Скачать или смотреть Finding the Exact Value of Trig Functions Using the Sum Identities (MathAngel369)

✨ Finding the Exact Value of Trig Functions Involving Inverse Trig Functions Using the Sum Identities (MathAngel369)✨

In this discussion, we are going to learn how to find the exact value of trig functions involving inverse trig functions using the sum identities.

In our example, we are given

sin[arctan(-3/5) + arccos(4/5)]

We want to find the exact value of the given expression.

The first thing we want to do is work from the inside out.

Let u=arctan(-3/4), where (-pi/2,pi/2), since the range of inverse tangent is form -pi/2 to pi/2.

Let v=arccos(4/5), where [pi,0], since the range of inverse cosine is between 0 and pi.

Then sin[arctan(-3/5) + arccos(4/5)]

= sin[u+v]

= sinucosv+cosusinv by the sum formula for sine.


Next, we need to find the value of sinu, cosu, and sinv, and we do this by drawing a triangle.

Now, u=arctan(-3/4) means that tanu=-3/4

and v=arccos(4/5) means that cosv=4/5.

Since the angle u is between-pi/2 to pi/2, and tanu= -3/4=y/x is negative, we draw our triangle for u in the fourth quadrant.

Since the angle v is between 0 and pi and cosv=4/5=x/r is positive, we draw the triangle for v in the first quadrant.


Now we use Pythagorean Theorem to find the r value for the angle u and the y value for the angle v.

For the angle u, we know that x^2 +y^2 =r^2.
So, (4)^2 +(-3)^2 =r^2
16+ 9 = r^2
25=r^2
5=r

For the angle v, we know that x^2+y^2=r^2.
So, (4)^2 +y^2 =5^2
16+y^2=25
y^2=9
y=3


Now, sinu=y/r=-3/5
and cosu=x/r=4/5

Similarly, sinv=y/r=3/5
and it is given that cosv=x/r=4/5.

Thus, we can plug in our values into sinucosv+cosusinv.

Hence, we have


sin[arctan(-3/5) + arccos(4/5)]

= sin[u+v]

= sinucosv+cosusinv

= [-3/5][4/5]+[4/5][3/5]

=-12/25+12/25

=0


We have found the exact value of the given trig expression using the sum identity.

Thank you for reading!

I hope that I can assist you on your math journey!

Sincerely,

➕MathAngel369➕




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🎥 Suggested Video and Playlist:

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