Trigonometric Functions PDF

Chapter 3 TRIGONOMETRIC FUNCTIONS vA mathematician knows how to solve a problem, he can not solve it. – MILNE v 3.1 Introduction The word ‘trigonometry’ is derived from the Greek wor...

Chapter 3 TRIGONOMETRIC FUNCTIONS vA mathematician knows how to solve a problem, he can not solve it. – MILNE v 3.1 Introduction The word ‘trigonometry’ is derived from the Greek words ‘trigon’ and ‘metron’ and it means ‘measuring the sides of a triangle’. The subject was originally developed to solve geometric problems involving triangles. It was studied by sea captains for navigation, surveyor to map out the new lands, by engineers and others. Currently, trigonometry is used in many areas such as the science of seismology, designing electric circuits, describing the state of an atom, predicting the heights of tides in the ocean, analysing a musical tone and in many other areas. In earlier classes, we have studied the trigonometric Arya Bhatt ratios of acute angles as the ratio of the sides of a right (476-550) angled triangle. We have also studied the trigonometric identities and application of trigonometric ratios in solving the problems related to heights and distances. In this Chapter, we will generalise the concept of trigonometric ratios to trigonometric functions and study their properties. 3.2 Angles Angle is a measure of rotation of a given ray about its initial point. The original ray is Vertex Fig 3.1 2024-25 44 MATHEMATICS called the initial side and the final position of the ray after rotation is called the terminal side of the angle. The point of rotation is called the vertex. If the direction of rotation is anticlockwise, the angle is said to be positive and if the direction of rotation is clockwise, then the angle is negative (Fig 3.1). The measure of an angle is the amount of rotation performed to get the terminal side from the initial side. There are several units for measuring angles. The definition of an angle Fig 3.2 suggests a unit, viz. one complete revolution from the position of the initial side as indicated in Fig 3.2. This is often convenient for large angles. For example, we can say that a rapidly spinning wheel is making an angle of say 15 revolution per second. We shall describe two other units of measurement of an angle which are most commonly used, viz. degree measure and radian measure. th  1  3.2.1 Degree measure If a rotation from the initial side to terminal side is   of  360  a revolution, the angle is said to have a measure of one degree, written as 1°. A degree is divided into 60 minutes, and a minute is divided into 60 seconds. One sixtieth of a degree is called a minute, written as 1′, and one sixtieth of a minute is called a second, written as 1″. Thus, 1° = 60′, 1′ = 60″ Some of the angles whose measures are 360°,180°, 270°, 420°, – 30°, – 420° are shown in Fig 3.3. Fig 3.3 2024-25  TRIGONOMETRIC FUNCTIONS 45 3.2.2 Radian measure There is another unit for measurement of an angle, called the radian measure. Angle subtended at the centre by an arc of length 1 unit in a unit circle (circle of radius 1 unit) is said to have a measure of 1 radian. In the Fig 3.4(i) to (iv), OA is the initial side and OB is the terminal side. The figures show the 1 1 angles whose measures are 1 radian, –1 radian, 1 radian and –1 radian. 2 2 (i) (ii) (iii) (iv) Fig 3.4 (i) to (iv) We know that the circumference of a circle of radius 1 unit is 2π. Thus, one complete revolution of the initial side subtends an angle of 2π radian. More generally, in a circle of radius r, an arc of length r will subtend an angle of 1 radian. It is well-known that equal arcs of a circle subtend equal angle at the centre. Since in a circle of radius r, an arc of length r subtends an angle whose measure is 1 l radian, an arc of length l will subtend an angle whose measure is radian. Thus, if in r a circle of radius r, an arc of length l subtends an angle θ radian at the centre, we have l θ = or l = r θ. r 2024-25 46 MATHEMATICS P 3.2.3 Relation between radian and real numbers 2 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 1 measure of the angle which the arc will subtend at the centre of the circle. Consider the line PAQ which is 1 A 0 tangent to the circle at A. Let the point A represent the O 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° π π π π 3π Radian π 2π 6 4 3 2 2 2024-25  TRIGONOMETRIC FUNCTIONS 47 Notational Convention Since angles are measured either in degrees or in radians, we adopt the convention that whenever we write angle θ°, we mean the angle whose degree measure is θ and whenever we write angle β, we mean the angle whose radian measure is β. Note that when an angle is expressed in radians, the word ‘radian’ is frequently π π omitted. Thus, π = 180° and = 45° are written with the understanding that π and 4 4 are radian measures. Thus, we can say that π 180 × Radian measure = Degree measure 180 Degree measure = × Radian measure π Example 1 Convert 40° 20′ into radian measure. Solution We know that 180° = π radian. 