precalculus final monday please help me remember the corrects steps to solve these questions as i have lost my notes! Thanks!

Question

1) determine exact value of each trigonometric ratio

a)sec 315°

b)cos 5pi/6

c)cot 270

d)sin(-4pi/3)

 

2) find exact roots

a)tan θ = √3, -180° ≤ θ<180°

b)sec θ = 2/√3, -2Ω ≤ θ  < 2Ω

Answer 1

For part 1, you will rewrite the trig functions as an addition/subtraction angle identity.  This means that you will find two angles and add up to given angle.  These angles must be easily obvious to identify sine and cosine of in a unit circle.

 

I will do the first and second one for you, so you see the steps.  Then try the rest on your own.

 

a)

 

sec(315) = 1 / cos(315) = 1 / cos(360 - 45)

 

Notice that 360 and 45 are angles on a unit circle we can easily find sine and cosine of.  Use the subtraction angle identity for cosine.

 

1 / [cos(360)cos(45) + sin(360)cos(45)]

 

1 / [√(2)/2] =

 

2 / √2 =

 

√2

 

 

sec(315) = √2

 

 

b)

 

cos(5π/6) = cos(6π/6 - π/6) = cos(π - π/6)

 

 

Use the subtraction angle identity for cosine.

 

cos(π)cos(π/6) + sin(π)sin(π/6) =

 

-1(√(3)/2)) =

 

-√(3)/2

 

 

cos(5π/6) = -√(3)/2

 

 

2)

 

Let x=θ

 

 

tanx = √3

 

x = tan^-1(√3)

 

 

Since tangent is positive in the 1st and 3rd quadrants,

 

x = 180 + tan^-1(√3)

 

 

For part b, I do not know what you mean by Ω.