In the previous section, we used addition and subtraction formulas for trigonometric functions. Now, we take another look at those same formulas. The double-angle formulas are a special case of the sum formulas, where α=β. Deriving the double-angle formula for sine begins with the sum formula,
sin(α+β)=sinαcosβ+cosαsinβ
If we let α=β=θ, then we have
sin(θ+θ)=sinθcosθ+cosθsinθsin(2θ)=2sinθcosθ
Deriving the double-angle for cosine gives us three options. First, starting from the sum formula, cos(α+β)=cosαcosβ−sinαsinβ, and letting α=β=θ, we have
cos(θ+θ)=cosθcosθ−sinθsinθcos(2θ)=cos2θ−sin2θ
Using the Pythagorean properties, we can expand this double-angle formula for cosine and get two more interpretations. The first one is:
cos(2θ)=cos2θ−sin2θ=(1−sin2θ)−sin2θ=1−2sin2θ
The second interpretation is:
cos(2θ)=cos2θ−sin2θ=cos2θ−(1−cos2θ)=2cos2θ−1
Similarly, to derive the double-angle formula for tangent, replacing α=β=θ in the sum formula gives
The double-angle formulas are summarized as follows:
sin(2θ)=2sinθcosθ
cos(2θ)=cos2θ−sin2θ=1−2sin2θ=2cos2θ−1
tan(2θ)=1−tan2θ2tanθ
How To: Given the tangent of an angle and the quadrant in which it is located, use the double-angle formulas to find the exact value.
Draw a triangle to reflect the given information.
Determine the correct double-angle formula.
Substitute values into the formula based on the triangle.
Simplify.
Example 1: Using a Double-Angle Formula to Find the Exact Value Involving Tangent
Given that tanθ=−43 and θ is in quadrant II, find the following:
sin(2θ)
cos(2θ)
tan(2θ)
Solution
If we draw a triangle to reflect the information given, we can find the values needed to solve the problems on the image. We are given tanθ=−43, such that θ is in quadrant II. The tangent of an angle is equal to the opposite side over the adjacent side, and because θ is in the second quadrant, the adjacent side is on the x-axis and is negative. Use the Pythagorean Theorem to find the length of the hypotenuse:
(−4)2+(3)2=c216+9=c225=c2c=5
Now we can draw a triangle similar to the one shown in Figure 2.
Figure 2
Let’s begin by writing the double-angle formula for sine.
sin(2θ)=2sinθcosθ
We see that we to need to find sinθ and cosθ. Based on Figure 2, we see that the hypotenuse equals 5, so sinθ=53, and cosθ=−54. Substitute these values into the equation, and simplify.
Thus,
sin(2θ)=2(53)(−54)=−2524
Write the double-angle formula for cosine.
cos(2θ)=cos2θ−sin2θ
Again, substitute the values of the sine and cosine into the equation, and simplify.
cos(2θ)=(−54)2−(53)2=2516−259=257
Write the double-angle formula for tangent.
tan(2θ)=1−tan2θ2tanθ
In this formula, we need the tangent, which we were given as tanθ=−43. Substitute this value into the equation, and simplify.
This example illustrates that we can use the double-angle formula without having exact values. It emphasizes that the pattern is what we need to remember and that identities are true for all values in the domain of the trigonometric function.
Using Double-Angle Formulas to Verify Identities
Establishing identities using the double-angle formulas is performed using the same steps we used to derive the sum and difference formulas. Choose the more complicated side of the equation and rewrite it until it matches the other side.
Example 3: Using the Double-Angle Formulas to Establish an Identity
Establish the following identity using double-angle formulas:
1+sin(2θ)=(sinθ+cosθ)2
Solution
We will work on the right side of the equal sign and rewrite the expression until it matches the left side.
This process is not complicated, as long as we recall the perfect square formula from algebra:
(a±b)2=a2±2ab+b2
where a=sinθ and b=cosθ. Part of being successful in mathematics is the ability to recognize patterns. While the terms or symbols may change, the algebra remains consistent.
