How do I find sintheta = -cos^2theta -1 in radians from [0, 2pi]?

How do I find sintheta = -cos^2theta -1 in radians from [0, 2pi]?

3 Answers
Apr 19, 2018

{3pi}/2

Explanation:

sin theta = -cos^2 theta-1 = -(1-sin^2 theta)-1 = sin^2theta-2 implies

sin^2theta-sintheta-2=0 implies (sin theta-2)(sin theta +1)=0

Since -1 <= sin theta <= +1, the factor sin theta +2 can not be zero. Thus

sin theta +1 = 0 implies theta = sin^-1(-1) = {3pi}/2

(note that {3pi}/2 is the only value of theta in [0,2pi] that satisfies sin theta = -1)

Apr 19, 2018

theta=(3pi)/2

Explanation:

We can get this equation in terms of one trigonometric function using trig identities. In this case, we know the identity

sin^2theta+cos^2theta=1

-cos^2theta=sin^2theta-1

So, we apply it to the right side:

sintheta=sin^2theta-1-1

Move everything to the right. This gives us the following:

0=sin^2theta-sintheta-2

Or,

sin^2theta-sintheta-2=0

This strongly resembles a quadratic equation; however, instead of ax^2+bx+c, we observe the form asin^2theta+bsintheta+c.

As a result, we can factor this just as we would factor a quadratic:

(sintheta-2)(sintheta+1)=0

We then solve the following:
sintheta-2=0
sintheta+1=0

For sintheta-2=0:

sintheta=2

Tells us this one has no solutions, as -1<=sintheta<=1 for all theta.

sintheta+1=0

sintheta=-1

Holds true for theta=(3pi)/2 in the interval [0, 2pi]

Apr 19, 2018

Theta = (3pi)/2

Explanation:

We can use one of the Pythagorean identities to write our equation in terms of a single trigonometric function. In particular, we can use:

sin^2theta+cos^2theta = 1, rewritten as cos^2theta = 1-sin^2theta.

We get:

sintheta = -cos^2theta -1

sintheta = -(1-sin^2theta)-1

sintheta = -1+sin^2theta-1

sintheta = sin^2theta-2

We'll solve as follows.

-sin^2theta+sintheta+2 = 0

sin^2theta-sintheta-2=0 <--Multiplying Both Sides By Negative One

(sintheta+1)(sintheta-2) = 0 <--Factoring the left-hand side

The product of the factors sintheta + 1 and sintheta - 2 is zero.
So if the equation has a solution, at least one of the factors must be zero.

sintheta+1 = 0
sintheta = -1

or

sintheta-2 = 0
sintheta = 2

Sine has the value -1 at (3pi)/2.
It NEVER has the value 2.