3.5 Double Angle Identities

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1 3.5. Double Angle Identities Double Angle Identities Learning Objectives Use the double angle identities to solve other identities. Use the double angle identities to solve equations. Deriving the Double Angle Identities One of the formulas for calculating the sum of two angles is: sin(α+β)sinαcosβ+cosαsinβ If α and β are both the same angle in the above formula, then sin(α+α)sinαcosα+cosαsinα sinαsinαcosα This is the double angle formula for the sine function. The same procedure can be used in the sum formula for cosine, start with the sum angle formula: cos(α+β)cosαcosβ sinαsinβ If α and β are both the same angle in the above formula, then cos(α+α)cosαcosα sinαsinα cosαcos α sin α This is one of the double angle formulas for the cosine function. Two more formulas can be derived by using the Pythagorean Identity, sin α+cos α1. sin α1 cos α and likewise cos α1 sin α Using sin α1 cos α : Using cos α1 sin α : cosαcos α sin α cosαcos α sin α cos α (1 cos α) (1 sin α) sin α cos α 1+cos α 1 sin α sin α cos α 1 1 sin α 6

2 Chapter 3. Trigonometric Identities and Equations Therefore, the double angle formulas for cosa are: cosαcos α sin α cosαcos α 1 cosα1 sin α Finally, we can calculate the double angle formula for tangent, using the tangent sum formula: tan(α+β) tanα+tanβ 1 tanαtanβ If α and β are both the same angle in the above formula, then tan(α+α) tanα+tanα 1 tanαtanα tanα tanα 1 tan α Applying the Double Angle Identities Example 1: If sina 5 13 and a is in Quadrant II, find sina, cosa, and tana. Solution: To use sinasinacosa, the value of cosa must be found first.. cos a+sin a1 ( ) 5 cos a cos a cos a ,cosa±1 13 However since a is in Quadrant II, cosa is negative or cosa For cosa, use cos(a)cos a sin a ( ) ( 5 sinasinacosa 1 ) sina ( cos(a) 1 ) ( ) 5 or cos(a)

3 3.5. Double Angle Identities For tana, use tana tana 5. From above, tana 1 tan 13 a 1 13 Example : Find cos4θ. tan(a) ( 5 1 Solution: Think of cos4θ as cos(θ+θ). ) cos4θcos(θ+θ)cosθcosθ sinθsinθcos θ sin θ Now, use the double angle formulas for both sine and cosine. For cosine, you can pick which formula you would like to use. In general, because we are proving a cosine identity, stay with cosine. (cos θ 1) (sinθcosθ) 4cos 4 θ 4cos θ+1 4sin θcos θ 4cos 4 θ 4cos θ+1 4(1 cos θ)cos θ 4cos 4 θ 4cos θ+1 4cos θ+4cos 4 θ 8cos 4 θ 8cos θ+1 Example 3: If cotx 4 3 and x is an acute angle, find the exact value of tanx. Solution: Cotangent and tangent are reciprocal functions, tanx 1 cotx and tanx 3 4. tanx tanx 1 tan x ( ) Example 4: Given sin(x) 3 and x is in Quadrant I, find the value of sinx. Solution: Using the double angle formula, sin x sin x cos x. Because we do not know cos x, we need to solve for cosx in the Pythagorean Identity, cosx 1 sin x. Substitute this into our formula and solve for sinx. 8 sinxsinxcosx 3 sinx 1 sin x ( ) ) ( sinx 1 sin x sin x(1 sin x) 4 9 4sin x 4sin 4 x