Try It 2
Establish the identity: cos4θ−sin4θ=cos(2θ).
Solution
Example 4: Verifying a Double-Angle Identity for Tangent
Verify the identity:
tan(2θ)=cotθ−tanθ2
Solution
In this case, we will work with the left side of the equation and simplify or rewrite until it equals the right side of the equation.
tan(2θ)=1−tan2θ2tanθ=(1−tan2θ)(tanθ1)2tanθ(tanθ1)=tanθ1−tanθtan2θ2=cotθ−tanθ2Double-angle formulaMultiply by a term that results in desired numerator.Use reciprocal identity for tanθ1.
Analysis of the Solution
Here is a case where the more complicated side of the initial equation appeared on the right, but we chose to work the left side. However, if we had chosen the left side to rewrite, we would have been working backwards to arrive at the equivalency. For example, suppose that we wanted to show
When using the identities to simplify a trigonometric expression or solve a trigonometric equation, there are usually several paths to a desired result. There is no set rule as to what side should be manipulated. However, we should begin with the guidelines set forth earlier.
Try It 3
Verify the identity: cos(2θ)cosθ=cos3θ−cosθsin2θ.
Solution
Use Reduction Formulas to Simplify an Expression
The double-angle formulas can be used to derive the reduction formulas, which are formulas we can use to reduce the power of a given expression involving even powers of sine or cosine. They allow us to rewrite the even powers of sine or cosine in terms of the first power of cosine. These formulas are especially important in higher-level math courses, calculus in particular. Also called the power-reducing formulas, three identities are included and are easily derived from the double-angle formulas.
We can use two of the three double-angle formulas for cosine to derive the reduction formulas for sine and cosine. Let’s begin with cos(2θ)=1−2sin2θ. Solve for sin2θ:
The last reduction formula is derived by writing tangent in terms of sine and cosine:
tan2θ=cos2θsin2θ=21+cos(2θ)21−cos(2θ)=(21−cos(2θ))(1+cos(2θ)2)=1+cos(2θ)1−cos(2θ)Substitute the reduction formulas.
A General Note: Reduction Formulas
The reduction formulas are summarized as follows:
sin2θ=21−cos(2θ)
cos2θ=21+cos(2θ)
tan2θ=1+cos(2θ)1−cos(2θ)
Example 5: Writing an Equivalent Expression Not Containing Powers Greater Than 1
Write an equivalent expression for cos4x that does not involve any powers of sine or cosine greater than 1.
Solution
We will apply the reduction formula for cosine twice.
cos4x=(cos2x)2=(21+cos(2x))2=41(1+2cos(2x)+cos2(2x))=41+21cos(2x)+41(21+cos2(2x))=41+21cos(2x)+81+81cos(4x)=83+21cos(2x)+81cos(4x)Substitute reduction formula for cos2x. Substitute reduction formula for cos2x.
Analysis of the Solution
The solution is found by using the reduction formula twice, as noted, and the perfect square formula from algebra.
Example 6: Using the Power-Reducing Formulas to Prove an Identity
Use the power-reducing formulas to prove
sin3(2x)=[21sin(2x)][1−cos(4x)]
Solution
We will work on simplifying the left side of the equation:
sin3(2x)=[sin(2x)][sin2(2x)]=sin(2x)[21−cos(4x)]=sin(2x)(21)[1−cos(4x)]=21[sin(2x)][1−cos(4x)]Substitute the power-reduction formula.
Analysis of the Solution
Note that in this example, we substituted
21−cos(4x)
for sin2(2x). The formula states
sin2θ=21−cos(2θ)
We let θ=2x, so 2θ=4x.
Try It 4
Use the power-reducing formulas to prove that 10cos4x=415+5cos(2x)+45cos(4x).
