Find the exact value of each expression

Question

Find the exact value of each expression:\\

(a) sin^-1(√(3)/2) =

(b) cos^-1(-1) =

 

 

Answer 1

Hi Dalia,

 

This question is testing whether you know and understand your trig tables.  Here is our table:

 

Angle     |  Sin (Angle) |  Cos(Angle)  |  Tan (Angle)

Degrees

0           |       0            |         1           |       0

30         |   1/2             |        √3/2       |     1/√3

45         |      1/√2        |         1/√2      |       1

60         |     √3/2         |           1/2      |     √3

90         |           1        |        0            |   undefined

 

Let's try part a:

 

sin-1(√3/2) = 

 

we go to our table and we look to see when the sin is √3/2...it occurs at 60 degrees.  Therefore 

 

sin-1(√3/2) =60

 

part b:

 

cos-1(-1) = 

 

This is a cool question because it makes us recall some pretty hidden info.  It turns out that if we add 180 to an angle we will negate its trig value.  An example of this is...

 

cos(0) = 1 so cos(180)=-1

 

This means that :

 

cos-1(-1) = 180 degrees

 

Recall that degrees can also be written as radians using the conversion formula:

 

radians = degrees * (Π/180)

 

*Note* 

 

These answers are not the only answers.  Trig functions repeat themselves every 360 degrees.  So we can never truly give just one answer for these types of questions, the actual answer would have to include every value as a set...

 

cos-1(-1)  = {180n degrees| for all odd values of n}  n=1,3,5,...

 

This part is a bit advanced, and the first part may be enough for your class (double check with your teacher).  This is just some insight into the beautiful world of mathematics.  

 

Hope this helped!!

Answer 2

inverse trig functions basically spit out angles...... 

 

So, sin^-1(x) = θ    .... where theta (θ) is an angle..... 

 

 

it is also true that.....     sin(θ) = x 

 

So, do you remember the unit circle? 

 

Let's look at the first problem (a)...

 

sin^-1(√(3)/2)= θ ....... well, then sin(θ) = √(3)/2  .....

 

Do you remember an angle on the unit circle whose sin value is square root of 3 over 2?

 

60 degrees or π/3 is the answer. 

 

the same approach is true and will guide you to the answer for part b as well.... if you have any trouble with that one after trying, comment or reply to this post.