Two important Pre Cal Questions 1. If sin (θ − π/2)= 0.73, find cos (−θ). 2. Find secθ and cscθ 1 if cot θ = − 1/2 and cos θ < 0.

Question

Please show me step by step help I genuinely don't understand this.

Answer 1

This is an exercise in trigonometric identities

1.

0.73 = sin(θ − π/2)

= -sin(π/2 - θ)

= -cos(θ)

= -cos(-θ)

Therefore cos(-θ) = -0.73

2.

cot(θ) = -1/2

cos(θ)/sin(θ) = -1/2

cos²(θ)/sin²(θ) = (-1/2)²

(1 - sin²(θ))/sin²(θ) = 1/4

1/sin²(θ) - 1 = 1/4

1/sin²(θ) = 5/4 (1)

1/sin(θ) = √5/2

csc(θ) = √5/2

Note: We know that the solution to (1) is √5/2 and not -√5/2, because

cot(θ) < 0 and cos(θ) < 0 and therefore sin(θ) > 0

To calculate sec(θ) we start with (1)

1/sin²(θ) = 5/4

sin²(θ) = 4/5

1 - cos²(θ) = 4/5

cos²(θ) = 1/5 (2)

cos(θ) = -1/√5

sec(θ) = -√5

Note: We know that the solution to (2) is -1/√5 and not 1/√5, because

cos(θ) < 0