Solution
Using Half-Angle Formulas to Find Exact Values
The next set of identities is the set of half-angle formulas, which can be derived from the reduction formulas and we can use when we have an angle that is half the size of a special angle. If we replace θ with 2α, the half-angle formula for sine is found by simplifying the equation and solving for sin(2α). Note that the half-angle formulas are preceded by a ± sign. This does not mean that both the positive and negative expressions are valid. Rather, it depends on the quadrant in which 2α terminates.
The half-angle formula for sine is derived as follows:
Notice that we used only the positive root because sin(15o) is positive.
How To: Given the tangent of an angle and the quadrant in which the angle lies, find the exact values of trigonometric functions of half of the angle.
Draw a triangle to represent the given information.
Determine the correct half-angle formula.
Substitute values into the formula based on the triangle.
Simplify.
Example 8: Finding Exact Values Using Half-Angle Identities
Given that tanα=158 and α lies in quadrant III, find the exact value of the following:
sin(2α)
cos(2α)
tan(2α)
Solution
Using the given information, we can draw the triangle shown in Figure 3. Using the Pythagorean Theorem, we find the hypotenuse to be 17. Therefore, we can calculate sinα=−178 and cosα=−1715.
Figure 3
Before we start, we must remember that, if α is in quadrant III, then 180∘<α<270∘, so 2180∘<2α<2270∘. This means that the terminal side of 2α is in quadrant II, since 90∘<2α<135∘.To find sin2α, we begin by writing the half-angle formula for sine. Then we substitute the value of the cosine we found from the triangle in Figure 3 and simplify.
We choose the positive value of sin2α because the angle terminates in quadrant II and sine is positive in quadrant II.
To find cos2α, we will write the half-angle formula for cosine, substitute the value of the cosine we found from the triangle in Figure 3, and simplify.
We choose the negative value of cos2α because the angle is in quadrant II because cosine is negative in quadrant II.
To find tan2α, we write the half-angle formula for tangent. Again, we substitute the value of the cosine we found from the triangle in Figure 3 and simplify.
We choose the negative value of tan2α because 2α lies in quadrant II, and tangent is negative in quadrant II.
Try It 5
Given that sinα=−54 and α lies in quadrant IV, find the exact value of cos(2α).
Solution
Example 9: Finding the Measurement of a Half Angle
Now, we will return to the problem posed at the beginning of the section. A bicycle ramp is constructed for high-level competition with an angle of θ formed by the ramp and the ground. Another ramp is to be constructed half as steep for novice competition. If tanθ=35 for higher-level competition, what is the measurement of the angle for novice competition?
Solution
Since the angle for novice competition measures half the steepness of the angle for the high level competition, and tanθ=35 for high competition, we can find cosθ from the right triangle and the Pythagorean theorem so that we can use the half-angle identities.
32+52=34c=34
Figure 4
We see that cosθ=343=34334. We can use the half-angle formula for tangent: tan2θ=1+cosθ1−cosθ. Since tanθ is in the first quadrant, so is tan2θ. Thus,
Double-angle identities are derived from the sum formulas of the fundamental trigonometric functions: sine, cosine, and tangent.
Reduction formulas are especially useful in calculus, as they allow us to reduce the power of the trigonometric term.
Half-angle formulas allow us to find the value of trigonometric functions involving half-angles, whether the original angle is known or not.
Glossary
double-angle formulas
identities derived from the sum formulas for sine, cosine, and tangent in which the angles are equal
half-angle formulas
identities derived from the reduction formulas and used to determine half-angle values of trigonometric functions
reduction formulas
identities derived from the double-angle formulas and used to reduce the power of a trigonometric function
Section Exercises
1. Explain how to determine the reduction identities from the double-angle identity cos(2x)=cos2x−sin2x.
2. Explain how to determine the double-angle formula for tan(2x) using the double-angle formulas for cos(2x) and sin(2x).
