If θ is an angle in standard position and tan θ = −7/2, in which quadrant can θ be located?
by definition tan θ = y/x
y/x = -7/2 <----- y is negative, x is positive, so quadrant IV
or
y/x = 7/-2 <---- y is positive, x is negative, so quadrant II
Find out about a very useful rule called the CAST rule, which
tells you where the 3 main trig ratios are positive.
Very hard to do any of these questions without knowing that rule.
since tan θ = y/x, x and y must have different signs.
QII, QIV
To determine the quadrant in which θ is located, we need to look at the signs of the trigonometric functions in each quadrant.
We are given that tan θ = -7/2, which means that the tangent of the angle is negative. In standard position, the tangent is negative in the 2nd and 4th quadrants.
Now let's go through the steps to find θ in the 2nd and 4th quadrants:
1. 2nd Quadrant: In the 2nd quadrant, both sine and cosine are positive, and the tangent is negative. If we assume that θ is in the 2nd quadrant, we can use the formula tan θ = sin θ / cos θ to find the angle. Since the tangent is negative, it implies that both sine and cosine need to have different signs in order for the division to result in a negative value. However, sine and cosine are both positive in the 2nd quadrant, so tan θ cannot equal -7/2 in this quadrant.
2. 4th Quadrant: In the 4th quadrant, both sine and cosine are positive, and the tangent is negative. Using the same logic as before, tan θ = sin θ / cos θ. Again, we need sine and cosine to have different signs to obtain a negative value. In the 4th quadrant, the sine is positive and the cosine is negative, satisfying the condition. Therefore, tan θ = -7/2 is possible in the 4th quadrant.
Therefore, θ can be located in the 4th quadrant.