Trigonometry - The Unit Circle, Angles, & Right Triangles
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Trigonometry - The Unit Circle, Angles, & Right Triangles

Analytical Trig, Verifying Trigonometric Identities, Double Angle Formulas, Inverse Trigonometric Functions & More.
4.5 (1 rating)
Instead of using a simple lifetime average, Udemy calculates a course's star rating by considering a number of different factors such as the number of ratings, the age of ratings, and the likelihood of fraudulent ratings.
14 students enrolled
Last updated 8/2017
English
Current price: $10 Original price: $200 Discount: 95% off
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Includes:
  • 19.5 hours on-demand video
  • Full lifetime access
  • Access on mobile and TV
  • Certificate of Completion
What Will I Learn?
  • Students should have a solid understanding of trigonometry if they work through all of the example problems.
View Curriculum
Requirements
  • Students should have a basic understanding of Algebra and Geometry
Description

This online trigonometry course is for high school and college students who are taking trig and may need help with their homework or simply to pass a test.  Here is a list of topics contained in this video.

1.  Introduction to Angles - Drawing In Standard Position

2.  Converting Angles From Degrees to Radians and Radians to Degrees

3.  Circular Arc Length, Linear Speed and Angular Speed

4.  Coterminal Angles and Reference Angles

5.  The Six Trigonometric Functions and The Unit Circle

6.  Finding Exact Values of Trigonometric Functions - Sine, Cosine, and Tangent

7.  Even and Odd Trigonometric Functions, Reciprocal Identities, and Quotient Identities

8.  Fundamental Pythagorean Identities With Sine and Cosine

9.  Right Triangle Trigonometry - SOHCAHTOA and Pythagorean Theorem

10.  30-60-90 and 45-45-90 Right Triangles

11.  Angle of Elevation and Angle of Depression Word Problems

12.  Trigonometric Functions of Any Angle - Using Reference Angles To Find The Exact Value

13.  Evaluating Trigonometric Functions of Quadrantal Angles

14.  The Signs of Trigonometric Functions In Quadrants 1 to 4

15.  Graphing Sine and Cosine Functions - Identifying The Amplitude, Period, Phase Shift, and Vertical Shift

16.  The Graphs and Secant and Cosecant Functions - Domain and Range

17.  Graphing Tangent and Cotangent Functions - Identifying The Vertical Asymptote

18.  Inverse Trigonometric Functions

19.  Inverse Sine Function From Restricted Sine Graph - One to One Function & Horizontal Line Test

20.  Finding The Exact Value of an Inverse Sine and Cosine Function

21.  Graphing The Inverse Sine, Cosine, and Tangent Function

22.  Domain and Range of Inverse Trig Functions

23.  Evaluating a Composite Trigonometric Expression With Inverse Functions

24.  Applications of Trigonometric Functions - Solving Two Right Triangle Systems

25.  Introduction to Bearings - Word Problems

26.  Verifying Trigonometric Identities - Changing Sines to Cosines and Factoring

27.  Sum and Difference Formulas of Sine, Cosine, and Tangent - Alpha and Beta Angles

28.  Finding Exact Values With Sum and Difference Formulas 

29.  Verifying Trig Identities With Sum and Difference Formulas

30.  Using Double Angle Formulas For Sine, Cosine, and Tangent To Find Exact Values

31.  Simplifying Trigonometric Expressions With Power Reducing Formulas

32.  Half Angle Formulas and Identities

33.  Verifying Trigonometric Identities With Double Angle Formulas

34.  Inverse Trigonometric Functions With Double Angle Formulas and Half Angle Identities



Who is the target audience?
  • This course is for high school and college students who wish to do well in Trigonometry.
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Curriculum For This Course
165 Lectures
19:16:55
+
Introduction
20 Lectures 01:44:39

In this trigonometry course, I recommend pausing the video and working through as many examples as possible.  This will help you to learn the material.

Preview 00:30

This lesson provides an introduction into angles.  It explains how angles are formed between two rays joined by a common endpoint also known as a vertex.  The two rays are known as the initial side and the terminal side.  This video also describes the difference between a segment, ray, and a line.

