1. Power reducing formulas can be used to rewrite 8sin^2 xcos^2 x as 1-cos4x. True or False?
2. Find the exact value of cos15 degrees using a suitable half angle formula.
square root of 2+root3 / 2?
3. Find the exact value of cos^2(pi/8) using a suitable power reducing formula.
2- root2 / 4, I think..?
4. If secx = -3, and x lies in quadrant II find tan x/2
I got -root2
5. Power reducing formulas can be used to rewrite sin^2 xcos^2 x as 1/8 + cos4x / 4
True?
if a question is simply a True or False, and no work has to be shown, a good method for your #1 would be to take an unusual value of x and test it with your calculator.
e.g. let x = 32.7° (use your calculator memory to store intermediate answers
I get 8sin^2 (32.7°)cos^2 (32.7°) = appr 1.6534..
1 - cos(4(32.7)) = 1.6534
YUP, there is a very high probability that it is TRUE
for #2, your answer > 1, but the cosine of "anything" cannot be > 1 , so you should have known your answer to be incorrect
cos 15° ..... notice 15 is half of 30
cos 30 = 2cos^2 15° - 1
√3/2 + 1 = 2cos^2 15°
(√3 + 2)/4 = cos^2 15°
cos 15° = √(√3+2)/2 , check with a calculator, it works!
4. if secx = -3
then cosx = -1/3 and in II
make a sketch of a triangle, to find
sinx = √8/3
tanx = -√8
so tanx = 2tan(x/2) / (1 - tan^2 (x/2)
let tan (x/2) = t
-√8 = 2t/(1 - t^2)
-√8 + √8t^2 = 2t
√8 t^2 - 2t - √8 = 0
t = (2 ± √(2 + 32)/(2√8)
= 4/√8 or -2/√8
or t = √2 or t = -√2/2 after rationalizing
but if x is in II, the x/2 must be in quadrant I
and tan (x/2) = + √2
hint for #5
sin^2 x cos^2 x
= (sinxcosx)^2
= ( (1/2)sin 2x)^2
= (1/4) sin^2 (2x)
again, why don't you pick a weird angle and test it ?
1. False. The correct rewriting of 8sin^2(x)cos^2(x) using power reducing formulas is 4sin^2(2x).
2. To find the exact value of cos(15 degrees), we can use the half-angle formula for cosine: cos^2(x/2) = (1 + cos(x))/2. Plugging in x = 30 degrees, we get cos^2(15 degrees) = (1 + cos(30 degrees))/2.
Since cos(30 degrees) = sqrt(3)/2, we can substitute this value in: cos^2(15 degrees) = (1 + sqrt(3)/2)/2 = (2 + sqrt(3))/4.
Hence, the exact value of cos(15 degrees) is sqrt(2 + sqrt(3))/2.
3. To find the exact value of cos^2(pi/8), we can use the power reducing formula: cos^2(x) = (1 + cos(2x))/2. Substituting x = pi/8, we get:
cos^2(pi/8) = (1 + cos(pi/4))/2 = (1 + sqrt(2)/2)/2 = (2 + sqrt(2))/4 = (1/2)(sqrt(2) + 1).
So, the exact value of cos^2(pi/8) is (1/2)(sqrt(2) + 1).
4. If sec(x) = -3 and x lies in Quadrant II, we can use the fact that sec(x) is the reciprocal of cosine, so cosine is -1/3. Since x lies in Quadrant II, cosine is negative in that quadrant.
Using the half-angle formula for tangent: tan(x/2) = sqrt((1 - cos(x))/(1 + cos(x))), we can substitute -1/3 for cos(x):
tan(x/2) = sqrt((1 - (-1/3))/(1 + (-1/3))) = sqrt((4/3)/(2/3)) = sqrt(2).
So, the value of tan(x/2) is -sqrt(2).
5. False. The correct rewriting of sin^2(x)cos^2(x) using power reducing formulas is sin^2(x)(1 - sin^2(x)).
1. False. To rewrite 8sin^2 xcos^2 x using power reducing formulas, we can use the identity sin^2 x = (1 - cos2x)/2 and cos^2 x = (1 + cos2x)/2. Substituting these values, we get:
8sin^2 xcos^2 x = 8((1 - cos2x)/2)((1 + cos2x)/2)
= 8(1 - cos^2 2x)/4
= 2(1 - cos^2 2x)
Therefore, the correct expression is 2(1 - cos^2 2x), not 1 - cos4x.
2. The half angle formula for cosine is cos(x/2) = ±√[(1 + cos x)/2]. To find the exact value of cos15 degrees using this formula, we substitute x = 30 degrees (twice the angle) into the formula:
cos(30/2) = ±√[(1 + cos 30)/2]
= ±√[(1 + √3/2)/2]
= ±√[(2 + √3)/4]
= ±√((2 + √3)/4)
= ±(√2 + √3)/2
So, the exact value of cos15 degrees using the half angle formula is ±(√2 + √3)/2.
3. To find the exact value of cos^2(pi/8) using a power reducing formula, we can use the double angle formula for cosine: cos(2x) = 2cos^2 x - 1. Rearranging this formula, we get:
cos^2 x = (1 + cos(2x))/2
Now, substituting x = pi/8, we have:
cos^2(pi/8) = (1 + cos(2 * (pi/8)))/2
= (1 + cos(pi/4))/2
= (1 + 1/√2)/2
= (2/√2 + 1)/2
= √2 + 1)/2√2
= (√2 + 1)/(2√2) * (√2/√2) [Rationalizing the denominator]
= (√2 + √2/√2)/2√2
= (2√2/√2)/2√2
= 1/√2
= √2/2
Therefore, the exact value of cos^2(pi/8) using a power reducing formula is √2/2.
4. If secx = -3 and x lies in quadrant II, we can find the value of tan(x/2) using the half-angle formula for tangent: tan(x/2) = ±√[(1 - cosx)/(1 + cosx)].
Given secx = -3, we know that cosx = -1/3 (since secx = 1/cosx). Substituting this value into the half-angle formula, we get:
tan(x/2) = ±√[(1 - (-1/3))/(1 + (-1/3))]
= ±√[(1 + 1/3)/(1 - 1/3)]
= ±√[(4/3)/(2/3)]
= ±√(4/2)
= ±√2
Since x lies in quadrant II (where the tangent is negative), the value of tan(x/2) is -√2.
5. False. Power reducing formulas cannot be used to rewrite sin^2 xcos^2 x as 1/8 + cos4x / 4. Power reducing formulas are used to rewrite expressions involving powers of trigonometric functions, not in combination with addition or division. The correct way to rewrite sin^2 xcos^2 x using power reducing formulas is explained in question 1 above.