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Series: Trigonometry: Trigonometric Identities

About this Series

  • Lessons: 23
  • Total Time: 3h 5m
  • Use: Watch Online & Download
  • Access Period: Unlimited
  • Created At: 08/13/2009
  • Last Updated At: 06/01/2011

In this 23-lesson series, we'll look at an assortment of trigonometric identities that will be tremendously helpful in evaluating trig expressions, solving trig equations, and simplifying trigonometric expressions. We'll start with the derivation and explanation of the four fundamental trig identities (tan x = sin x / cos x; cos^2 x + sin^2 x = 1; 1 + tan^2 x = sec^2 x; and 1+ cot^2 x = csc^2 x), and we'll learn how we can use these identities to find the values of the other trigonometric functions given the value of a trigonometric function for an angle and the quadrant that the angle lies in. We'll then generate proofs based upon these identities to establish extension identities that also hold.

We'll then look at several techniques for simplifying trig expressions - factoring, multiplying binomial products, restating expressions in terms of only sines and cosines, etc. We can apply several of these same techniques and others to equations to solve trig identities, which we'll also practice.

We'll next turn to sum and difference identities. These trigonometric identities will help you find trig values for the sums and differences of angles. The sine of the sum of two angles, x and y, is equal to sin (x + y) = sin x cos y + cos x sin y. We'll go through this identity as well as identities for differences of angles and cosine and tangent.

We'll also cover the double-angle identities. Double-angle identities can help you solve trigonometric equations like our example: cos 2x = sin x (where x is between 0 and 2*pi). They are also commonly needed to simplify trig expressions or solve word problems like the one we'll go through.

Last, we'll look at a few additional advanced trigonometric identities that can be used to help you solve trig equations or simplify trig expressions: cofunction identities, power-reducing identities and half-angle identities. The cofunction identities relate sine to cosine and tangent to cotangent: cos (x -pi/2) = sin x and sin (x - pi/2) = -cos x and tan (x - pi/2) = -cot x. Power-reducing identities allow you to write trig expressions with powers as equivalent expressions without powers: cos^2 x = (1 + cos (2x) ) / 2 and sin ^2 x = (1 - cos (2x)) / 2. These formulas are useful in solving trig equations and simplifying trig expressions that contain powers of trig functions (including powers in excess of 2). Half-angle identities allow you to write trig functions of an angle x/2 (half of the angle x) in terms of trig functions of the angle x: cos x/2 = +/- ((1+cos x)/2))^(1/2) and sin x/2 = +/- ((1-cos x)/2))^(1/2) and tan x/2 = +/- (1-cos x)/sin x

Taught by Professor Edward Burger, this bundle of lessons was selected from a broader, comprehensive course, Trigonometry. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers an algebra review, trigonometric functions, trigonometric identities, applications of trig, complex numbers, polar coordinates, exponential functions, logarithmic functions, conic sections, and more.

About this Author

2174 lessons

Founded in 1997, Thinkwell has succeeded in creating "next-generation" textbooks that help students learn and teachers teach. Capitalizing on the power of new technology, Thinkwell products prepare students more effectively for their coursework than any printed textbook can. Thinkwell has assembled a group of talented industry professionals who have shaped the company into the leading provider of technology-based textbooks. For more information about Thinkwell, please visit or visit Thinkwell's Video Lesson Store at

Thinkwell lessons feature a star-studded cast of outstanding university professors: Edward Burger (Pre-Algebra through...

Lessons Included

Good but abstract
~ Nina3

Still too abstract to get students (not math lovers;) exited....

~ greenshoes


Great lesson, but could use more real application
~ sharon5

When would you have a real problem where you would be looking for a position of a spring? I know that is what my students are going to be asking.

Easy way to identify even and odd functions!
~ brittanie

Great online video tutorial on how to verify if a trig function like sin or cos is even or odd. This question comes up on many college prep assessment tests and it's great to have these videos available for easy review online.

~ hmt79

This lesson walks you through several different proofs of trig identities - he does a good job at going step by step and so I didn't get lost anywhere along the way. Great lesson!

Very Well done
~ isioma

Still a little confused but not because of tutorial.

~ Yaa

This video was incredible! I was studying for ACTs when this question came up about odd functions...which I don't think I've ever encountered yet. But this is straight-forward and worthwhile to watch. Thanks for making this tutorial - simply can't thank you enough!!! I'll definitely be looking out for some more tutorials :D

Below are the descriptions for each of the lessons included in the series:

Supplementary Files: