Section 3.3 Double-Angle and Half-Angle Formulas
Suppose we want to accurately position the Hawaiian Star Compass on the Unit Circle. In Figure 1.1.4, the house for Manu is located halfway between Hikina and ʻĀkau, resulting in an angle of By applying right triangle trigonometry, we can determine the exact coordinates of Manu as However, as we move to the house of ʻĀina, located halfway between Manu and Hikina, we encounter a problem. The angle for ʻĀina is which is not explicitly listed in Table 1.5.18. Therefore, we must resort to a calculator for numerical approximations.
In this section, we will learn about the double and half-angle formulas for trigonometry. These formulas allow us to determine exact trigonometric function values for angles that are double or half of the common angles. This will enable us to use our existing knowledge of trigonometric functions at and apply the half-angle formulas to obtain exact values at
Subsection 3.3.1 Double-Angle Formulas
Consider the case when the two angles are equal. We will call this angle so let and Then Eq (3.3.1) becomes
Thus we obtain a formula for sine of twice the angle
The proofs for the Double-Angle Formulas for Cosine and Tangent are left as exercises (Exercise 3.3.4.6-Exercise 3.3.4.9).
Remark 3.3.2.
Notice that there are three variations of the double-angle formula for cosine. All three equations give the correct answer, however, one version may be more convenient depending on the given information. For example, if we are given the value of it may be easier to select the version that solely involves and does not include
Example 3.3.3.
Given and lies in Quadrant III, find the exact value of:
(a)
Solution.
By the double-angle formula, we have We are given the value of but we do not have To find we will draw the triangle formed from where lies in Quadrant III.
(b)
Solution.
To compute notice there are three different formulas: or Using any of the three equations will give us the correct answer. However, given that we know it may be easier to use since the other two equations require us to know
(c)
Solution.
Example 3.3.4.
Write in terms of
Solution.
Subsection 3.3.2 Reducing Powers Formulas
The double-angle formula for cosine expresses a trigonometric function in terms of the square of another trigonometric function. By rearranging the terms, we can derive formulas for reducing the powers of sine, cosine, and tangent in expressions with even powers to terms involving only cosine. These formulas are particularly useful in calculus.
Definition 3.3.5. Formulas for Reducing Powers.
Proof.
To prove the first formula, solve for in the double-angle formula: The second formula is obtained similarly by solving for in the formula The first two formulas can be used to obtain the third formula:
Example 3.3.6.
Write as an expression that does not involve powers of sine or cosine greater than 1.
Solution.
Subsection 3.3.3 Half-Angle Formulas
Another set of useful formulas are the half-angle formulas.
Definition 3.3.7. Half-Angle Formulas.
The choice of the + or - sign depends on the Quadrant in which lies.
Proof.
We take the square root of both sides of the Formulas for Reducing Powers (Definition 3.3.5) and halve the angle ( becomes and becomes ) to arrive at our formulas.
Example 3.3.8. Locating ʻĀina.
We are now ready to revisit the problem posed at the start of this section where we were asked to determine the exact coordinates of the house ʻĀina on the Unit Circle.
Solution.
We know the coordinates are at
Since the half-angle formula has we check the quadrant. In this case, our angle is which is in Quadrant I. Therefore, we choose the positive value.
Example 3.3.9.
Given and lies in Quadrant III, find the exact value of:
(a)
Solution.
Notice the Half-Angle Formulas all require us to know Since the given information describes the same triangle as in Example 3.3.3, we refer to that problem to get
Next, since is in Quadrant III, dividing by 2 gives us or Therefore, we conclude that lies in Quadrant II.
To calculate we first note that because lies in Quadrant II, so we will choose the positive (+) sign in the Half-Angle Formula:
(b)
Solution.
Since is in Quadrant II, we know that so we will choose the negative (-) sign in the Half-Angle Formula:
(c)
Solution.
Since is in Quadrant II, we know that so we will choose the negative (-) sign in the Half-Angle Formula:
Definition 3.3.10. Half-Angle Formulas for Tangent.
Proof.
We begin by first multiplying both sides of the Sine formula for Reducing Powers by 2 and halving the angle:
and applying the double-angle formula to:
Example 3.3.11.
Calculate from Example 3.3.9 using the above formula.
Solution.
Note: We obtained the same result for as we did in Example 3.3.9. In this example, we did not have to determine whether was positive or negative, however, we need to know the values of both and
Exercises 3.3.4 Exercises
Exercise Group.
The house ʻĀina is located at and the house Nā Leo is located at Use the half-angle formulas to evaluate the exact value of the given expression at each of these houses.
6.
Use the Addition Formula, to prove the double angle formula for cosine:
7.
Use the Pythagorean Identity ( ) and the result from Exercise 3.3.4.6 to prove
8.
Use the Pythagorean Identity ( ) and the result from Exercise 3.3.4.6 to prove
9.
Use the addition formula, to prove the double angle formula for tangent:
Exercise Group.
Use the figure below to find the exact values for each of the following exercises.
10.
Answer.
11.
Answer.
12.
Answer.
13.
Answer.
14.
Answer.
15.
Answer.
16.
Answer.
17.
Answer.
18.
Answer.
Exercise Group.
Use the figure below to find the exact values for each of the following exercises.
19.
Answer.
20.
Answer.
21.
Answer.
22.
Answer.
23.
Answer.
24.
Answer.
25.
Answer.
26.
Answer.
27.
Answer.
Exercise Group.
Find the exact value of each expression given and is in Quadrant III.
Exercise Group.
Find the exact value of each expression given and
Exercise Group.
Find the exact value of each expression given and
Exercise Group.
Use the Half-Angle Formula to find the exact value of each of the following
Exercise Group.
Write each of the following as expressions that do not involve powers of sine or cosine greater than 1.
64.
Write expressions that does not involve powers of sine or cosine greater than 1.
(a)
Using the Reducing Powers Formula (Definition 3.3.5)
Answer.
(b)
Using the Double Angle Formula (Definition 3.3.1) where
Answer.
65.
Find the exact value of
(a)
Evaluate using the Reducing Powers Formula (Definition 3.3.5)
Answer.
(b)
Evaluate using the Half-Angle Formula (Definition 3.3.7)
Answer.
Exercise Group.
Use half angle to find the exact deviation for the indicated angle,