4 Chapter 3. Trigonometric Identities and Equations At this point we need to get rid of the fraction, so multiply both sides by the reciprocal. ( ) sin x 4sin 4 x 19sin x 9sin 4 x 09sin 4 x 9sin x+1 Now, this is in the form of a quadratic equation, even though it is a quartic. Set a sin x, making the equation 9a 9a+10. Once we have solved for a, then we can substitute sin x back in and solve for x. In the Quadratic Formula, a9,b 9,c1. 9± ( 9) 4(9)(1) (9) 9± ± ± ± 5 6 So, a or This means that sin x or.173 so sinx or sinx.357. Example 5: Prove tanθ 1 cosθ sinθ Solution: Substitute in the double angle formulas. Use cosθ1 sin θ, since it will produce only one term in the numerator. tanθ 1 (1 sin θ) sinθcosθ sin θ sinθcosθ sinθ cosθ tanθ Solving Equations with Double Angle Identities Much like the previous sections, these problems all involve similar steps to solve for the variable. trigonometric function, using any of the identities and formulas you have accumulated thus far. Example 6: Find all solutions to the equation sinxcosx in the interval[0,π] Solution: Apply the double angle formula sin x sin x cos x Isolate the 9

5 3.5. Double Angle Identities sinxcosxcosx sinxcosx cosxcosx cosx sinxcosx cosx0 cosx(sinx 1)0 Factor out cosx Then cosx0 or sinx 10 cosx0 or sinx sinx 1 sinx 1 The values for cosx0 in the interval[0,π] are x π and x 3π and the values for sinx 1 in the interval[0,π] are x π 6 and x 5π 6. Thus, there are four solutions. Example 7: Solve the trigonometric equation sin x sin x such that( π x < π) Solution: Using the sine double angle formula: sinxsinx sinxcosx sinx sinxcosx sinx0 sinx(cosx 1)0 ց cosx 10 cosx1 sinx0 x0, π cosx 1 x π 3, π 3 Example 8: Find the exact value of cosx given cosx if x is in the second quadrant. Solution: Use the double-angle formula with cosine only. 30

6 Chapter 3. Trigonometric Identities and Equations cosxcos x 1 ( cosx 13 ) 1 14 ( ) 169 cosx ( ) 338 cosx cosx cosx Example 9: Solve the trigonometric equation 4sinθcosθ 3 over the interval[0,π). Solution: Pull out a from the left-hand side and this is the formula for sinx. 4sinθcosθ 3 (sinθcosθ) 3 (sinθcosθ)sinθ sinθ 3 3 sinθ The solutions for θ are π 3, π 3, 7π 3, 8π 3, dividing each of these by, we get the solutions for θ, which are π 6, π 3, 7π 6, 8π 6. Points to Consider Are there similar formulas that can be derived for other angles? Can technology be used to either solve these trigonometric equations or to confirm the solutions? Review Questions 1. If sinx 4 5 and x is in Quad II, find the exact values of cosx,sinx and tanx. Find the exact value of cos 15 sin Verify the identity: cos3θ4cos 3 θ 3cosθ 4. Verify the identity: sint tant tant cost 5. If sinx 9 41 and x is in Quad III, find the exact values of cosx,sinx and tanx 6. Find all solutions to sinx+sinx0 if 0 x<π 7. Find all solutions to cos x cosx0 if 0 x<π 8. If tanx 3 4 and 0 < x<90, use the double angle formulas to determine each of the following: a. tanx b. sin x 31

7 3.5. Double Angle Identities c. cosx 9. Use the double angle formulas to prove that the following equations are identities. a. cscxcsc xtanx b. cos 4 θ sin 4 θcosθ c. sinx 1+cosx tanx 10. Solve the trigonometric equation cosx 1sin x such that[0,π) 11. Solve the trigonometric equation cos x cos x such that 0 x < π 1. Prove cscxtanxsec x. 13. Solve sinx cosx1 for x in the interval[0,π). 14. Solve the trigonometric equation sin x cosx such that 0 x<π Review Answers 1. If sinx 4 5 and in Quadrant II, then cosine and tangent are negative. Also, by the Pythagorean Theorem, the third side is 3(b 5 4 ). So, cosx 3 5 and tanx 4 3. Using this, we can find sinx,cosx, and tan x. sinxsinxcosx This is one of the forms for cosx. cosx1 sin x tanx tanx 1 tan x ( ) ( ) cos 15 sin 15 cos(15 ) cos Step 1: Use the cosine sum formula cos3θ4cos 3 θ 3cosθ cos(θ+θ)cosθcosθ sinθsinθ Step : Use double angle formulas for cosθ and sinθ (cos θ 1)cosθ (sinθcosθ)sinθ 3

