Section 7.3 Double Angle Identities

Transcription

1 Section 7.3 Double Angle Identities 3 Section 7.3 Double Angle Identities Two special cases of the sum of angles identities arise often enough that we choose to state these identities separately. Identities The double angle identities sin( α sin( αcos( α cos(α cos ( α sin sin ( α ( α cos ( α These identities follow from the sum of angles identities. Proof of the sine double angle identity sin( α sin( α + α Apply the sum of angles identity sin( α cos( α + cos( αsin( α Simplify sin( αcos( α Establishing the identity Try it Now. Show cos(α cos ( α sin ( α by using the sum of angles identity for cosine. For the cosine double angle identity, there are three forms of the identity stated because the basic form, cos(α cos ( α sin ( α, can be rewritten using the Pythagorean Identity. Rearranging the Pythagorean Identity results in the equality cos ( α sin ( α, and by substituting this into the basic double angle identity, we obtain the second form of the double angle identity. cos(α cos ( α sin ( α Substituting using the Pythagorean identity cos(α sin ( α sin ( α Simplifying cos(α sin ( α

2 3 Chapter 7 Example 3 If sin( θ and θ is in the second quadrant, find exact values for sin( θ and 5 cos( θ. To evaluate cos( θ, since we know the value for sin( θ, we can use the version of the double angle that only involves sine. 3 cos(θ sin ( θ Since the double angle for sine involves both sine and cosine, we ll need to first find cos(θ, which we can do using the Pythagorean Identity. sin ( θ + cos ( θ 3 + cos ( θ 5 9 cos ( θ 5 6 cos( θ ± ± 5 5 Since θ is in the second quadrant, we know that cos(θ < 0, so cos( θ 5 Now we can evaluate the sine double angle 3 sin( θ sin( θ cos( θ Example Simplify the expressions a cos ( b 8 sin( 3x cos( 3x a Notice that the expression is in the same form as one version of the double angle identity for cosine: cos(θ cos ( θ. Using this, cos ( cos( cos( b This expression looks similar to the result of the double angle identity for sine. 8 sin( 3x cos( 3x Factoring a out of the original expression sin( 3x cos( 3x Applying the double angle identity sin(6x

3 Section 7.3 Double Angle Identities 33 We can use the double angle identities to simplify expressions and prove identities. Example cos(t Simplify. cos( t sin( t With three choices for how to rewrite the double angle, we need to consider which will be the most useful. To simplify this expression, it would be great if the denominator would cancel with something in the numerator, which would require a factor of cos( t sin( t in the numerator, which is most likely to occur if we rewrite the numerator with a mix of sine and cosine. cos(t cos( t sin( t Apply the double angle identity cos ( t sin ( t cos( t sin( t Factor the numerator ( cos( t sin( t ( cos( t + sin( t cos( t sin( t Cancelling the common factor cos( t + sin( t Resulting in the most simplified form Example 3 sec ( α Prove sec( α. sec ( α Since the right side seems a bit more complicated than the left side, we begin there. sec ( α Rewrite the secants in terms of cosine sec ( α cos ( α cos ( α At this point, we could rewrite the bottom with common denominators, subtract the terms, invert and multiply, then simplify. Alternatively, we can multiple both the top and bottom by cos ( α, the common denominator: cos ( α cos ( α Distribute on the bottom cos ( α cos ( α

4 3 Chapter 7 cos ( α cos ( α Simplify cos ( α cos ( α cos ( α Rewrite the denominator as a double angle cos ( α Rewrite as a secant cos(α sec( α Establishing the identity Try it Now. Use an identity to find the exact value of cos ( 75 sin ( 75. As with other identities, we can also use the double angle identities for solving equations. Example Solve cos( t cos( t for all solutions with 0 t < π. In general when solving trig equations, it makes things more complicated when we have a mix of sines and cosines and when we have a mix of functions with different periods. In this case, we can use a double angle identity to rewrite the cos(t. When choosing which form of the double angle identity to use, we notice that we have a cosine on the right side of the equation. We try to limit our equation to one trig function, which we can do by choosing the version of the double angle formula for cosine that only involves cosine. cos( t cos( t Apply the double angle identity cos ( t cos( t This is quadratic in cosine, so make one side 0 cos ( t cos( t 0 Factor cos( t + cos( t Break this apart to solve each part separately ( ( 0 cos( t + 0 or cos( t 0 cos( t or cos( t π π t or t 3 3 or t 0 Looking at a graph of cos(t and cos(t shown together, we can verify that these three solutions on [0, π seem reasonable.

5 Section 7.3 Double Angle Identities 35 Example 5 A cannonball is fired with velocity of 00 meters per second. If it is launched at an angle of θ, the vertical component of the velocity will be 00sin( θ and the horizontal component will be 00cos( θ. Ignoring wind resistance, the height of the cannonball will follow the equation h ( t.9t + 00sin( θ t and horizontal position will follow the equation x ( t 00cos( θ t. If you want to hit a target 900 meters away, at what angle should you aim the cannon? To hit the target 900 meters away, we want x( t 900at the time when the cannonball hits the ground, when h ( t 0. To solve this problem, we will first solve for the time, t, when the cannonball hits the ground. Our answer will depend upon the angleθ. h ( t 0.9t + 00sin( θ t 0 Factor t.9t + 00sin( θ Break this apart to find two solutions ( 0 t 0 or.9t + 00sin( θ 0 Solve for t.9t 00sin( θ 00sin( θ t.9 This shows that the height is 0 twice, once at t 0 when the cannonball is fired, and again when the cannonball hits the ground after flying through the air. This second value of t gives the time when the ball hits the ground in terms of the angle θ. We want the horizontal distance x(t to be 900 when the ball hits the ground, in other words 00sin( θ when t..9 Since the target is 900 m away we start with x ( t 900 Use the formula for x(t 00 cos( θ t 900 Substitute the desired time, t from above 00sin( θ 00 cos( θ Simplify 00 cos( θ sin( θ Isolate the cosine and sine product 900(.9 cos( θ sin( θ 00