Preview 01:26

This video discusses the difference between positive and negative angles.  Positive angles are formed due to counterclockwise rotation of the angle from the initial side to the terminal side.  Negative angles are formed due to clockwise rotation.  This video lesson also explains what it means when an angle is placed in its standard position, that is, when the common endpoint / vertex is placed at the origin of rectangular x - y coordinate system

Positive and Negative Angles
01:23

This video explains the difference between acute angles, right angles, obtuse angles, and straight angles.

Acute Angles, Right Angles, Obtuse Angles, and Straight Angles
01:18

This video discusses the 4 quadrants on a typical graph and the angles associated with them.

Angles and Quadrants
01:36

This video explains how to graph angles in standard position in degrees.  Examples include positive and negative angles and how to place them in the appropriate quadrants.

Drawing Angles In Standard Position In Degrees
06:17

This video discusses the definition of the radian and how it relates to the arc length of a circle and its radius.

Definition of the Radian
02:26

This lecture explains how to calculate the value of an angle in radians using the arc length of a circle and its radius.

Calculating Radian Measure
02:04

This video explains how to convert degrees to radians using a simple conversion technique.

Preview 03:12

This video discusses the process of converting radians to degrees using values such as pi and 180.

Converting Radians to Degrees
02:53

This tutorial explains how to draw angles in standard position with the angle having the unit radians.

Drawing Angles In Standard Position In Radians
01:49

This lesson explains how to draw common angles in radians such as pi/4, pi/3, and pi/6 with many more examples.

Drawing Common Angles In Radian
04:25

Coterminal Angles are angles with different values that have the same terminal side when placed in their respective standard positions.  To find a coterminal angle - add or subtract 360 degrees or 2pi radians.

Coterminal Angles
06:18

This video lecture explains how to convert dms - degrees minutes and seconds into decimal degrees by dividing by numbers such as 60 and 3600 considering that 60 minutes = 1 hour and 3600 seconds = 1 hr.

Preview 03:11

This video explains the process of converting decimal degrees to DMS - Degrees Minutes and Seconds.

Converting Decimal Degrees to DMS
06:42

This video tutorial explains how to calculate the arc length of a circle given the radius and angle in both radians and degrees.

Arc Length
07:48

This video explains how to calculate the area of the sector of a circle given the radius and the angle in both degrees and radians.

Area of the Sector of a Circle
06:05

This video explains how to solve trigonometry word problems that involve linear speed and angular speed.  Linear speed tells you how fast an object is moving in the forward direction and it has common units such as meters per second (m/s), feet per second (ft/s), miles per hour (mph), or kilometers per hour (km/hr).  Linear speed of a wheel / disk or any circular object can be calculated by dividing the arc length by the time.  Angular speed tells how fast something is spinning or rotating.  Common units of angular speed or angular velocity are radians per second (rad/s) and revolutions per minute (rpm).  The linear speed equals the angular speed multiplied by the radius of the circle.  The symbol for angular speed is omega.

Preview 18:57

This video explains how to calculate the angle measure in degrees between the minute hand the hour hand of an analog clock given the time.

Clock Time to Angle Measure
12:18

This video quiz contains 10 multiple choice questions that cover most of the topics taught in this section.

Video Quiz - Angle Measure
14:01
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The Unit Circle
9 Lectures 01:33:16

This video lesson provides a basic introduction into the unit circle and it explains how to use it in order to evaluate trigonometric functions such as sine and cosine given an angle in degrees or radians that can be in any of the four quadrants.

Introduction to the Unit Circle
12:48

This video explains how to find the reference angle of any angle in quadrants 1, 2, 3, or 4.  It explains how to find the reference angle in degrees and in radians.  The reference angle is an acute angle less than 90 degrees that forms between the terminal side of the angle and the x-axis.

Reference Angles
12:27

This video tutorial provides an introduction into the six trigonometric functions sine, cosine, tangent, cotangent, cosecant, and secant in addition to formulas associated with the unit circle in terms of the variables x and y.

The Six Trigonometric Functions
02:01

This video provides the formulas and equations associated with the reciprocal identities sin, cos, tan, csc, cot, sec.  This video tutorial also explains how to evaluate functions like secant and cosecant using the reciprocal identities and the unit circle.

Reciprocal Identities
14:38

This video provides the equations and formulas associated with the quotient identities of the tangent and cotangent functions.  It explains how to evaluate tangent and cotangent given the values of sine and cosine and how to evaluate it using the unit circle.  It discusses what happens when evaluating cot and tan given a quadrantal angle and how to know when the function will be undefined.  Examples include positive and negative angles in degrees and radians.