8 Chapter 3. Trigonometric Identities and Equations Step 3: Distribute and simplify. cos 3 θ cosθ sin θcosθ cosθ( cos θ+sin θ+1) cosθ[ cos θ+(1 cos θ)+1] cosθ[ cos θ+ cos θ+1] cosθ( 4cos θ+3) 4cos 3 θ 3cosθ Substitute 1 cos θ for sin θ 4. Step 1: Expand sint using the double angle formula. Step : change tant and find a common denominator. 5. If sinx 9 41 So, 6. Step 1: Expand sin x sint tant tant cost sint cost tant tant cost sint cost sint cost sint cos t sint cost sint(cos t 1) cost sint cost (cos t 1) tant cost and in Quadrant III, then cosx and tanx 9 40 (Pythagorean Theorem, b 41 ( 9) ). cosx cos x 1 ( sinxsinxcosx 40 ) 1 tanx sinx 41 cosx Step : Separate and solve each for x sinx+sinx0 sinxcosx+sinx0 sinx(cosx+1)0 cosx sinx0 cosx 1 x0,π or x π 3, 4π 3 33

9 3.5. Double Angle Identities 7. Expand cosx and simplify cos x cosx0 cos x (cos x 1)0 cos x+10 cos x1 cosx±1 cosx1 when x0, and cosx 1 when xπ. Therefore, the solutions are x0,π. 8. a b c a. cscx x sinx cscx x sinxcosx 1 cscx x ( sinxcosx )( ) sinx 1 cscx x sinx sinxcosx cscx x sinx sin xcosx cscx x 1 sin x sinx cosx cscx xcsc xtanx b. cos 4 θ sin 4 θ(cos θ+sin θ)(cos θ sin θ) cos 4 sin 4 θ1(cos θ sin θ) cosθcos θ sin θ cos 4 θ sin 4 θcosθ 34 c. sinx 1+cosx sinxcosx 1+(1 sin x) sinx 1+cosx sinxcosx sin x sinx 1+cosx sinxcosx (1 sin x) sinx 1+cosx sinxcosx cos x sinx 1+cosx sinx cosx sinx 1+cosx tanx

10 Chapter 3. Trigonometric Identities and Equations 10. cosx 1sin x (1 sin x) 1sin x sin xsin x 03sin x 0sin x 0sinx x0,π 11. cosxcosx cos x 1cosx cos x cosx 10 (cosx+1)(cosx 1)0 ց ց cosx+10 or cosx 10 cosx 1 cosx1 cosx 1 1. cosx1 when x0 and cosx 1 when x π 3. cscxtanxsec x sinx sinx cosx 1 cos x sinxcosx sinx cosx 1 cos x 1 cos x 1 cos x 35

11 3.5. Double Angle Identities sinx cosx1 sinxcosx (1 sin x)1 sinxcosx 1+sin x1 sinxcosx+sin x sinxcosx+sin x1 sinxcosx1 sin x sinxcosxcos x ( ) ± 1 cos x cosxcos x ( 1 cos x ) cos xcos 4 x cos x cos 4 xcos 4 x cos x cos 4 x0 cos x(1 cos x)0 ւ ց 1 cos x0 cos x0 cos x 1 cosx0 or cos x 1 x π, 3π cosx± x π 4, 5π 4 Note: If we go back to the equation sinxcosx cos x, we can see that sinxcosx must be positive or zero, since cos x is always positive or zero. For this reason, sinx and cosx must have the same sign (or one of them must be zero), which means that x cannot be in the second or fourth quadrants. This is why 3π 4 and 7π 4 are not valid solutions. 14. Use the double angle identity for cosx. sin x cosx sin x cosx sin x 1 sin x 3sin x3 sin x1 sinx±1 x π, 3π 36

3.1 Fundamental Identities

3.1 Fundamental Identities www.ck.org Chapter. Trigonometric Identities and Equations. Fundamental Identities Introduction We now enter into the proof portion of trigonometry. Starting with the basic definitions of sine, cosine,