Download (Sections 5.1-5.5) Practice Test

Name: ______________________________
Tuesday, October 16, 2012
MTH 113 Test 2 (Chapter 5)
1. (X pts.) Verify the following trigonometric identities…
sin x – sin x cos2 x = sin3 x
cot(θ) cos(θ) + sin(θ) = csc(θ)
sin x – sin x(1 – sin2x)
sin x – sin x + sin3x
sin3x
2. (X pts.) Suppose that sin α =
[cos x / sin x]cos x + sin x
cos2x / sin x + sin x
(cos2x + sin2x) / sin x
1 / sin x = csc x
𝟑
𝟓
for a quadrant II angle α and sin β =
exact value of the following (hint: find cos α and cos β first)…
cos (α – β)
𝟏
𝟐
for a quadrant I angle β. Find the
sin (α + β)
cos a = - 4/5 , cos b = sqrt(3)/2
cos a cos b + sin a sin b
(- 4/5)(sqrt(3)/2) + (3/5)(1/2)
[- 4sqrt(3) + 3] / 10
sin a cos b + cos a sin b
(3/5)(sqrt(3)/2) + (- 4/5)(1/2)
[3sqrt(3) - 4] / 10
3. (X pts.) Find the exact value of cos 30o using only 90o and 60o angles
cos 30o = cos (90o – 60o) = cos 90o cos 60o + sin 90o sin 60o = 0 + 1(sqrt(3)/2) = sqrt(3) / 2
4. (X pts.) Verify the trigonometric identity
𝐜𝐨𝐬(𝜶−𝜷)
𝐜𝐨𝐬 𝜶 𝐜𝐨𝐬 𝜷
= 1 + tan(α) tan(β)
(cos a cos b) / (cos a cos b) + (sin a sin b) / cos a cos b = 1 + tan a tan b
5. (X pts.) If sin θ = 4/5 and θ lies in quadrant II, find the exact value of the following…
sin (2θ)
tan (2θ)
cos a = - 3/5 , tan a = - 4/3
2 sin a cos a
2(4/5)(- 3/5)
- 24 / 25
2 tan a / (1 – tan2 a)
2(- 4/3) / (1 – (16/9))
(- 8/3) / (-7/9)
(- 8/3) * -(9/7)
24 / 7
6. (X pts.) Verify the following trigonometric identities…
tan θ =
𝐬𝐢𝐧 𝟐𝛉
𝟏+𝐜𝐨𝐬 𝟐𝛉
(2 sin a cos a) / (1 + (2cos2a – 1))
(2 sin a cos a) / (2cos2a)
[2/2] [sin a / cos a] [cos a / cos a]
tan a
𝒙
𝒔𝒆𝒄 𝒙−𝟏
tan( ) =
𝟐
𝒕𝒂𝒏 𝒙
[(1/cos x) – 1] / [sin x / cos x]
[(1 – cos x)/cos x] / [sin x / cos x]
[(1 – cos x)/cos x] * [cos x / sin x]
(1 – cos x )/ sin x = tan (x/2)
7. (X pts.) Use the half-angle formulas to find the exact value of the function cos 105o -- use radical signs as
necessary
cos 105o = cos (210o/2) = - sqrt[(1 + cos 210o)/2] = - sqrt[(1 + (-sqrt(3)/2)) / 2] = - sqrt[(2 – sqrt(3))/2 / 2]
= - sqrt [(2 – sqrt(3)) / 4]
8. (X pts.) Express each sum or difference as a product (if possible, find this product’s exact value)
sin 8x – sin 3x
cos 75o + cos 15o
2 sin [(8x – 3x)/2] cos[(8x + 3x)/2]
2 sin(5x/2) cos(11x/2)
2 cos[(75+15)o/2] cos[(75-15)o/2]
2 cos(45o) cos(30o)
2 (sqrt(2)/2) (sqrt(3)/2)
sqrt(6) / 2
9. (X pts.) Express each product as a sum or difference
sin 7x sin 3x
½[cos(4x) – cos(10x)]
½cos(4x) – ½cos(10x)
10. (X pts.) Verify the trigonometric identity
cos 4x sin x
½[sin(5x) – sin(3x)]
½sin(5x) – ½sin(3x)
𝐜𝐨𝐬 𝟑𝐱 –𝐜𝐨𝐬 𝐱
𝐬𝐢𝐧 𝟑𝐱+𝐬𝐢𝐧 𝐱
= – tan x
-2 sin(2x)sin(x) / 2 sin(2x)cos(x) = - sin(x) / cos(x) = - tan(x)
11. (X pts.) Solve each equation on the interval [0, 2π)
tan 2x = √𝟑
4 cos2 x – 3 = 0
2x = pi/3
x = pi/6 + npi
x = {pi/6, 7pi/6}
cos2 x = 3/4
cos x = +/- sqrt(3) / 2 + 2npi
x = {pi/6, 5pi/6, 7pi/6, 11pi/6}
cos 2x + sin x = 0
1 – 2sin2x + sin x = 0
2sin2x – sin x – 1 = 0
(2sin x + 1)(sin x – 1) = 0
sin x = -1/2
sin x = 1
x = 7pi/6, 11pi/6
x = pi/2
12. (X pts.) Potential word problem??? See chapter review at end of chapter 5 in book (pg 641 - #’s 68/69)
-- EXTRA CREDIT -????