Trigonometry questions trying to get answers........?

Question

1. Using your calculator with only acute angles, evaluate the sine, cosine and 
tangent of each of the following angles, correct to 3 sig. fig. 

(a) 276.1° 
(b) 135.7° 
(c) 27.9° 
(d) 92.4° 
(e) 350.1° 
(f) 239.0° . 


2. If angle α is reflex, and cos α = -9/41, without using a calculator, evaluate 

(a) sin α 
(b) tan α 
(c) cos (α - 180°) .

Answer 1

• 1).

(a). sin(276.1⁰) = sin(270 + 6.1⁰) = - cos(6.1⁰) = - 0.994338,

cos(276.1⁰) = cos(270 + 6.1⁰) = sin(6.1⁰) = 0.106264,

tan(276.1⁰) = tan(270 + 6.1⁰) = - cot(6.1⁰) = - 9.35724.

(b). sin(135.7⁰) = sin(90 + 45.7⁰) = cos(45.7⁰) = 0.698415,

cos(135.7⁰) = cos(90 + 45.7⁰) = - sin(45.7⁰) = - 0.715693,

tan(135.7⁰) = tan(90 + 45.7⁰) = - cot(45.7⁰) = - 0.975859.

(c). sin(27.9⁰) = sin(90 - 62.1⁰) = cos(62.1⁰) = 0.467930,

cos(27.9⁰) = cos(90 - 62.1⁰) = sin(62.1⁰) = 0.8837656,

tan(27.9⁰) = tan(90 - 62.1⁰) = cot(62.1⁰) = 0.529473.

(d). sin(92.4⁰) = sin(90 + 2.4⁰) = cos(2.4⁰) = 0.9991228,

cos(92.4⁰) = cos(90 + 2.4⁰) = - sin(2.4⁰) = - 0.0418757,

tan(92.4⁰) = tan(90 + 2.4⁰) = - cot(2.4⁰) = - 23.8593.

(e). sin(350.1⁰) = sin(360 - 9.9⁰) = - sin(9.9⁰) = - 0.171929,

cos(350.1⁰) = cos(360 - 9.9⁰) = cos(9.9⁰) = 0.985109,

tan(350.1⁰) = tan(360 - 9.9⁰) = - tan(9.9⁰) = - 0.174528.

(f). sin(239.0⁰) = sin(270 - 31⁰) = - cos(31⁰) = - 0.857167,

cos(239.0⁰) = cos(270 - 31⁰) = - sin(31⁰) = - 0.515038,

tan(239.0⁰) = tan(270 - 31⁰) = cot(31⁰) = 1.66428.

• 2).

cos α = - 9/41.

Phythagoras identity : sin² α + cos² α = 1 .

sin² α = 1 - cos² α.

sin α = √(1 - cos² α)

= √(1 - (- 9/41)²)

= √(1 - 81/1681)

= √(1 - 81/1681)

= √(1681 - 81/1681) = √(1600/1681)

= 40/41.

(a). sin α = 40/41.

(b). tan α = sin α/cos α = (40/41)/(- 9/41) =  - 40/9.

(c). cos(α - 180) = cos(- (180 - α)) = cos(180 - α) = - cos α = - (- 9/41) = 9/41. (From the given data)