Lesson Overview:

​

Inverse trig functions can either be read as sin−1(x) or arcsin(x). In this lesson we will mostly use sin−1(x).  They are used to find the unknown measure of a right triangle when two side lengths are known. They are the inverse of regular trig functions.

• Identify and use basic trigonometric identities to find trigonometric values.

• Use basic trigonometric identities to simplify and rewrite trigonometric expressions.

​

Part A: Identify and use basic trigonometric identities to find trigonometric values  

Before we can simplify trigonometric expressions, we must know the basic trig identities.

These identities include; reciprocal identities, ratio identities, and Pythagorean identities.

​

There are the reciprocal trig identities:

​

​

​

​

​

​

​

​

​

​

​

​

These are the Pythagorean Trig Identities:

 

 

 

 

​

​

​

​

​

These videos will help you better understand how we derive these different types of trig identities that will be essential when you are rewriting and simplifying trig expressions.

Using this video, you will learn how reciprocal trig identities.

 

 

 

 

 

 

 

​

​

​

​

​

 

Using these videos, you will learn how we derive ratio identities.

​

​

​

​

​

​

​

​

​

​

​

​

​

 

Using this video, you will learn how Pythagorean trig identities are derived.

 

 

 

 

 

 

 

​

​

​

 

​

​

​

Here is a very useful foldable you should create to ensure that you are able to remember all these trigonometric identities.

                     Cover of the Foldable:                                                                        Contents of the Foldable:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

​

​

​

​

 

 

Another important part of simplifying trigonometric expressions is understanding the formulas for the operations of fractions as well as the formula for complex fractions.

These are the formulas operations of fractions:

 

 

 

 

 

 

 

​

​

​

​

​

 

These are the formulas for simplifying complex fractions:

​

​

​

​

​

​

Examples of operations of fractions and complex fractions using variables and numbers:

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

Use this video to get a better step by step understanding on how to simplify complex fractions:

​

​

​

​

​

​

​

​

​

​

​

​

​

Part B: Use basic trigonometric identities to simplify and rewrite trigonometric expressions.

Using your knowledge of basic trig identities, you are now able to simplify and rewrite trigonometric expressions.

Here are some tips on how to simplify and rewrite trig identities:

​

​

​

​

​

​

​

​

​

​

​

​

​

​

Here are a couple examples on how to solve different types of problems:

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

Use this video to get more information on how to simplify trig expressions

​

​

​

​

​

​

​

​

​

​

​

​

​

​

Simplifying trig expressions on the AP Exam:

Most tests like the IB and AP exams will not require you to simplify and solve for the answer. Instead they will require that you use your simplification skills to prove that one trig expression is equal to another. You will have to look back at the different types of trig identities to solve these types of problems.

​

​

​

 

​

​

​

​

​

​

​

​

Trig Identities and Expressions in the Real World

​

​

​

​

​

​

​

Conclusion:

In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios.