sum and difference identities

Submitted by diachicago on 19. April 2007 - 3:12.

how do u do this

cos42=cos(30+12)

i did it already but i got the answer in decimals

Submitted by Tutor on 19. April 2007 - 3:35.

cos42=cos(30+12)

Do it as cos42=cos(60-18)

We already know the value of Cos60 and Sin 60

We need to find the value of Cos18 and Sin 18

Here is how we can do it

$\theta=18$
$2\theta=90- 3\theta$
$Sin2\theta=Sin(90 - 3\theta ) = Cos3\theta$
$2Sin\theta*Cos\theta=4Cos^3\theta-3Cos\theta$

$Cos\theta(4(1-Sin^2\theta)-3-2Sin\theta)=0$
$4Sin^2\theta+2Sin\theta-1=0$

Solving the quadratic Equation we have

$Sin\theta=\frac{{\sqrt5-1}}{4}$

$Cos\theta=\sqrt{1-Sin^2\theta }$

$Cos\theta= \frac{{\sqrt {10+2\sqrt5}}}{4}$

So now we have values of

$Sin18=\frac{{\sqrt5-1}}{4}$

$Cos18= \frac{{\sqrt {10+2\sqrt5}}}{4}$

Now just use the Cosine subtraction formula , put these values to get answer , you already know cos60 and Sin60

Hope this helps

Online Tutor