If sin(theta) < 0 and tan(theta) > 0 then: in degrees. Ok I need help with this I couldn't understand the lesson and want to know how to do problems like this. Please help me by showing me step by step! Thank u in advance.

I needed help figuring it out in degrees like 0 degrees < theta < 90 degrees. Thanks

quadrant 3 is the lower left, 180 degrees to 270 degrees

180 </= theta </= 270

sin has the same sign as cos in quadrants 1 and 3 so that is where tan >0

but the sin is negative in quadrants 3 and 4 only
so theta must be in quadrant 3

sorry I forgot to put my name.

Of course, I'd be happy to help you understand this problem step by step!

First, let's review the conditions given: sin(theta) < 0 and tan(theta) > 0.

When sin(theta) < 0, it means that the sine of theta is negative. In trigonometry, sine is negative in the third and fourth quadrants of a unit circle.

When tan(theta) > 0, it means that the tangent of theta is positive. In trigonometry, tangent is positive in the first and third quadrants of a unit circle.

To determine the possible values of theta, we need to find the overlapping region where both conditions are satisfied.

Step 1: Analyze the sign of sine and tangent in each of the quadrants:

- In the first quadrant, both sine and tangent are positive.
- In the second quadrant, sine is positive but tangent is negative.
- In the third quadrant, both sine and tangent are negative.
- In the fourth quadrant, sine is negative but tangent is positive.

Step 2: Identify the potential quadrants that satisfy sin(theta) < 0 and tan(theta) > 0:

Based on the conditions given, we know that sine is negative and tangent is positive. This only occurs in the third quadrant. Therefore, theta must lie in the third quadrant.

Step 3: Determine the range of theta within the third quadrant:

In the third quadrant, the angle ranges from 180 degrees to 270 degrees. So, theta must be between 180 degrees and 270 degrees.

Therefore, the possible values for theta, in degrees, that satisfy the given conditions are angle measurements between 180 and 270.

I hope this step-by-step explanation helps you understand how to approach problems like this. If you have any further questions, feel free to ask!