an initial side x and a terminal side r, where r is the radius and hypotenuse of the right triangle. (see figure below) The used to label a non-right angle. The six trigonometric functions can be used to find the ratio of the side lengths. The six functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). Below you will see the ratios formed by these functions. sin ߠ , also referred to as ௦௧ ௦ௗ
cos ߠ , also referred to as ௗ௧ ௦ௗ
tan ߠ௫ , also referred to as ௦௧ ௦ௗ
These three functions have 3 reciprocal functions
csc ߠ ௬ , which is the reciprocal of sin ߠ 13 sec ߠ ௫ ,which is the reciprocal of ...ߠ cot ߠ ௬ , which is the reciprocal of -ߠ You may recall a little something called SOH-CAH-TOA to help your remember the functions! Example: Find the values of the trigonometric ratios of angle ߠ Before we can find the values of the six trig ratios, we need to find the length ofPythagorean Theorem)
Now we can find the values of the six trig functions sin ɽ с ௦௧ ଵଷ csc ɽ с ௬௧௨௦ cos ɽ с ௗ௧ ଵଷ sec ɽ с ௬௧௨௦ tan ɽ с ௦௧ ହ cot ɽ с ௗ௧ 14Example 5
a) Use the triangle below to find the six trig ratiosExample 6
Use the triangle below to find the six trig ratios Need more help? Click below for a Khan Academy VideoKhan Academy video 2
First use Pythagorean Theorem to find the hypotenuse a² + b² = c², where a and b are legs of the right triangle and c is the hypotenuse 15 Find the value of trig functions given an angle measureSuppose you know the value of ߠ
the six trigonometric functions? First way: You can familiarize yourself with the unit circle we talked about. An ordered pair along the unit circle (x, y) can also be known as (cos ߠ, sin ߠ since the r value on the unit circle is always 1. So to find the trig function values With that information we can easily find the values of the reciprocal functions We can also find the tangent and cotangent function values using the quotient identities 16 tan 45° = ୱ୧୬ସହι = 1 cot 45° = 1Example 7
Find ...ቀగ
Example 8
Find -ቀగ
Example 9
Using this method limits us to finding trig function values for angles that are accessible on the unit circle, plus who wants to memorize it!!! Second Way: If you are given a problem that has an angle measure of 45°, 30°, or60°, you are in luck! These angle measures belong to special triangles.
If you remember these special triangles you can easily find the ratios for all the trig functions. Below are the two special right triangles and their side length ratios 17 How do we use these special right triangles to find the trig ratios? If the ɽ you are giǀen has one of these angle measures it's easy͊Example 10 Example 11 Example 12
Third way: This is not only the easiest way, but also this way you can find trig values for angle measures that are less common. You can use your TI Graphing calculator. First make sure your TI Graphing calculator is set to degrees by pressing mode 18Next choose which trig function you need
After you choose which function you need type in your angle measureExample 13 Example 14 Example 15
19Find a missing side length given an angle measure
Suppose you are given an angle measure and a side length, can you find the remaining side lengths? Yes. You can use the trig functions to formulate an equation to find missing side lengths of a right triangle.Example 16
Let's see another edžample,
Example 17
Need more help? Click below for a Khan Academy videoKhan Academy video 3
First we know that ߠ
ǡ therefore ͵-ൌ௫Next we solve for x, ͷڄ
Use your TI calculator to compute ͷڄ
We are given information about the opposite and adjacent sides of the triangle, so we will use tan 20Find an angle measure using trig functions
Wait a minute, what happens if you have the trig ratio, but you are asked to find the angles measure? Grab your TI Graphing calculator and notice that above the trigonometric functions, also known as arcsine, arccosine, and arctangent. If you use these buttons in conjunction with your trig ratio, you will get the angle measure for ߠLet's see some edžamples of this.
Example 18
How about another
Example 19
We know that -ߠ
So to find the ǀalue of ɽ, press 2nd tan on your calculator and then type in (8/6) We are given information about the adjacent side and the hypotenuse, so we will use the cosine function 21Practice Problems
2223
24
Solutions
25Back to Table of Contents.
26USING DEFINITIONS AND FUNDAMENTAL IDENTITIES OF TRIG
FUNCTIONS
Fundamental Identities
Reciprocal Identities
sin ߠ = 1/(csc ߠ) csc ߠ = 1/(sin ߠ cos ߠ = 1/(sec ߠ) sec ߠ = 1/(cos ߠquotesdbs_dbs17.pdfusesText_23[PDF] cos sin tan formulas
[PDF] cos sin tan graph
[PDF] cos sin tan rules
[PDF] cos sin tan table
[PDF] cos2x formula
[PDF] cos2x sin2x identity
[PDF] cos2x sin2x
[PDF] cosine calculator
[PDF] cosine series expansion
[PDF] cosinus definition
[PDF] cosinus formule
[PDF] cosinus joint
[PDF] cosinus sinus tangens
[PDF] cosinusoidal function