If \(\sin \theta = - \frac{1} {2}\) and \(\tan \theta = \frac{1} {{\sqrt 3 }}\) , then in which quadrant does θ lie?
Third
Determining the Quadrant of an Angle
The question asks us to identify the quadrant in which an angle \(\theta\) lies, given the values of its sine and tangent.
We are given:
\(\sin \theta = - \frac{1} {2}\)
\(\tan \theta = \frac{1} {{\sqrt 3 }}\)
To find the quadrant of \(\theta\), we need to consider the signs of the trigonometric functions in each quadrant.
Analyzing the Sign of Sine
The value of \(\sin \theta\) is \(-\frac{1}{2}\), which is negative. The sine function is negative in the following quadrants:
Third Quadrant
Fourth Quadrant
So, based on the sine value, \(\theta\) must lie in either the Third or the Fourth Quadrant.
Analyzing the Sign of Tangent
The value of \(\tan \theta\) is \(\frac{1}{\sqrt{3}}\), which is positive. The tangent function is positive in the following quadrants:
First Quadrant
Third Quadrant
So, based on the tangent value, \(\theta\) must lie in either the First or the Third Quadrant.
Combining the Information to Find the Quadrant
For \(\theta\) to satisfy both conditions (sine is negative AND tangent is positive), it must lie in a quadrant that is common to the possibilities derived from analyzing both signs.
From \(\sin \theta < 0\), possible quadrants are Third and Fourth.
From \(\tan \theta > 0\), possible quadrants are First and Third.
The quadrant that appears in both lists is the Third Quadrant.
Therefore, the angle \(\theta\) lies in the Third Quadrant.
Signs of Trigonometric Functions by Quadrant
Here is a summary of the signs of the main trigonometric functions in each quadrant:
| Quadrant | Interval (Degrees) | Interval (Radians) | sin \(\theta\) | cos \(\theta\) | tan \(\theta\) |
|---|---|---|---|---|---|
| First | \(0^\circ < \theta < 90^\circ\) | \(0 < \theta < \frac{\pi}{2}\) | + | + | + |
| Second | \(90^\circ < \theta < 180^\circ\) | \(\frac{\pi}{2} < \theta < \pi\) | + | - | - |
| Third | \(180^\circ < \theta < 270^\circ\) | \(\pi < \theta < \frac{3\pi}{2}\) | - | - | + |
| Fourth | \(270^\circ < \theta < 360^\circ\) | \(\frac{3\pi}{2} < \theta < 2\pi\) | - | + | - |
From the table, we can see that only in the Third Quadrant are both \(\sin \theta\) negative and \(\tan \theta\) positive.
Conclusion
Given \(\sin \theta < 0\) and \(\tan \theta > 0\), the angle \(\theta\) must lie in the Third Quadrant.
Revision Table: Quadrant Determination
Review the signs of sine and tangent to determine the correct quadrant.
If \(\sin \theta > 0\), \(\theta\) is in Quadrant I or II.
If \(\sin \theta < 0\), \(\theta\) is in Quadrant III or IV.
If \(\tan \theta > 0\), \(\theta\) is in Quadrant I or III.
If \(\tan \theta < 0\), \(\theta\) is in Quadrant II or IV.
Find the common quadrant based on the given signs.
Additional Information: Understanding Trigonometric Functions
The trigonometric functions sine, cosine, and tangent relate an angle in a right-angled triangle to the ratios of its sides. When extended to the unit circle, they can be defined for any angle. The signs of these functions depend on which quadrant the terminal arm of the angle falls into, which is determined by the x and y coordinates of the point on the unit circle.
In Quadrant I, both x and y are positive, so sin, cos, and tan are all positive.
In Quadrant II, x is negative and y is positive, so sin is positive, cos is negative, and tan (y/x) is negative.
In Quadrant III, both x and y are negative, so sin is negative, cos is negative, and tan (y/x) is positive.
In Quadrant IV, x is positive and y is negative, so sin is negative, cos is positive, and tan (y/x) is negative.
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The distance of the highest point on the graph of the function y = √3 cos x + sin x from the x-axis is:
If A = π / 6 and B = π / 3, then consider the following statements:
I. sin A + sin B = cos A + cos B
II. tan A + tan B = cot A + cot B
Which of the above statements is / are correct?
If \(\sin A = \frac{1}{{\sqrt 2 }}\) and \({\mathop{\rm Cos}\nolimits} B = \frac{{\sqrt 3 }}{2}\), then, find the value of (A + B)º.
A. 60º
B. 75º
C. 105º
D. 90º
