Solution to Chapter 8 Problems:

Note: Calculators normally do not have the cotangent function built in them.  You may reciprocate the value of tangent to get the value for cotangent.

1) {a = 43.0°}:  With calculator in degrees mode,

Ans:  sin a = 0.682   ; cos a = 0.731     ; tan a = 0.933     ; cot a = 1.07

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2) {b = 2.90 radians}: With calculator in radians mode,

Ans:  sin b =  0.239    ; cos b = -0.971     ; tan b = -0.246     ; cot b = -4.06

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3) {c = 10π/9 radians}: With calculator in radians mode,

Ans:  sin c =  -0.342    ; cos c = -0.934     ; tan c = 0.364     ; cot c = 2.75

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4){d = 3981°}:  With calculator in degrees mode,

Ans:  sin d =  0.3584    ; cos d = 0.9336     ; tan d = 0.3839     ; cot d = 2.605

Note: 10(2π) radians = 10(360°) = 3600° = 10 full turns.

11(2π) radians = 11(360°) = 3960° = 11 full turns.

Taking 11 full turns out of 3981° that means 3981° - 3960° leaves us with 21°

Redoing this problem with d = 21°, we will get the same results as follows:

{d = 21°}: sin d =  0.3584    ; cos d = 0.9336     ; tan d = 0.3839     ; cot d = 2.605 (Verify all).

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5){e = 3,070 grads}: With calculator in grads mode,

Ans:  sin e-0.891    ; cos e = -0.454     ; tan e = 1.96     ; cot e = 0.501

Note: 7(2π) radians = 7(360)° = 7(400)grads = 7 full turns.

7(400)grads = 2800grads = 7 full turns.

Taking 7 full turns out of 3070grads that means (3070 - 2800)grads leaves us with 270grads

Redoing this problem with d = 270grads, we will get the same results as follows:

{d = 270grads}: sin e-0.891    ; cos e = -0.454     ; tan e = 1.96     ; cote = 0.501

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6){f = 3.90π radians}:  With calculator in radians mode,

Ans:  sin f-0.309    ; cos f = 0.951     ; tan f = -0.325     ; cot f = -3.08

Note: Taking 1 full turn or 2π out, will leave us with 3.90π - 2π = 1.90π . Redoing for {f = 1.9π radians} will end up the same results as follows: (Verify.)

Ans:  sin f-0.309    ; cos f = 0.951     ; tan f = -0.325     ; cot f = -3.08

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7){g = 135.45°}: With calculator in degrees mode,

Ans:  sin g =  0.70153    ; cos g = -0.71264     ; tan g= -0.98441     ; cot g = -1.0158

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8) Given [sin A = 1/2], determine all angles that satisfy this property.

Solution: Since sin30° = 1/2 or sin (π/6) = 1/2 ; therefore, A = π/6.

One solution set is of course, [k(2π) + (π/6)].

The other solution set is of course, [k(2π) + π - (π/6)] because

sin( π - A) = sin A.

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9) Given [cos B = 2/2], determine all angles that satisfy this property.

Solution: Since cos45° = 2/2 or cos(π/4) = 2/2 ; therefore, B = π/4.

The two solution sets are of course:  [k(2π) ± (π/4)].

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10) Given [tan C = 3], determine all angles that satisfy this property.

Solution: Since tan60° = 3 or tan (π/3) = 3 ; therefore, C = π/3.

The solution set is of course:  [k(π) + (π/3)].

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11) Given [cot D= -1], determine all angles that satisfy this property.

Solution: Since cot (-45)° = -1 or cot (π/4) = -1 ; therefore, D = -π/4.

The solution set is of course:  [k(π) + (-π/4)].  This is because of the fact

that cot (-C) =  -cot (C).

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12) Given [sin x = -√3/2], determine all angles that satisfy this property.

Solution: Since sin (- 60)° = -√3/2 or sin(-π/3) = -√3/2 ; therefore, x = -π/3.

The solution sets are:  [2k(π) + (-π/3)] and  [2k(π) + π - (-π/3)].  This is because of the fact  that sin(-C) = -sin (C).

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13) Given [cos y = -1/2], determine all angles that satisfy this property.

Solution: Since cos 120° = -1/2  or  cos (/3) = -1/2 ; therefore, y = 2π/3.

The two solution sets are of course:  [k(2π) ± (2π/3)].  This is because of the fact

that  cos (2π/3) = cos (π - π/3) = - cos(π/3) = -1/2

or,  cos(120°) =  cos(180° - 60°) = - cos60° = - 1/2.

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14) Given [tan z = -√3/3], determine all angles that satisfy this property.

Solution: Since tan(-30°) = -√3/3 or tan (-π/3) = -√3/3 ; therefore, z = -π/6.

The solution set is of course:  [k(π) -π/3].  This is because of the fact

that tan (-C) = -tan (C).

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15) Given [sin A = 0.545], determine all angles that satisfy this property.

Solution: Since 0.545 is not the sine of any of the special angles: 0, 30°, 45°, 60°, or 90°, we need to first use a calculator to find the inverse sine of 0.545 .   We first write:

sin-1 (0.545) = 33.0°  (found by a calculator)Now that the angle is found, we write the general solution sets as :  [2k(π) + 33.0°] and  [2k(π) + π - 33.0°].

Of course, π = 180°.

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16) Given [cos B = - 0.927], determine all angles that satisfy this property.

Solution:  First B = cos-1 (-0.927) = 158°.

The two solution sets are of course:  [k(2π) ± 158°].

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17) Given [tan C = -2.100], determine all angles that satisfy this property.

Solution: First, C = tan-1 (-2.100) = - 64.54°.

The solution set is :  [k(π) -64.54°].

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18) Given [cotD = 5.237], determine all angles that satisfy this property.

Solution: First, reciprocate the cotangent to get the tangent.

tanD = 1/5.237 = 0.1909 .

D = tan-1 0.1909 = 10.81°.

The solution set is :  [k(π) +10.81°].

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19)  Given [4cosB = 7/5], determine all angles that satisfy this property.

Solution: First, solve for cos B to get :  cosB = 7/20 = 0.35; thus,

B = cos-1(0.35) = 70°

The two solution sets are of course:  [k(2π) ± 70°].

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20)  Given [1/sinA = -2.5], determine all angles that satisfy this property.

Solution: First, solve for sin A to get :  sin A = 1/(-2.50) = - 0.40; thus,

A = sin-1(-0.40) = -24°

Now that the angle is found, we write the general solution sets as :

[2k(π) - 24°] and  [2k(π) + π -(- 24°)].