Finding Trig Identities: How Do I Solve Trig Equations with Given Values?

[Mike_Winegar]

Homework Statement

Problem 6. If you know that tan(theta) = -4/5 and sin(theta) > 0, find:
(a) sin(theta)
(b) cos(theta)
(c) tan(theta + pi)

Homework Equations

cos^2(t)+sin^2(t)=1
tan(t)=sin(t)/cos(t)

The Attempt at a Solution

My teacher went over this today, but likes to skip over steps that I just don't understand. What I've done so far...

-4/5=sin(t)/cos(t)

sin(t)=(-4cos(t)/5)
solved for sin(t)

((-4cos(t)/5)/cos(t))
plugged sin(t) equation back into original equation

ends up being
(-4cos^2(t)/5)=-4/5

We then have the identity that cos^2 + sin^2 = 1

Here's where I get lost. My teacher jumped it directly to:
(-4/5 cos(t))^2 +cos^2(t)=1

I assume you would plug in our original sin(t)=(-4cos(t)/5) into the above identity, but I have no clue as to how she did that without having an additional 4/5th where she substituted the cos side of the equation in for the sin portion of the identity.

Any help would be appreciated.

Lollol

I think I just figured it out actually. Was my teacher using the Pythagorean theorem and not a trig identity?
So:
(-4/5 cos)^2 + cos^2 =r^2
(-4/5 cos)^2 + cos^2 =1?

 

[TylerH]

To do these, it will help if you recall the geometric definitions of the trig functions. tan is opposite over adjacent. So, think of an angle in a triangle, with opposite=-4 and adjacent=5. Find the hypotenuse, then use SOH-CAH-TOA.