Question

If \(\tan \phi<0\) and \(\sin \phi>0\), state the quadrant in which \(\phi\) lies.

Short answer

Answer: Quadrant II

Step-by-step solution

Understanding the signs of trigonometric functions

Recall that in the coordinate plane, any angle can be represented in terms of its position relative to the four quadrants: - Quadrant I: All angles between \(0\) and \(90\) degrees (0 and \(\frac{\pi}{2}\) radians) - Quadrant II: All angles between \(90\) and \(180\) degrees (\(\frac{\pi}{2}\) and \(\pi\) radians) - Quadrant III: All angles between \(180\) and \(270\) degrees (\(\pi\) and \(\frac{3\pi}{2}\) radians) - Quadrant IV: All angles between \(270\) and \(360\) degrees (\(\frac{3\pi}{2}\) and \(2\pi\) radians) Trigonometric functions have specific signs depending on which quadrant they are: | Quadrant | sin | cos | tan | | ---------- | ----------- | ----------- | ----------- | | I | Positive | Positive | Positive | | II | Positive | Negative | Negative | | III | Negative | Negative | Positive | | IV | Negative | Positive | Negative | We are given that \(\tan \phi<0\) (Negative) and \(\sin \phi>0\) (Positive).

Identifying the quadrant

From the table above, we can see that the only quadrant where \(\tan \phi\) is negative and \(\sin \phi\) is positive is Quadrant II. Thus, the angle \(\phi\) lies in Quadrant II.

Key concepts

Quadrants

In trigonometry, the coordinate plane is divided into four sections known as quadrants. These quadrants help us determine the signs of different trigonometric functions based on the angle's position.

• Quadrant I: Angles range from 0 to 90 degrees, or in radians, from 0 to \(\frac{\pi}{2}\). Here, the sine, cosine, and tangent functions are all positive.
• Quadrant II: Angles are between 90 to 180 degrees, or \(\frac{\pi}{2}\) to \(\pi\) radians. Sine is positive, cosine and tangent are negative.
• Quadrant III: This includes angles from 180 to 270 degrees, or \(\pi\) to \(\frac{3\pi}{2}\) radians. Sine and cosine are negative; however, tangent is positive.
• Quadrant IV: Angles cover 270 to 360 degrees, or \(\frac{3\pi}{2}\) to \(2\pi\) radians. In this quadrant, sine is negative, cosine is positive, and tangent is negative.

Each quadrant provides unique conditions for the behavior of sine, cosine, and tangent. Recognizing these patterns is essential for solving trigonometric problems.

Trigonometric Functions

Trigonometric functions relate the angles and sides of a right triangle and include sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). These functions are not only fundamental in geometry but are also widely used in various fields such as physics, engineering, and navigation.

• Sine (\(\sin\)): This function represents the ratio of the length of the opposite side of an angle to the hypotenuse in a right triangle.
• Cosine (\(\cos\)): It calculates the ratio of the adjacent side to the hypotenuse.
• Tangent (\(\tan\)): Tangent is the ratio of the opposite side to the adjacent side.

Each function gives different insights depending on the angle and helps in determining various characteristics of the triangle.

Angle Signs

The sign of trigonometric functions changes depending on which quadrant the angle lies in. This concept is vital for solving trigonometric equations and understanding geometry.

Quadrant Sign Patterns:

Using the knowledge of quadrants, you can predict the sign of trigonometric functions:

• In Quadrant I, all trigonometric functions are positive.
• In Quadrant II, only sine is positive; cosine and tangent are negative.
• In Quadrant III, tangent is positive, while sine and cosine are negative.
• In Quadrant IV, cosine is positive, and sine and tangent are negative.

Understanding these sign changes can be incredibly helpful. For instance, if given that \(\sin \phi > 0\) and \(\tan \phi < 0\), one can determine that the angle must lie in Quadrant II. This systematic approach simplifies the problem-solving process and aids in visualizing the angle's exact position on the plane.