1 π 121 121π Hence 40° 20′ = 40 degree = × radian = radian. 3 180 3 540 121π Therefore 40° 20′ = radian. 540 Example 2 Convert 6 radians into degree measure. Solution We know that π radian = 180°. 180 1080 × 7 Hence 6 radians = × 6 degree = degree π 22 7 7 × 60 = 343 degree = 343° + minute [as 1° = 60′] 11 11 2 = 343° + 38′ + minute [as 1′ = 60″] 11 = 343° + 38′ + 10.9″ = 343°38′ 11″ approximately. Hence 6 radians = 343° 38′ 11″ approximately. Example 3 Find the radius of the circle in which a central angle of 60° intercepts an 22 arc of length 37.4 cm (use π = ). 7 2024-25 48 MATHEMATICS 60π π Solution Here l = 37.4 cm and θ = 60° = radian = 180 3 l Hence, by r = , we have θ 37.4×3 37.4×3×7 r= = = 35.7 cm π 22 Example 4 The minute hand of a watch is 1.5 cm long. How far does its tip move in 40 minutes? (Use π = 3.14). Solution In 60 minutes, the minute hand of a watch completes one revolution. Therefore, 2 2 in 40 minutes, the minute hand turns through of a revolution. Therefore, θ = × 360° 3 3 4π or radian. Hence, the required distance travelled is given by 3 4π l = r θ = 1.5 × cm = 2π cm = 2 × 3.14 cm = 6.28 cm. 3 Example 5 If the arcs of the same lengths in two circles subtend angles 65°and 110° at the centre, find the ratio of their radii. Solution Let r1 and r2 be the radii of the two circles. Given that π 13π θ1 = 65° = × 65 = radian 180 36 π 22π and θ2 = 110° = × 110 = radian 180 36 Let l be the length of each of the arc. Then l = r1θ1 = r2θ2, which gives 13π 22π r1 22 × r1 = × r2 , i.e., r = 36 36 2 13 Hence r1 : r2 = 22 : 13. EXERCISE 3.1 1. Find the radian measures corresponding to the following degree measures: (i) 25° (ii) – 47°30′ (iii) 240° (iv) 520° 2024-25  TRIGONOMETRIC FUNCTIONS 49 2. Find the degree measures corresponding to the following radian measures 22 (Use π = ). 7 11 5π 7π (i) (ii) – 4 (iii) (iv) 16 3 6 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 subtends an angle of 2π radian at the π centre of the circle, ∠AOB = , Fig 3.6 2 2024-25 50 MATHEMATICS 3π π ∠AOC = π and ∠AOD =. All angles which are integral multiples of are called 2 2 quadrantal angles. The coordinates of the points A, B, C and D are, respectively, (1, 0), (0, 1), (–1, 0) and (0, –1). Therefore, for quadrantal angles, we have cos 0° = 1 sin 0° = 0, π π cos =0 sin =1 2 2 cosπ = − 1 sinπ = 0 3π 3π cos =0 sin = –1 2 2 cos 2π = 1 sin 2π = 0 Now, if we take one complete revolution from the point P, we again come back to same point P. Thus, we also observe that if x increases (or decreases) by any integral multiple of 2π, the values of sine and cosine functions do not change. Thus, sin (2nπ + x) = sin x , n ∈ Z , cos (2nπ + x) = cos x , n ∈ Z Further, sin x = 0, if x = 0, ± π, ± 2π , ± 3π,..., i.e., when x is an integral multiple of π π 3π 5π and cos x = 0, if x = ± ,± ,± ,... i.e., cos x vanishes when x is an odd 2 2 2 π multiple of. Thus 2 π, where n is any integer sin x = 0 implies x = nπ, π cos x = 0 implies x = (2n + 1) , where n is any integer. 2 We now define other trigonometric functions in terms of sine and cosine functions: 1 cosec x = , x ≠ nπ, where n is any integer. sin x 1 π sec x = , x ≠ (2n + 1) , where n is any integer. cos x 2 sin x π tan x = , x ≠ (2n +1) , where n is any integer. cos x 2 cos x cot x = , x ≠ n π, where n is any integer. sin x 2024-25  TRIGONOMETRIC FUNCTIONS 51 We have shown that for all real x, sin2 x + cos2 x = 1 It follows that 1 + tan2 x = sec2 x (why?) 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 and sin (– x) = – sin x Since for every point P (a, b) on the unit circle, – 1 ≤ a ≤ 1 and Fig 3.7 2024-25 52 MATHEMATICS – 1 ≤ b ≤ 1, we have – 1 ≤ cos x ≤ 1 and –1 ≤ sin x ≤ 1 for all x. We have learnt in π previous classes that in the first quadrant (0 < x < ) a and b are both positive, in the 2 π second quadrant ( < x