In this explainer, we will learn how to use the Pythagorean identity and double-angle formulas to evaluate trigonometric values.
Recall the Pythagorean identity for trigonometric functions:
Dividing through by gives the identity . Using instead gives .
It is often a good strategy to rewrite more complicated trigonometric expressions in terms of and when seeking simplifications.
Example 1: Using Pythagorean Identities to Simplify Trigonometric Expressions
Using the Pythagorean identity on the numerator, we get
Since and , we get resulting in
The double-angle formulas give and in terms of and . We recall these.
Definition: Double Angle Formulas
For any real number , the following formulas hold:
We can use these two to derive a formula for too. In this case, purely in terms of , giving us and
Example 2: Simplifying Trigonometric Expressions Using Double-Angle Identities
We notice that we can factor out a :
Now, we will see examples on using the double-angle formulas in the opposite direction.
Example 3: Simplifying Trigonometric Expressions Using Double-Angle Identities
We use the same formula in the numerator and denominator: where we used the Pythagorean identity. In the denominator, now with the identity ,
A more challenging example is the following.
Example 4: Using Trigonometric Identities to Find Exact Values of Trigonometric Expressions
Find the value of given , where .
What we must evaluate is
We are given the numerator: . It is therefore enough to know the value of for this specific .
To do this, we can square the sum of the sine and cosine, because so that, using our known sum, which means
A second example of such an evaluation follows.
Example 5: Using Double Angle Identities to Evaluate a Trigonometric Expression
Find, without using a calculator, the value of given , where .
In order to apply the double-angle formulas, we need to know the value of . The Pythagorean identity can be solved to give
So is either or . The information about the location of tells us it is in the 4th quadrant, where the sine is negative. Therefore, and
Of course, an alternative is to observe that we are being asked to evaluate and use the double-angle formula for tangents.