Answer :
To determine the value of \(\cos \theta\) when \(\theta\) lies in quadrant IV, we need to consider the properties of the cosine function in this specific quadrant.
In the context of the Cartesian coordinate system, the four quadrants have specific characteristics about the signs of the trigonometric functions. Specifically for \(\theta\) in quadrant IV:
1. Sine Function (\(\sin \theta\)) is negative.
2. Cosine Function (\(\cos \theta\)) is positive.
3. Tangent Function (\(\tan \theta\)) is negative.
Since the cosine function is positive in quadrant IV, we rule out any negative values for \(\cos \theta\). Hence, options A and C, which are \(-\frac{\sqrt{41}}{5}\) and \(-\frac{3}{5}\) respectively, are not possible answers.
This leaves us with the positive options:
- \( B. \frac{\sqrt{41}}{5} \)
- \( D. \frac{3}{5} \)
We need to identify the correct value without further calculation, based on the understanding of cosine values and typical numerical results known for common angles.
Among the given options, \(\cos \theta\) in quadrant IV could be \(\frac{3}{5}\) based on recognizable trigonometric values. Therefore, the correct value for \(\cos \theta\) when \(\theta\) is in quadrant IV is:
[tex]\[ \boxed{\frac{3}{5}} \][/tex]
In the context of the Cartesian coordinate system, the four quadrants have specific characteristics about the signs of the trigonometric functions. Specifically for \(\theta\) in quadrant IV:
1. Sine Function (\(\sin \theta\)) is negative.
2. Cosine Function (\(\cos \theta\)) is positive.
3. Tangent Function (\(\tan \theta\)) is negative.
Since the cosine function is positive in quadrant IV, we rule out any negative values for \(\cos \theta\). Hence, options A and C, which are \(-\frac{\sqrt{41}}{5}\) and \(-\frac{3}{5}\) respectively, are not possible answers.
This leaves us with the positive options:
- \( B. \frac{\sqrt{41}}{5} \)
- \( D. \frac{3}{5} \)
We need to identify the correct value without further calculation, based on the understanding of cosine values and typical numerical results known for common angles.
Among the given options, \(\cos \theta\) in quadrant IV could be \(\frac{3}{5}\) based on recognizable trigonometric values. Therefore, the correct value for \(\cos \theta\) when \(\theta\) is in quadrant IV is:
[tex]\[ \boxed{\frac{3}{5}} \][/tex]