Quotient Identities
12:33

This video lesson discusses the even and odd properties of trigonometric functions.  cosine and secant are even functions.  For example cos(-x) = cos(x).  tangent, cotangent, sine, and cosecant are odd functions.  sin(-x) = - sin(x).  This video explains how to evaluate trigonometric functions using the even and odd properties of sine and cosine.

Even and Odd Trigonometric Functions
10:29

This video lecture explains how to find the value sine given the value of cosine using the pythagorean identity formula.  sin^2 + cos^2 = 1.  This video also helps you to determine the sign of the missing trig function given information that can be used to determine the quadrant of the angle.

Pythagorean Identities
07:42

This video lesson discusses how to use periodic properties to evaluate trig functions.  An angle and its coterminal angle will have the same value of any trig function.  Coterminal angles differ by 360 degrees or 2pi radians and therefore are periodic with respect to each other.  Coterminal angles differ in their angle measure but share the same trigonometric value.

Periodic Properties
04:47

This video quiz contains 10 multiple choice practice problems that cover the topics listed above in this section.

Video Quiz - The Unit Circle
15:51
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Right Triangle Trigonometry
11 Lectures 52:35

This video explains the term SOHCAHTOA and how to use it to evaluate the six trigonometric functions.  SOH means sine is equal to the ratio of the opposite side relative to the angle and the hypotenuse of the right triangle.  CAH means that cosine is equal to the adjacent side divided by the hypotenuse.  TOA means tangent is the ratio of the opposite side and the adjacent side.  Cosecant is the reciproccal of sine.  Therefore, cosecant is the ratio of the hypotenuse to the opposite side.  Secant is equal to the hypotenuse divided by the adjacent and cotangent is the reciprocal of the tangent function.

SOH CAH TOA
02:16

This video explains how to find the value of the six trigonometric functions using soh cah toa.  It explains how to find the missing of the right triangle using the pythagorean theorem.  In addition, it discusses how to simplify radicals and how to rationalize the denominator which is all part of the process of simplifying the final answer.

Preview 16:33

This video explains how to use the 45-45-90 right triangle to evaluate sin(45) and cos(45) which is equal to square root 2 / 2.  The 45 45 90 right triangle is an isosceles right triangle since two sides have the same length.  Make sure to commit to memory this special reference triangle.

45 45 90 Reference Isosceles Right Triangle
02:42

This video provides the 30-60-90 special reference right triangle and its corresponding values.  it explains how to use it to evaluate sin(30), cos(60), tan(30), cot(60), sec(30), csc(60), and so forth.

Preview 03:18

This video explains how to use soh cah toa to find the value of the missing side of a right triangle.  You need to know which trig function to use when setting up the equation - sine, cosine, or tangent.

Finding The Value of the Missing Side of a Right Triangle
04:08

This video explains how to find the value of the missing angle of a right triangle using inverse trig functions such as inverse sine, arctan, or arccos.  SOH CAH TOA is still needed in this lesson.

Finding The Value of the Missing Angle of a Right Triangle
02:36

This video explains the difference between the angle of elevation and angle of depression with respect to a horizontal line.  Elevation is associated with the word up and depression is associated with the word down.

Angle of Elevation vs Angle of Depression
02:03

This applications of trigonometry video lesson explains how to find the height of a building using right triangle trigonometry.  It discusses how to set up the picture given such a word problem.  To find the height of the building, you need the angle of elevation and the distance between the building and the point where the angle of elevation is measured with respect to the ground.

Finding The Height of a Building - Applications of Trigonometry
01:43

This video explains how to calculate the angle of elevation of the sun using the inverse tangent function given the height of the building and the length of the shadow that the sun produces against the building.

How To Find The Angle of Elevation
02:01

This video provides a basic introduction into the cofunction identities of trigonometric functions.  Cofunctions are complentary which means their angles add up to 90 degrees or pi/2 radians.  cosine is the cofunction of sine.  cosecant is the cofunction of secant.  Cotangent is the cofunction of tangent.  sin(30) = cos(60), tan(20) = cot(70) and sec(10) = csc(80).

Cofunction Identities
04:05

This multiple choice video quiz contains 10 practice problems that cover most of the topics taught in this section.

