persamaan trigonometri A COS X + B SIN X

makalah trigonometri

1. TRIGONOMETRIC EQUATIONS OF THE FORM A COS X + B SIN X =C Lecturer: Wa Ode Indrawati, S.Pd., M.Pd. by: group 5

2. Members of the group 1. Nina Astuti (A1I123019) 2. Rahma ayu (A1I123021) 3. Eky Tri Putri (A1I123043) 4. Komang Trima sari (A1I123088) 5. Nuryasti (A1I123094) 6. Viky Arditya (A1I123098) 7. Calan (A1I120086)

3. 2.1 Understanding Trigonometric Equations Trigonometric equations are equations that contain comparisons between trigonometric angles in the form x. Solve this equation by finding all the values ​​of the angles x, so that the equation is true for a certain area of ​​origin. Solving trigonometric equations in degree form in the range 0o to 360o or in radian form in the range o to 2π

4. 2.2 Express the form a cos x + b sin x into the form k cos (x-a) F(x)= a cos x+b sin x can be transformed into the form k cos (x - α), with k a setting and 0 ≤ a ≤ 360. The values of k and & are determined by the values of a and b with the process as follows: a cos x + b sin x = k cos (x – α) a cos x + b sin x = k (cos x cos α + sin x sin α) a cos x + b sin x = k cos x . cos α + k sin x . sin α From the equation above, the cos x coefficient on the left side must be the same as the cos x coefficient on the right side and the same goes for the sin x coefficient. Therefore. we get the relationship: Cos α = a ……..(1) Sin α = b………(2)

5. 1. Determinate the value of k: Note that if we square and add equations (1) and (2), we get: k² cos2 α + k² sin² α = a² + b² k² (cos2 α + sin² α ) = a² + b² k² = a2 + b², because (cos2α + sin2α = 1) k = √a² + b²,taken k > 0 2.Determinate the size of α Divide equation (2) by equation (1).

6. Determining the description above, we can conclude: Cos x + sin x = k cos (x - α) the relationship applies The size of the angle depends on the sign (positive / negative) of the coefficients cos x and sin x according to the following table:

7. Example : Convert the form cos x + √3 sin x into the form k cos (x - α) Answer : Cos x + √3 sin x = k cos (x - α) cos x + √3 sin x = k cos x.cos α + k sin x . sin α Obtained : k cos α = 1 → a = 1 K sin α = √3 → b = √3 Value of k : k = = 2 The size of angle α : dan α terletak di kuadran I 𝒂 𝟏 Maka α = 60° So, cos x + √3 sin x = 2 cos (x - 60)°

8. 2.3 Trigonometric Equation a cos x + b sin x = C One use of converting the form a cos x + b sin x into the form k cos (x-α) is to determine the solution to the trigonometry equation in the form cos x + sin x = C, with a, b, and c being real numbers. not zero. First, the life side of the equation is converted into be in the form k cos (x - α), with k = √a² + b² and tan α = 𝒃. Then by substituting a cos x + b sin x 𝒂 with k cos (x - α), the equation becomes: k cos (x - α) = C 𝒄 cos(x - α) = 𝒌

9. Example: Determinate the values of x in the interval 0 ≤ x ≤ 360 that satisfy the equation 3 cos x + 4 sin x = 2 3 cos x + 4 sin x = k cos (x - α) 3 cos x + 4 sin x = k cos x . cos α + k sin x sin α

10. 2.4 Maximum and Minimum Value of the function f(x) = a cos x + b sin x Remembering that the form a cos x + b sin x can be changed to form k cos (x - a). Then trigonometric function y = f(x) = a cos x + b sin x can be converted into the form y = f(x) = k cos (x - α) with k as s positive constant and 0 ≤ α ≤ 360. For example a cos x+ b sin x = k cos (x - α), Then by the composing the right side obtained : a cos x + b sin x = k cos (x - α) a cos x + b sin x = k (cos x cos a + sin x sin α) a cos x + b sin x = k cos x cos α + k sin x sin α

11. By equating the coefficients cos x and sin x we ​​obtain:

12. Based on the form of the function above, we can determine the stationary values (maximum and minimum value)of trigonometric function.The stationary values in question are :

13. Example: Determine the maximum and minimum values for each of the following trigonometric function : 1) y = 2 cos x + 3 sin x 2) y = 4 cos x -3 sin x

14. 2.5 Maximum and Minimum Value of the function y = f(x) = a Cos x + b sinx + c

15. Example : 1) Determine the maximum and minimum values ​​for each of the following trigonometric functions.y = f(x) = 2 cos x + 3 sinx - 1. 2) y = f(x) = √7 cos x - 5 sin x - √3.

16. 2.6 Graphing the Function y = f(x) = a cos x + b sin x

17. Example : Draw a sketch of the trigonometric graph y = cos x + √3 sin x in the interval 0 ≤ x ≤ 360.

18. 3. Titik-titik A (60,2), B(240,-2),C(150,0),D(330,0) dan E(0,1). Dilukiskan pada bidang cartesius. Titik-titik td di hubungkan dengan kurva yang mulus, sehingga diperoleh sketsa grafik fungsi y = cos x + √3 sin x, perhatikan gambar berikut.

19. 2.7 The Function Graph y = f(x) = a cos x + b sin x +c A sketch of a trigonometric graph in the form y = f(x) = a cos x + b sin x + c where a, b and c are non zero real number can be obtained from a sketch of the function graph y = f(x) a cos x + b sin x which is translated vertically as far : (i) C satuan ke atas, jika e positif; atau (ii) C satuan ke bawah, jika e negatif.

20. Example : Draw a sketch of the graph of each of the following trigonometric functions in the interval (0 ≤ x ≤360) 1. Y = f(x) = cosx + √3 sin x° +2 2. Y= f(x) = cos x + √3 sin x°-1

21. Answer : In the previous example we have sketched the graph of the function Y = f(x) = cos x + √3 sin x 1. Sketch the graph of the function Y = f(x) = cos x° + √3 sin x° +2 obtained from the sketch of the graph of the function Y = f(x) = cos x + √3 sin x which is translated vertically 2 unit upwards 2. Sketch the graph of the function Y = f(x) = cos x + √3 sin x° - 1 obtained from the sketch of the graph of the function Y = f(x) = cos x + √3 sin x which is translated vertically 1 unit downwards

22. Thank You