Trigonometric Functions | Class 11 Mathematics Notes

Fundamental Identities

Fundamental identities are equations involving trigonometric functions that hold true for all values of the variable within the domain. The most important identity is the Pythagorean identity:

cos² x + sin² x = 1

This identity is derived from the equation of the unit circle x² + y² = 1, where x = cos x and y = sin x.

Other important identities include:

1 + tan² x = sec² x 1 + cot² x = cosec² x

These identities are essential tools for simplifying trigonometric expressions and solving equations. They also form the basis for proving more complex identities.

The section explains the derivation and applications of these identities with examples. Understanding these identities helps in manipulating trigonometric expressions and is fundamental for advanced mathematics.

📊 Diagram: Figure 18 on page 15

🧪 Activity: Activity: Verify fundamental identities for specific angle values.

🔗 Connection: These identities are used in the next section to derive formulas for sum and difference of angles.

Table on page 14 (1×2)

| trigonometric functions. 5 12 Solution Since cot x = – , we have tan x = – 12 5 144 169 Now sec2 x = 1 + tan2 x = 1 + = 25 25 13 Hence sec x = ± 5 Since x lies in second quadrant, sec x will be negative. Therefore 13 sec x = – , 5 which also gives 5 cosx = − 13 Further, we have 12 5 12 sin x = tan x cos x = (– ) ×(– ) = 5 13 13 1 13 and cosec x = = . sin x 12 31π Example 8 Find the value of sin . 3 Solution We know that values of sin x repeats after an interval of 2π. Therefore | |

Table on page 7 (1×2)

| 3. A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second? 4. Find the degree measure of the angle subtended at the centre of a circle of 22 radius 100 cm by an arc of length 22 cm (Use π = ). 7 5. In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord. 6. If in two circles, arcs of the same length subtend angles 60° and 75° at the centre, find the ratio of their radii. 7. Find the angle in radian through which a pendulum swings if its length is 75 cm and th e tip describes an arc of length (i) 10 cm (ii) 15 cm (iii) 21 cm 3.3 Trigonometric Functions In earlier classes, we have studied trigonometric ratios for acute angles as the ratio of sides of a right angled triangle. We will now extend the definition of trigonometric ratios to any angle in terms of radian measure and study them as trigonometric functions. Consider a unit circle with centre at origin of the coordinate axes. Let P (a, b) be any point on the circle with angle AOP = x radian, i.e., length of arc AP = x (Fig 3.6). We define cos x = a and sin x = b Since ∆OMP is a right triangle, we have OM2 + MP2 = OP2 or a2 + b2 = 1 Thus, for every point on the unit circle, we have a2 + b2 = 1 or cos2 x + sin2 x = 1 Since one complete revolution | |

Table on page 9 (1×2)

| 1 + cot2 x = cosec2 x (why?) In earlier classes, we have discussed the values of trigonometric ratios for 0°, 30°, 45°, 60° and 90°. The values of trigonometric functions for these angles are same as that of trigonometric ratios studied in earlier classes. Thus, we have the following table: π π π π 3π 0° π 2π 6 4 3 2 2 1 1 3 sin 0 1 0 – 1 0 2 2 2 3 1 1 cos 1 0 – 1 0 1 2 2 2 1 not not tan 0 1 3 0 0 3 defined defined The values of cosec x, sec x and cot x are the reciprocal of the values of sin x, cos x and tan x, respectively. 3.3.1 Sign of trigonometric functions Let P (a, b) be a point on the unit circle with centre at the origin such that ∠AOP = x. If ∠AOQ = – x, then the coordinates of the point Q will be (a, –b) (Fig 3.7). Therefore cos (– x) = cos x | |