Video Quiz - Right Triangle Trigonometry
11:10
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Trigonometric Functions of any Angle
7 Lectures 35:48

This video provides the formulas of the six trigonometric functions in terms of x, y, and R.

Trigonometric Functions of any Angle
01:35

This video explains the signs of the trigonometric functions in certain quadrants.  For instance, the sine function has a positive value in quadrants 1 and 2 and has a negative value in quadrants 3 and 4.

The signs of the Trigonometric Functions
01:39

This video explains how to evaluate trigonometric functions given a point on the terminal side of the angle.  You need to use the pythagorean theorem to find the missing side of the right triangle unless you the special right triangle numbers.  You can use SOHCAHTOA to evaluate the six trigonometric functions given the terminal point.

Evaluating Trigonometric Functions Given a Point on the Terminal Side
07:11

This video lecture explains how to identify the quadrant in which an angle lies in.

Identifying The Quadrant In Which an Angle Lies
01:59

This video tutorial explains how to find the exact value of the five remaining trigonometric functions given the value of one of the six trigonometric functions.  You need to determine the quadrant in which the angle lies and draw the right triangle in that quadrant with the appropriate signs.

Preview 07:10

This video lecture explains how to use reference angles to evaluate trigonometric functions.  Examples include trig functions with positive and negative angles in degrees and radians.

Using Reference Angles to Evaluate Trigonometric Functions
10:58

This video quiz contains 5 multiple choice questions.

Video Quiz
05:16
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Graphing Trigonometric Functions
9 Lectures 01:29:55

This video lesson provides an introduction into the graphs of the sine and cosine function.  It discusses what happens when a negative sign is placed in front of sine and cosine and how it reflects over the x-axis.  It explains how to graph two full periods and how to identify the key points needed to graph these functions.

Introduction to Sine and Cosine Graphs
04:25

This video tutorial explains how to identify the amplitude and period of a sine and cosine wave.  The amplitude can shrink or stretch the sinusoidal function vertically.  The period can cause a horizontal stretch or shink.  The period of sine and cosine functions is 2pi/B.  This video explains how to graph sine and cosine functions using the amplitude and period.  In addition, it discusses the domain and range of sine and cosine.

Preview 08:41

This video tutorial explains how to graph trigonometric functions using the vertical shift as well as the amplitude and period of the function.  This video describes how to find the range of such functions.

Graphing Trig Functions With a Vertical Shift
04:18

This video explains how to graph trig functions by identifying the phase shift, vertical shift, amplitude, and period.  This lesson contains many examples and practice problems.

Graphing Trigonometric Functions With a Phase shift
10:40

This video lecture explains how to identify the amplitude, period, phase shift, and vertical shift directly from a graph in order to write the equation of the sine or cosine function.

Writing The Equation Given The Graph
07:47

This video explains how to solve a word problem that contains a sinusoidal function that describes the fluctuation of temperature that occurs in a typical day.  It explains how to identify the amplitude, maximum and minimum temperature, and how to calculate the temperature at a certain time of day.

Sinusoidal Function Word Problem
05:39

This video lesson explains how to graph secant and cosecant functions by first tracing the graph of their corresponding reciprical functions such as sin and cos.  It explains how to identify the amplitude, period, and vertical shift of these graphs and how to determine the domain and range of the sec and csc functions.  In addition, it discusses how to identify the vertical asymptotes as well.

Graphing Secant and Cosecant Functions
14:41

This video discusses how to graph the tangent and cotangent functions by identifying the amplitude, period, vertical shift and horizontal shift or phase shift of the functions.  It explains how to identify the vertical asymptotes of tan and cot and provides the range of these functions as well.

How to Graph Tangent and Cotangent Functions
17:42

This 10 question multiple choice video quiz covers topics such as identifying the amplitude, period, phase shift and vertical shift of sinusoidal functions, identify the graph of sin, cos, tan, cot, sec, and csc graphs and determining the range as well as the maximum and minimum values of these functions.

Video Quiz - Graphing Trigonometric Functions
16:02
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Inverse Trigonometric Functions
11 Lectures 01:13:55

This video tutorial explains how to graph of the inverse sine function from the restricted sine function.  The normal sine function is not a one-to-one function since it does not pass the horizontal line test which means the inverse curve is not a function since it does not pass the vertical line test.  However, an inverse function exists for the restricted sine function.  This video provides the domain and range for the inverse sine function.  It also discusses the inverse function notation sin^-1(x) and distinguishes from the reciprocal function of (sinx)^-1.

