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

AI Thread Summary
To solve for sin(theta) and cos(theta) given tan(theta) = -4/5 and sin(theta) > 0, the relationship sin(t) = (-4/5)cos(t) is established. Using the Pythagorean identity cos^2(t) + sin^2(t) = 1, the equation can be transformed into a solvable form by substituting sin(t) into the identity. The discussion highlights the importance of understanding geometric definitions of trigonometric functions, particularly in visualizing the triangle formed by the given tangent values. The user initially struggles with the steps but ultimately realizes the connection to the Pythagorean theorem. This approach aids in finding the required trigonometric values effectively.
Mike_Winegar
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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?
 
Last edited:
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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.
 
I picked up this problem from the Schaum's series book titled "College Mathematics" by Ayres/Schmidt. It is a solved problem in the book. But what surprised me was that the solution to this problem was given in one line without any explanation. I could, therefore, not understand how the given one-line solution was reached. The one-line solution in the book says: The equation is ##x \cos{\omega} +y \sin{\omega} - 5 = 0##, ##\omega## being the parameter. From my side, the only thing I could...
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