Download 1. Evaluate each of the following EXACTLY

Math107 – Questions to try (Ch. 7) – NEW MATERIAL for FINAL EXAM
1. Evaluate each of the following EXACTLY (No Decimals). Give any angles in radians.



3
 = ____________
3 
(a) arctan  −
(d) arccos (−1) = __________
(b) arcsin  −



2
 = ____________
2 
(c) arctan 0 = __________
(e) arccos ( −
3
) = _____________
2
(f) tan(arccos
1
) = _________
2
(g) arcsin(sin
π
) = ____________
3
(h) cos(arccos
6
) = ____________
7
(i) csc(arcsin
6
) = _________
5
(j) arctan (tan
9π
) = ____________
(k) sec(arcsin
2
) = ____________
2
(l) cot(arctan
15
) = _________
12
5π
) = ____________
6
(n) arcsin(cos
4π
) = ____________
3
(m) arccos (sin
(o) tan(arcsin 1) = ____________
2. Draw a triangle for each of the following expressions. Write an algebraic expression or evaluate accordingly.
(a) sin(arccos 2x) = ________________________
(b) sec(arctan
9
) = ________________________
2
5
) = ________________________
13
(d) cot(arccos
x
) = ________________________
9
(c) tan(arcsin
1
3. Solve the right triangles. Solve for any angles in radians and degrees. Round angles to two decimal places.
c = __________________
β
(A)
5 3
3
θ = __________________
θ
β = __________________
5
a = __________________
β
(B)
36
θ = __________________
β = __________________
θ
18 2
(C)
β
b = __________________
6
2
θ = __________________
θ
β = __________________
(D)
β
x = __________________
y = __________________
10
y
β = __________________
32°
x
2
4. Find all θ in the interval 0° ≤ θ ≤ 360° for each of the following. Round to nearest whole angle. Give answers in degrees.
(a) sin θ = .7071067812
Reference Angle: __________
Quadrants: __________
θ = _________________
(b) tan θ = −.4663076582
Reference Angle: __________
Quadrants: ____________
θ = _________________
(c) cot θ = 1.732050808
Reference Angle: __________
Quadrants: ____________
θ = _________________
(d) cos θ = −.6427876097
Reference Angle: __________
Quadrants: ____________
θ = __________________
(e) csc θ = −1.624269245
Reference Angle: __________
Quadrants: ____________
θ = __________________
(f) cos θ = −1
θ = ______________________
(g) csc θ = undefined
θ = ______________________
(h) cot θ = 0
θ = ______________________
5. Find all θ for each of the following. Give answers in degrees.
(a) sin θ = .7071067812
θ = _____________________________
(a) θ = arcsin (.7071067812)
θ = ____________
(b) tan θ = −.4663076582
θ = _____________________________
(b) θ = arctan (−.4663076582)
θ = ____________
(c) cot θ = 1.732050808
θ = _____________________________
(c) θ = arctan (.577350269)
θ = ____________
(d) cos θ = −.6427876097
θ = _____________________________
(d) θ = arccos (−.6427876097)
θ = ____________
(e) csc θ = −1.624269245
θ = _____________________________
(e) θ = arcsin (−.6156614755)
θ = ____________
3
6. Using trigonometric identities, show that the following are true statements by manipulating only one side of the equation.
SHOW ALL STEPS NEATLY AND CLEARLY FOR CREDIT. The second your work is hard to follow, it is WRONG.
(a) (csc β + cot β )(csc β − cot β ) = 1
(d) sin2 θ − cos2 θ = 2sin2 θ − 1
π
 sec 2 β
(g) cot  − β  −
= − cot β
2
 tan β
(i) cscψ − cot ψ =
sin ψ
1 + cosψ
π

(c) sin 2 α + sin 2  − α  = 1
2

(b) cos θ (sec θ − cos θ) = sin2 θ
(e) sin 4 α − cos 4 α = sin 2 α − cos 2 α
(h)
(f) cot (− φ ) cos(− φ ) + sin (− φ ) = − csc φ
1
1
+
=1
sin x + 1 csc x + 1
(j) tan x +
cos x
= sec x
1 + sin x
4
7. For the following right triangle, find sin (2θ) and cos (2θ).
θ
4
sin (2θ) = ______________
1
cos (2θ) = ______________
8. For the right triangle above, find tan (2θ). First use the Double Angle formula, then use sin (2θ) and cos (2θ).
tan (2θ) = ______________
tan (2θ) = ______________
2
9. For the triangle above, verify that sin
10. Given that sin x
=−
θ + cos2 θ = 1. Then, verify that sin2 (2θ) + cos2 (2θ) = 1.
24
in Quadrant IV, find csc(2x), sec (2x), and cot (2x).
25
csc (2x) = ______________
sec (2x) = ______________
cot (2x) = ______________
5
11. Let csc θ =
61
5
and sec β = , where θ and β are in Quadrant I.
60
3
(a) Find sin (θ + β) and cos (θ + β). Show your work.
sin (θ + β) = __________
cos (θ + β) = __________
2
(b) Verify that sin
(θ + β) + cos2 (θ + β) = 1.
(c) Find sin (θ − β) and cos (θ− β). Show your work.
sin (θ− β) = __________
cos (θ− β) = __________
2
(d) Verify that sin
(θ − β) + cos2 (θ − β) = 1.
(e) Find tan (θ + β) by first using the Addition Formula, then by using sin (θ + β) and cos (θ + β).
(f) Find tan (θ − β) by first using the Subtraction Formula, then by using sin (θ − β) and cos (θ − β).
6
12. Evaluate each of the following completely and exactly without a calculator. (No Decimals) Show your work.
(a) cos
11π
= ___________
12
π
π
+ tan
12
4 = ______________
(b)
π
π
1 − tan tan
12
4
tan
(c) sin
(d)
13π
= ___________
12
tan 64 ° − tan 4°
= ______________
1 + tan 64° tan 4°
(e) tan
 19π 
−
 = ___________
 12 
(f)
sin 36° cos 9° + cos 36° sin 9°
(g)
cos
= ______________
5π
π
5π
π
cos + sin
sin = ______________
6
3
6
3
13. Evaluate each of the following using the Addition and Subtraction formulas.
(a) sin
π

 − x  = ___________
2

(b) cos
π

 − x  = ___________
2

(c) tan
π

 − x  = ___________
2

7
14. Find all solutions of the following equations. Give answers in radians. BOX YOUR ANSWERS.
(a) tan4 x – 4tan2 x +3 = 0
(b) sin (2x) – sin x = 0
(c) 2sin2 (5x) = 2 + cos (5x)
(d) cot x csc x = cot x
(e) sin x = cos x
(f) −2 + 2cos x + cos (2x) = 0
8
 x
(g) 2 sin   + 3 = 0
7
(h) 2tan x – 4 = sec2 x – 8
x π 
x π 
(k) 6 cos 2  +  − 7 cos +  + 2 = 0
3 4 
3 4
(i) sin x + 1= cos x
(j) tan (2x – 6) – 1= 0

π

π



(l)  3 sec  x −  − 2  2 sin  x −  + 2  = 0
6
6





9