Preview 05:50

This video tutorial explains how to evaluate inverse sine functions.  It explains how to find the exact value of sin^-1(-1/2), sin^-1(0), sin^-1(-1) and so forth.  The inverse sine function only exists in quadrants 1 and 4 and has a restricted range of [-pi/2, pi/2] or [-90, 90].

Evaluating Inverse Sine Functions
08:02

This video explains how to graph the arccos function using the restricted cosine function which is a one-to-one function that passes the horizontal line test.  The inverse cosine function exists in quadrants 1 and 2 and has a restricted range from [0, pi]

Graphing The Inverse Cosine Function
02:11

This video explains how to find the exact value of the arc cosine function such as cos^-1(1/2) using the unit circle, 30-60-90 special right triangles and reference angles.

Finding The Exact Value of the Arc Cosine Function
05:16

This video tutorial explains how to graph the inverse tangent function using just one full period of the normal tangent function.  The vertical asymptotes of the tan graph become horizontal asymptotes in the arctan function.  The domain of the inverse tangent function is all real numbers but the range is restricted to [-pi/2, pi/2] just like the arcsine function and exists only in quadrants 1 and 4.

Graphing The Inverse Tangent Function
02:49

This video tutorial explains how to evaluate the inverse tan function using reference triangles and the unit circle.  It provides examples and practice problems such as tan^-1(0), tan^-1(1), and tan^-1(-1).

Evaluating The Inverse Tan Function
07:34

This video explains how to evaluate a composition of inverse trigonometric functions such as sin^-1[sin pi/3] and cos^-1[cos 2pi/3] using reference angles, coterminal angles, and special right triangles like the 30-60-90 triangle.

Composition of Inverse Trigonometric Functions
10:36

This video tutorial explains how to evaluate a composition of different trig functions such as sin^-1[cos pi/3] and tan^-1[sin pi/2].  

Preview 01:45

This video lecture explains the process of evaluating a composite trigonometric expression using right triangles placed in appropriate quadrants.  The pythagorean theorem is needed in this lesson along with the ability to simplify radical expressions and to rationalize the denominator of a fraction.  Practice problems include examples such as sin[cos^-1 3/5], cos[tan^-1 8/15] and csc[cos^-1 (-7/9)].

Evaluating a Composite Trigonometric Expression
07:19

This video explains the process of simplifying composite trigonometric expressions with inverse functions.  It explains how to evaluate it using SOHCAHTOA, the pythagorean theorem formula, and placing right triangles in the appropriate quadrants.

Simplifying Composite Trigonometric Expressions With Inverse Functions
06:39

This 10 question multiple choice video quiz covers the graphs of inverse trig functions and how to evaluate & simplify composite inverse trigonometric functions.

Video Quiz - Inverse Trigonometric Expressions
15:54
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Applications of Trigonometric Functions
4 Lectures 46:14

This video lesson explains how to solve systems with two right triangles using the tangent function and SOHCAHTOA along with some basic geometry principles.

Solving Two Right Triangle Systems
15:13

This video provides a basic introduction into bearings.  It explains how to draw it given an example such as N 30 E or how to identify it given a drawing.

Introduction to Bearings
05:49

Bearing Practice Problems
12:29

This video quiz contains 4 multiple choice questions that covers bearing word problems and angles.

Video Quiz
12:43
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Verifying Trigonometric Identities
9 Lectures 01:11:02

This video tutorial provides a list of formulas and equations that you need to verify trigonometric identities.

Fundamental Identities - List of Formulas & Equations
02:10

This video tutorial explains how to verify trigonometric identities by first converting any tangent, secant, cosecant, or cotangent expressions into sine and cosine.

Preview 09:49

This video explains how to verify trig identities by using the pythagorean identities.

Pythagorean Identities
05:21

This video explains how to use factoring techniques to verify the trigonometric identity.  Techniques include removing the GCF - greatest common factor, factoring trinomials and difference of perfect squares.

Factoring
08:12

This video tutorial explains when you should obtain common denominators in order to verify the trig identity.

Common Denominators
06:03

This video tutorial explains the process of splitting fractions in order to expand 1 term into 2 terms in order to verify a trigonometric identity.

Splitting Fractions
03:40

This video tutorial explains when you should multiply by the conjugate in order to verify a trigonometric expression.

