In this explainer, we will learn how to use the double-angle and half-angle identities to evaluate trigonometric values.
The double-angle identities give and in terms of and . They are a special case of the sum identities, and , namely when . The double-angle identities can be simply derived from them as shown in the following:
Identity: Double-Angle Identities
For any real number , we have
For , we have
It is worth pausing here for a moment and reflecting on the respective signs of , , and as functions of . As shown in the following figure, if we let be in a half-quadrant of the unit circle, then is in a whole quadrant.
For instance, when (or ), we have (or ). In this case, , , , , , and are all nonnegative, and is not defined for .
Let us check that this is consistent with the double-angle identities. The first identity is . Since , we have , and thus, , which corresponds indeed to being nonnegative.
We derive from the second identity, , that is nonnegative if and have the same sign or one of them is zero. This is the case for .
Finally, since when , we have , which leads to , and thus, .
We can apply similar considerations for all 8 half-quadrants of the unit circle (for the 4 half-quadrants at the bottom of the circle, the values of are then greater than or rad) and show that the double-angle identities hold for any values of .
We can use these two to derive a formula for too. In this case, purely in terms of , giving us and
Let us look with our first example at how to use these double-angle identities to simplify trigonometric expressions.
Example 1: Simplifying Trigonometric Expressions Using Double-Angle Identities
Simplify
Answer
Recall that , which we can use to simplify the numerator and denominator: where we used the Pythagorean identity . In the denominator, now with the identity ,
Therefore,
Let us take a note of the two identities that we derived from in the previous example.
Identity: Identities Derived from the Double-Angle Identity cos 2𝜃 = cos2 𝜃 - sin2 𝜃
For any real number , we have
Now, we will see an example of how we can simplify a trigonometric expression by using the double-angle identity in the opposite direction.
Example 2: Simplifying Trigonometric Expressions Using Double-Angle Identities
Simplify .
Answer
We notice that we can factor out a : The Pythagorean identity gives . Substituting this in leads to Using the double-angle identity , we find that . Substituting in gives
Let us now evaluate a trigonometric expression using the double-angle identity.
Example 3: Using Double-Angle Identities to Evaluate a Trigonometric Expression
Find, without using a calculator, the value of given , where .
Answer
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,
Using the double-angle identities and , we find that
Of course an alternative is to observe that we are being asked to evaluate and use the double-angle formula for tangents.
So far, we have used the double-angle identities. The half-angle identities can be derived from them simply by realizing that the difference between considering one angle and its double and considering an angle and its half is just a matter of perspective.
Using the identity that we derived from the double-angle identity and substituting in and , we get
Starting from the other identity derived from the double-angle identity (i.e., ), we find that
Finally, to find , we use the expressions we just established for and by writing
With the three half-angle identities we have derived above, we see that the absolute values of the three trigonometric functions of half an angle depend only on the value of the cosine of this angle. However, the “” shows that their signs are not determined. This means that we will be able to write the correct sign in the identity only if we know in which quadrant the half angle is located.
This comes from the fact that an angle is not fully determined by its cosine value. Take for instance , , , and . While we have , we see that and , as shown in the following diagram.
Let us now see how the tangent half-angle identity can be rewritten so that the sign is determined.
Starting with we multiply the numerator and denominator of the fraction on the right-hand side by , which gives
The numerator can be rewritten as using the double-angle identities (with and ) and the denominator as using the identity (with and ) derived from the double-angle identity .
Hence, we have
Starting again from we now multiply the numerator and denominator of the fraction on the right-hand side by . We get
Using here the identity , we can rewrite as and, as before, as . Hence, we find that
It is worth noting that rearranging by multiplying the right-hand side of the equation by either or leads to and . Studying the respective signs of and as shown in the following table, we conclude that and always have the same sign. This allows us to establish the identities and .
| () | (0–90) | (90–180) | (180–270) | (270–360) | (360–450) | (450–540) | (540–630) | (630–720) |
|---|---|---|---|---|---|---|---|---|
| () | (0–45) | (45–90) | (90–135) | (135–180) | (180–225) | (225–270) | (270–315) | (315–360) |
Let us recap the half-angles identities that we have just derived.
Identity: Half-Angle Identities
For any real number , we have
For , where is an integer, we have
For , we have
Let us look at how to use the half-angle identities in the following example.
Example 4: Evaluating the Cosine Function for a Half Angle given the Cosine Function and Quadrant of the Angle
Find the value of given , where , without using a calculator.
Answer
Recall the half-angle identity
We are told that . Hence, and . This means that we have since
Let us work out the value of by substituting the value of into the above equation:
We can rationalize this fraction by multiplying it by , which gives
Finally, let us see with our last example how the double-angle and half-angle identities can be used to find the exact value of some trigonometric functions.
Example 5: Finding the Exact Value of a Trigonometric Expression
Using the half-angle formulas, or otherwise, find the exact value of .
Answer
The angle radians is not one of the special angles whose trigonometric ratios are well known. However, we notice that it is half the value of these special angles, namely half of radians. Using the half-angle identity , we find that
Since , we have
Multiplying the right-hand side by gives or is the exact value of .
We would of course find the same result using .
We could also use , which gives since we know that is positive.
Let us now summarize what we have learned in this explainer.
Key Points
- The double-angle identities state that, for any real number , we have and for , we have
- From the above identity , we can derive the following two identities for any real number :
- The half-angle identities state that, for any real number , we have For , we have and for , we have
