Trigonometric Functions | Class 11 Mathematics Notes

Sign of Trigonometric Functions

The sign of trigonometric functions depends on the quadrant in which the terminal side of the angle lies. The coordinate plane is divided into four quadrants:

• Quadrant I: 0° to 90° (0 to π/2 radians)
• Quadrant II: 90° to 180° (π/2 to π radians)
• Quadrant III: 180° to 270° (π to 3π/2 radians)
• Quadrant IV: 270° to 360° (3π/2 to 2π radians)

In the unit circle, the x-coordinate (cos x) and y-coordinate (sin x) determine the sign of sine and cosine functions respectively. The tangent function's sign depends on the signs of sine and cosine since tan x = sin x / cos x.

The signs of the six trigonometric functions in each quadrant are summarized as:

• Quadrant I: All six functions are positive.
• Quadrant II: sin and cosec are positive; cos, sec, tan, cot are negative.
• Quadrant III: tan and cot are positive; sin, cosec, cos, sec are negative.
• Quadrant IV: cos and sec are positive; sin, cosec, tan, cot are negative.

This knowledge is essential for solving trigonometric equations and understanding the behavior of functions across their domains. The section also explains the symmetry properties of sine and cosine functions with respect to positive and negative angles.

📊 Diagram: Figure 12 on page 9: (Fig 3.7). Therefore; Figure 13 on page 10: functions in different quadrants. In fact, we have the following table.

🧪 Activity: Activity: Determining signs of trigonometric functions for given angles in different quadrants.

🔗 Connection: Understanding signs leads to defining domain and range of trigonometric functions.

Table on page 10 (6×5)

Table on page 4 (4×11)

| 3.2.3 Relation between radian and real numbers Consider the unit circle with centre O. Let A be any point on the circle. Consider OA as initial side of an angle. Then the length of an arc of the circle will give the radian measure of the angle which the arc will subtend at the | | | | | | | | | 2 1 | |

| centre of the circle. Consider the line PAQ which is 1 A tangent to the circle at A. Let the point A represent the O 0 real number zero, AP represents positive real number and AQ represents negative real numbers (Fig 3.5). If we −1 rope the line AP in the anticlockwise direction along the circle, and AQ in the clockwise direction, then every real number will correspond to a radian measure and −2 conversely. Thus, radian measures and real numbers can Fig 3.5 Q be considered as one and the same. 3.2.4 Relation between degree and radian Since a circle subtends at the centre an angle whose radian measure is 2π and its degree measure is 360°, it follows that 2π radian = 360° or π radian = 180° The above relation enables us to express a radian measure in terms of degree measure and a degree measure in terms of radian measure. Using approximate value 22 of π as , we have 7 180° 1 radian = = 57° 16′ approximately. π π Also 1° = radian = 0.01746 radian approximately. 180 The relation between degree measures and radian measure of some common angles are given in the following table: Degree 30° 45° 60° 90° 180° 270° 360° | | | | | | | | | | |

| | Radian | π 6 | π 4 | π 3 | π 2 | π | 3π 2 | 2π | | |

Table on page 9 (3×9)

| | 0° | π 6 | π 4 | π 3 | π 2 | π | 3π 2 | 2π |

| sin | 0 | 1 2 | 1 2 | 3 2 | 1 | 0 | – 1 | 0 | | cos | 1 | 3 2 | 1 2 | 1 2 | 0 | – 1 | 0 | 1 | | tan | 0 | 1 3 | 1 | 3 | not defined | 0 | not defined | 0 |