3. We can determine the half-angle formula for tan(2x)=1+cosx1−cosx by dividing the formula for sin(2x) by cos(2x). Explain how to determine two formulas for tan(2x) that do not involve any square roots.
4. For the half-angle formula given in the previous exercise for tan(2x), explain why dividing by 0 is not a concern. (Hint: examine the values of cosx necessary for the denominator to be 0.)
For the following exercises, find the exact values of a) sin(2x), b) cos(2x), and c) tan(2x) without solving for x.
5. If sinx=81, and x is in quadrant I.
6. If cosx=32, and x is in quadrant I.
7. If cosx=−21, and x is in quadrant III.
8. If tanx=−8, and x is in quadrant IV.
For the following exercises, find the values of the six trigonometric functions if the conditions provided hold.
9. cos(2θ)=53 and 90∘≤θ≤180∘
10. cos(2θ)=21 and 180∘≤θ≤270∘
For the following exercises, simplify to one trigonometric expression.
11. 2sin(4π)2cos(4π)
12. 4sin(8π)cos(8π)
For the following exercises, find the exact value using half-angle formulas.
13. sin(8π)
14. cos(−1211π)
15. sin(1211π)
16. cos(87π)
17. tan(125π)
18. tan(−123π)
19. tan(−83π)
For the following exercises, find the exact values of a) sin(2x), b) cos(2x), and c) tan(2x) without solving for x.
20. If tanx=−34, and x is in quadrant IV.
21. If sinx=−1312, and x is in quadrant III.
22. If cscx=7, and x is in quadrant II.
23. If secx=−4, and x is in quadrant II.
For the following exercises, use Figure 5 to find the requested half and double angles.
Figure 5
24. Find sin(2θ),cos(2θ), and tan(2θ).
25. Find sin(2α),cos(2α), and tan(2α).
26. Find sin(2θ),cos(2θ), and tan(2θ).
27. Find sin(2α),cos(2α), and tan(2α).
For the following exercises, simplify each expression. Do not evaluate.
28. cos2(28∘)−sin2(28∘)
29. 2cos2(37∘)−1
30. 1−2sin2(17∘)
31. cos2(9x)−sin2(9x)
32. 4sin(8x)cos(8x)
33. 6sin(5x)cos(5x)
For the following exercises, prove the identity given.
34. (sint−cost)2=1−sin(2t)
35. sin(2x)=−2sin(−x)cos(−x)
36. cotx−tanx=2cot(2x)
37. 1+cos(2θ)sin(2θ)tan2θ=tanθ
For the following exercises, rewrite the expression with an exponent no higher than 1.
38. cos2(5x)
39. cos2(6x)
40. sin4(8x)
41. sin4(3x)
42. cos2xsin4x
43. cos4xsin2x
44. tan2xsin2x
For the following exercises, reduce the equations to powers of one, and then check the answer graphically.
45. tan4x
46. sin2(2x)
47. sin2xcos2x
48. tan2xsinx
49. tan4xcos2x
50. cos2xsin(2x)
51. cos2(2x)sinx
52. tan2(2x)sinx
For the following exercises, algebraically find an equivalent function, only in terms of sinx and/or cosx, and then check the answer by graphing both equations.
53. sin(4x)
54. cos(4x)
For the following exercises, prove the identities.
55. sin(2x)=1+tan2x2tanx
56. cos(2α)=1+tan2α1−tan2α
57. tan(2x)=2cos2x−12sinxcosx
58. (sin2x−1)2=cos(2x)+sin4x
59. sin(3x)=3sinxcos2x−sin3x
60. cos(3x)=cos3x−3sin2xcosx
61. sin(2t)−cost1+cos(2t)=2sint−12cost
62. sin(16x)=16sinxcosxcos(2x)cos(4x)cos(8x)
63. cos(16x)=(cos2(4x)−sin2(4x)−sin(8x))(cos2(4x)−sin2(4x)+sin(8x))