Multiplying By The Conjugate
09:08

This video contains plenty of mixed practice problems on verifying trigonometric identities.

Mixed Problems
17:33

This video quiz contains 7 multiple choice questions that cover the basic techniques taught in this section.

Video Quiz - Verifying Trigonometric Identities
09:06
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Sum and Difference Formulas
9 Lectures 01:27:29

This video tutorial provides a list of sum and difference formulas that you need to know.

Sum and Difference Identities
03:02

This video explains how to use the sum and difference formulas to evaluate expressions such as sin(15) and sin(105).

Using The Sum and Difference Formula From Sine
03:43

This video explains how to use the sum and difference identities for cosine.

Sum and Difference Formula For Cosine
04:16

This video explains how to find the exact value of trigonometric expressions using sum and difference formulas of sine, cosine, and tangent.

Find The Exact Value of the Trigonometric Sum and Difference Expression
06:12

This video tutorial explains how to use the sum and difference identities with tangent using the angles alpha and beta.

Sum and Difference Identities With Tangent
10:40

This video explains how to use the sum and difference identities to evaluate trigonometric expressions give sine alpha and sine beta.

Sum and Difference Problems Given Sine Alpha and Beta
17:11

This video explains how to use the sum and difference formulas to verify trigonometric identities.

Verifying Trigonometric Identities With Sum and Difference Formulas
10:57

This video tutorial explains how to simplify inverse trigonometric functions using sum and difference identities.

Simplifying Inverse Trigonometric Functions With Sum and Difference Identities
12:13

This video quiz contains 10 multiple choice practice problems that covers the topic taught in this section.

Video Quiz - Sum and Difference Identities
19:15
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Double Angle & Half Angle Identities and Power Reducing Formulas
12 Lectures 01:36:52

This video tutorial explains how to derive the double angle formulas of sine, cosine, and tangent.

Double Angle Formulas
02:53

This video explains how to use the double angle formulas to find exact values using right triangle trigonometry.  It explains how to evaluate sin(2x), cos(2x), and tan(2x).

Using Double Angle Formulas to Find Exact Values
11:28

This video explains how to solve alternative / reverse double angle trigonometry problems.

Alternative Double Angle Problems
03:32

This video tutorial explains how to use double angle formulas of sine, cosine, and tangent when verifying trigonometric identities.

Verifying Trigonometric Identities With Double Angle Formulas
15:18

This trigonometry video tutorial explains how to derive the power reducing formulas for sine, cosine, and tangent.

Power Reducing Formulas
02:36

This trigonometry video lesson explains how to use the power reducing formulas to simplify trigonometric expressions like sin^2(x) cos^2 (x) and sin^4(x).

Using The Power Reducing Formulas To Simplify Trigonometric Expressions
06:45

This video explains how to derive the half angle formulas.

Half Angle Identities
03:14

This video explains how to use the half angle formulas to find the exact value of trigonometric expressions using SOHCAHTOA and right triangle trigonometry.

Using Half Angle Formulas to Find The Exact Value
06:18

This video explains how to use right triangle trigonometry and half angle formulas when evaluating trigonometric expressions.

Right Triangle Trigonometry and Half Angle Identities
11:19

This video explains how to use the half angle formulas when verifying trigonometric identities.

Verifying Trigonometric Identities With Half Angle Formulas
05:12

This video explains how to simplify inverse trigonometric expressions with double angle formulas and half angle identities.

Inverse Trig Functions With Double Angle Formulas and Half Angle Identities
12:04

This 10 question multiple choice video quiz covers the main topics taught in this section.

Video Quiz - Double Angle Formulas & Half Angle Identities
16:13
8 More Sections
About the Instructor
Mr. Julio Gonzalez
4.6 Average rating
62 Reviews
548 Students
6 Courses
Math & Science Teacher

I started my education career by tutoring students right after high school and into college receiving a BA in Chemistry.  The request that I usually receive were hard core topics like Algebra, Geometry, Trigonometry, Precalculus, Calculus, General Chemistry, Physics, and Organic Chemistry.

Over time, I learned which teaching methods are most effective for helping students to not only understand the material but to retain it and to apply it during an exam to boost their score.  Now, I focus more on creating videos and online courses that help not just students in my locality but students worldwide who might be struggling with this course.

Best Regards,

Julio G.