Trig identities: how would you solve sinx+cosx=sinxcosx? both in radians and degree?

Answer 1

Squaring is one algebraic way to solve this equation.
Another way is to use the substitution x = t + 45
Some identities you need to know:
1) sin x + cos x = √2 sin (x + 45)
2) sin x cos x = 0.5 sin 2x
3) sin (x + 90) = cos x
4) cos 2x = 2 cos² x - 1

Thus your equation is √2 sin (x + 45) = 0.5 sin 2x [ identities 1 and 2]
√2 sin (t + 90) = 0.5 sin (2t+90) [ substituting x = t + 45]
√2 cos t = 0.5 cos 2t = 0.5 ( 2 cos² t - 1) = cos² t - 0.5 [identities 3 and 4]

You now have a quadratic in cosine
cos² t - √2 cos t - 0.5 = 0
cos t = [√2 - 2] / 2
t = 107 + 360 k
or t = 253 + 360 k
x = 298 + 360 k
or x = 152 + 360 k

Answer 2

there is a non graphing way to do this, but it takes some effort and being careful. The first few steps are pretty straightforward:

take your initial equation and square both sides:

sin^2x+2 sinxcosx +cos^2x = [sinxcosx]^2

the left becomes 1+2sinxcosx, and you wind up with a quadratic equation in the variable sinx*cos x:

(sinx cosx) - 2 sin x cos x -1=0

solving as you would for a quadratic, you get roots:

sinx*cosx=1+sqrt[2] and 1-sqrt[2]; since you know that sin and cos have values between -1 and 1, their product can never exceed 1, so you throw out the first root and are left with:

sinx*cosx=1-sqrt[2]

now, use the identity sin(2x)=sinxcosx, so our left hand side now becomes:

1/2 Sin(2x)=-0.4114
Sin(2x)=-0.8228
and this has two solutions one of which is:

x=152 degrees = 2.65 radians

x=298 degrees = 5.2 radians

hope this helps

Answer 3

Squaring both sides, we get

sin^2 x + 2 sin x cos x + cos^2 x = (sinx cosx )^2. Since sinx cos x = 1/2 sin(2x), we get

1/4 sin 2x - sin2x - 1 = 0 or

sin^2 2x - 4 sin 2x - 4 =0

So, we have 2 possibilities for sin 2x

(4 + sqrt(16 + 16))/2 = 2 + sqrt(2) and
2 - sqrt(2)

The first one is > 1, so leadas to no real solution, although there are complex ones.

The second leads to 2x = arc sin (2 - sqrt(2)). In [0, 2pi). So there are 2 possible values for 2x: One in the 1st quadrant, since 2 - sqrt(2) >0 and the other in the 2nd quadrant.

The value in the 1st quadrant is sure a solution of the original equation. But the second, you have to check, because when you square an equation you get solutions that may not be solution sof the original equation.

By the way, this is not a trigonometric identity.

Answer 4

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RE:
Trig identities: how would you solve sinx+cosx=sinxcosx? both in radians and degree?

Answer 5

Square both sides:

(sinx)^2 + 2(sinx)(cosx) + (cosx)^2 = (sinx cosx)^2

Now (sinx)^2 + (cosx)^2 = 1, and if we write p = sinx cosx,
the equation becomes
1 + 2p - p^2 = 0
or
p^2 - 2p - 1 = 0
Hence
p = 1 +/- sqrt(2)
Now 2p = sin 2x, and so we have
sin 2x = 2*[1 +/- sqrt(2)]
The larger root is greater than 1, which isn't possible for sine, and so the only solutions come from
sin 2x = 2*[1 - sqrt(2)]
which is about -0.828427
and so 2x = -0.97629 rad or -2.16530 or 5.30689 or 4.11788 etc.
Hence x = -0.48815 or -1.08265 or 2.65345 or 2.05894 etc

You can convert these to degrees by multiplying by
180/pi.

Answer 6

You need to solve graphically or numerically. You can simplify the equation a bit by dividing cos(x). This results in
tan(x) + 1 = sin(x)
From this you can see that the tan must be negative because otherwise the equation will not hold. Subtract sin(x) for getting
1 + tan(x) - sin(x) = 0
Draw or plot this function (1 + tan(x) + sin(x)). Where it cuts the x-axis the equation is fulfilled and you got the solution graphically.

For a numerical solution there are many different ways. One of the most common is named "Newtons method" and you will find it described at the link below. You will need a starting value, but we already found out, that the tan(x) must be negative, so the x must be negative as well. Solcing numerically results in
x = -1.082649525 radians = −62.0312485 degrees
To convert from radians to degrees multiply by 180 and devide by pi.

Oops, forgot the link.

Answer 7

I'm guessing it has something to do with the following identity: sin^2 + cos^2 = 1 1 - sin^2 = cos^2 [1-sin^2]^1/2 = cos ....

Answer 8

You would need a graphing calculator to solve that in my opinion, mainly because there are no trig identities that you could use to simplify it further.

Answer 9

you need a graphic calculator to solve