6 36 Chapter 7 The left side of this equation almost looks like the result of the double angle identity for sin( θ sin θ cos θ. sine: ( ( By dividing both sides of the double angle identity by, we get sin(α sin( αcos( α. Applying this to the equation above, 900(.9 sin(θ 00 Multiply by (900(.9 sin( θ 00 Use the inverse sine (900(.9 θ sin Divide by.080 θ 0.50, or about 30.9 degrees Power Reduction and Half Angle Identities Another use of the cosine double angle identities is to use them in reverse to rewrite a squared sine or cosine in terms of the double angle. Starting with one form of the cosine double angle identity: cos(α cos ( α Isolate the cosine squared term cos(α + cos ( α Add cos(α + cos ( α Divide by cos(α + cos ( α This is called a power reduction identity Try it Now 3. Use another form of the cosine double angle identity to prove the identity cos(α sin ( α. Example 6 Rewrite cos ( x without any powers. Since cos ( x ( cos ( x cos ( x ( cos ( x, we can use the formula we found above

7 Section 7.3 Double Angle Identities 37 cos(x + ( cos( x + Square the numerator and denominator Expand the numerator cos (x + cos(x + Split apart the fraction cos (x cos(x + + Apply the formula above to cos (x cos( x + cos ( x cos(x + cos(x + + Simplify cos(x + + cos(x Combine the constants cos(x 3 + cos(x The cosine double angle identities can also be used in reverse for evaluating angles that cos(α + are half of a common angle. Building from our formula cos ( α, if we let θ cos( θ α, then α this identity becomes cos θ +. Taking the square root, we obtain cos( θ + cos ±, where the sign is determined by the quadrant. This is called a half-angle identity. Try it Now. Use your results from the last Try it Now to prove the identity cos( θ sin ±. Example 7 Find an exact value for ( 5 cos. Since 5 degrees is half of 30 degrees, we can use our result from above:

8 38 Chapter 7 30 cos(5 cos ± cos(30 + We can evaluate the cosine. Since 5 degrees is in the first quadrant, we need the positive result. cos( Identities Half-Angle Identities cos( θ + cos ± sin ± cos( θ Power Reduction Identities cos(α + cos ( α cos(α sin ( α Since these identities are easy to derive from the double-angle identities, the power reduction and half-angle identities are not ones you should need to memorize separately. Important Topics of This Section Double angle identity Power reduction identity Half angle identity Using identities Simplify equations Prove identities Solve equations Try it Now Answers cos α cos( α + α (. cos( αcos( α sin( αsin( α cos ( α sin ( α

9 Section 7.3 Double Angle Identities 39. cos( cos( α ( α α cos ( sin ( α + α + α cos ( sin ( sin ( sin ( sin ( α α sin ( α. cos(α sin ( α cos(α sin( α ± θ α cos sin ± cos( θ sin ±

10 0 Chapter 7 Section 7.3 Exercises. If sin (. If cos( x and x is in quadrant I, then find exact values for (without solving for x: 8 tan x a. sin ( x b. cos( x c. ( x and x is in quadrant I, then find exact values for (without solving for x: 3 tan x a. sin ( x b. cos( x c. ( Simplify each expression. cos 8 sin (8 3. (. ( 5. sin (7 6. ( 7. cos ( 9x sin (9 x 8. ( x 9. sin ( 8x cos(8 x 0. ( x Solve for all solutions on the interval [0, π. 6sin t 9sin t 0 cos 37 cos 37 sin (37 cos 6 sin (6 x 6sin 5 cos(5 x. ( + (. sin ( t + 3cos( t cos( θ 9cos ( θ. ( α ( α 5. sin ( t cos( t 6. cos( t sin ( t 7. cos( 6x cos( 3x 0 8. sin ( x sin ( x 0 8cos 8cos Use a double angle, half angle, or power reduction formula to rewrite without exponents. 9. cos (5 x 0. cos (6 x. sin (8 sin 3x 3. cos x. ( xsin x. cos 5. If csc( 7 xsin x and 90 < x < 80, then find exact values for (without solving for x: a. sin b. cos c. tan 6. If sec( x and 90 < x < 80, then find exact values for (without solving for x: a. sin b. cos c. tan x

11 Section 7.3 Double Angle Identities Prove the identity. sin t cost sin t 7. ( ( 8. ( ( sin x cos x + sin x tan ( x 9. sin ( x + tan ( x sin ( x cos( x 30. tan ( x cos ( x 3. cot ( x tan ( x cot ( x sin ( θ 3. tan ( θ + cos( θ tan ( α 33. cos( α + tan ( α + cos( t cos( t 3. sin ( t cos( t sin ( t sin ( 3x 3sin ( x cos ( x sin ( x cos 3x cos 3 ( x 3sin ( xcos x 36. ( (