Mathematics Lecture. 3 Chapter. 1 Trigonometric Functions. By Dr. Mohammed Ramidh

Transcription

1 Mathematics Lecture. 3 Chapter. 1 Trigonometric Functions By Dr. Mohammed Ramidh

2 Trigonometric Functions This section reviews the basic trigonometric functions. Trigonometric functions are important because they are periodic. Radian Measure: The radian measure of the angle ACB at the center of the unit circle equals the length of the arc that ACB cuts from the unit circle. Figure shows that s = r θ is the length of arc cut from a circle of radius r when the subtending angle θ π radians = 180. Degrees to radians: multiply by π 180 Radians to degrees: multiply by 180 π For example: 45 in radian measure is 45 π π 6 in degree measure is 180 = π 4 π * 180 = π

3 Figure shows the angles of two common triangles in both measures. Angles measured counterclockwise from the positive x-axis are assigned positive measures;

4 angles measured clockwise are assigned negative measures, shows in Figure. Types and trigonometric relations: The Six Basic Trigonometric Functions, sin θ = opp hyp tan θ = opp adj sec θ = hyp adj cos θ = adj hyp cot θ = adj opp csc θ = hyp opp

5 the angle in standard position in a circle of radius r. We then define the trigonometric functions in terms of the coordinates of the point P(x, y) where the angle s terminal ray intersects the circle (shows in Figure). sine: sin θ = y r cosine: cos θ = x r tangent: tan θ = y x cotangent: cot θ = x y secant: sec θ = r x cosecant: csc θ = r y Notice also the following definitions, whenever the quotients are defined. tan θ = sin θ sec θ = cos θ 1 cos θ cot θ = csc θ = 1 tan θ 1 sin θ

6 The exact values of these trigonometric ratios for some angles can be read from the triangles in Figure. sin π 4 = 1 sin π 6 = 1 sin π 3 = 3 cos π 4 = 1 cos π 6 = 3 cos π 3 = 1 tan π 4 = 1 tan π 6 = 1 3 tan π 3 = 3 The CAST rule (shown in Figure) is useful for remembering when the basic trigonometric functions are positive or negative.

7 from the triangle in Figure, we see that. sin π 3 = 3 cos π 3 = 1 tan π 3 = 3 Using a similar method we determined the values of sin θ, cos θ, and tan θ shown in Table 1.4..

8 EXAMPLE 1: If tanθ = 3 and 0< θ < π, find the five other trigonometric functions of θ. Solution : From tanθ = 3 we construct the right triangle of height 3 (opposite) and base (adjacent) in Figure. The Pythagorean theorem gives the length of the hypotenuse, = 13. From the triangle we write the values of the other five trigonometric functions: cos θ = 13 sin θ = 3 13 sec θ = 13 csc θ = 13 3 cot θ = 3

9 Periodicity and Graphs of the Trigonometric Functions Periodic Function: A function ƒ(x) is periodic if there is a positive number p such that f(x+p) = f(x) for every value of x. The smallest such value of p is the period of ƒ. graph trigonometric functions in the coordinate plane: See Figure,

10 As we can see in Figure, the tangent and cotangent functions have period p = πthe other four functions have period π. Even and Odd Trigonometric functions: The symmetries in the Figure graphs functions, the cosine and secant functions are even and the other four functions are odd: Identities: The coordinates of any point P(x, y) in the plane can be expressed in terms of the point s distance from the origin and the angle θ that ray OP makes with the positive x-axis (shown in Figure). cos θ = x and sin θ = y We have r r x = r cos θ, y = r sin θ.

11 When r = 1 we can apply the Pythagorean theorem to the reference right triangle in Figure and obtain the equation. cos θ + sin θ = 1...(1) This equation, Dividing in turn by cos θ and sin θ gives 1 + tan θ = sec θ. 1 + cot θ = csc θ. (1 ) Addition Formulas: cos (A + B) = cos A cos B sin A sin B sin( A + B) = sin A cos B + cos A sin B There are similar formulas for...() cos (A B) = cos A cos B + sin A sin B sin(a B) = sin A cos B - cos A sin B

12 For example, substituting θ for both A and B in the addition formulas gives. Double-Angle Formulas: cos θ = cos θ sin θ sin θ = sin θ cos θ...(3) Additional formulas come from combining the equations cos θ + sin θ = 1, cos θ sin θ = cos θ. We add the two equations to get cos θ = 1 + cos θ, and subtract the second from the first to get sin θ = 1 cos θ, This results in the following identities, Half-Angle Formulas: cos θ = 1+cosθ sin θ = 1 cosθ...(4)...(5)

13 The Law of Cosines: If a, b, and c are sides of a triangle ABC and if u is the angle opposite c, then c = a + b - ab cos θ....(6) This equation is called the law of cosines. As in Figure The coordinates of A are (b, 0); the coordinates of B are (a cos θ, a sin θ). The square of the distance between A and B is therefore, C = (acos θ b) + (asinθ) C = a (cos θ + sin θ) + b abcosθ When cos θ + sin θ = 1 than, C = a + b abcosθ

14 EXERCISES 1.7: 1. One of sin x, cos x, and tan x is given. Find the other two if x lies in the specified interval. a. cos x = , x [π, π b. sin x = - 1 c. tan x = 1, x [ π, 3π x [π, 3π. Graph the following functions. What is the period of each function? a. cos πx b. sin πx 3 c. cos(x π ) d. sin(x π 4 ) + 1 e. S = - tan πt h. S = sec( πt ) 3. Use the addition formulas to derive the identities in the following Exercises. a. cos(x π ) = sin x b. sin(x π ) = cos x c. sin(π x) e. Evaluate cos 11π as cos( π + π ) 5π h. Evaluate sin Using the Double-Angle Formulas, Find the function values in following Exercises. a. cos π b. sin π 1 8 tan A + tan B 5.Derive the formula. tan(a + B) =,and derive the formula tan (A B). 1 tan A tan B 6.What happens if you take B =A in the identity cos(a B) = cos A cos B + sin A sinb?

15 7. Apply the law of cosines to the triangle in the accompanying figure to derive the formula for cos(a B). 8. Apply the formula for cos (A B) to the identity sin θ = π cos θ to obtain the addition formula for sin(a + B). 9. A triangle has sides a = and b = 3 and angle C = 60. Find the length of side c. 10. The law of sines says that if a, b, and c are the sides opposite the angles A, B, and C in a sin A triangle, then = sin B = sin C a b c Use the accompanying figures and the identity sin ( π - θ ) = sin θ. if required, to derive the law. 11. A triangle has side c = and angles A = π 4 and B = π 3. Find the length a of the side opposite A.