Education Resources. This section is designed to provide examples which develop routine skills necessary for completion of this section.

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1 Education Resources Trigonometry Higher Mathematics Supplementary Resources Section A This section is designed to provide examples which develop routine skills necessary for completion of this section. R I can convert radians to degrees and vice versa.. Convert the following angles from degrees to radians, giving you answer as an exact value (d) 90 (e) 80 (f) 60 (g) 0 (h) 5 (i) 50 (j) 0 (k) 5 (l) 40 (m) 70 (n) 00 (o) 5 (n) 0 (o) 540 (p) 70. Convert the following angles from degrees to radians, giving you answer to significant figures (d) 8 (e) 07 (f) 45

2 . Convert the following angles from radians to degrees. π radians π radians π radians (d) (g) (j) π radians (e) π radians (f) 5π radians π radians (h) π radians (i) 4π radians 5π radians (k) 7π radians (l) π 4 radians (m) π 4 radians (n) 5π 4 radians (o) 7π 4 radians (p) (s) π radians (q) 5π radians (r) 7π radians π 6 radians (t) π radians (u) 5π radians 4. Convert the following angles from radians to degrees, giving you answer to significant figures. radian radians radians (d) 4 radians (e) 4 radians (f) 7 radians R I can use and apply exact values.. Write down the exact value of sin 0 sin 60 sin 45 (d) sin 0 (e) sin 50 (f) sin 5 (g) sin 90 (h) sin 80 (i) sin 70 (j) sin 0 (k) sin 5 (l) sin 40 (m) sin 00 (n) sin 0 (o) sin 5

3 . Write down the exact value of cos 0 cos 60 cos 45 (d) cos 0 (e) cos 50 (f) cos 5 (g) cos 90 (h) cos 80 (i) cos 70 (j) cos 0 (k) cos 5 (l) cos 40 (m) cos 00 (n) cos 0 (o) cos 5. Write down the exact value of tan 0 tan 60 tan 45 (d) tan 0 (e) tan 50 (f) tan 5 (g) tan 90 (h) tan 80 (i) tan 70 (j) tan 0 (k) tan 5 (l) tan 40 (m) tan 00 (n) tan 0 (o) tan 5 4. Write down the exact value of sin π 6 sin π 4 sin π (d) sin π (e) sin π (f) sin π (g) sin 5π 6 (h) sin π 4 (i) sin π (j) sin 7π 6 (m) sin π 6 (k) sin 5π 4 (n) sin 7π 4 (l) sin 4π (o) sin 5π 5. Write down the exact value of cos π 6 cos π 4 cos π (d) cos π (e) cos π (f) cos π (g) cos 5π 6 (h) cos π 4 (i) cos π

4 (j) cos 7π 6 (m) cos π 6 (k) cos 5π 4 (n) cos 7π 4 (l) cos 4π (o) cos 5π 6. Write down the exact value of tan π 6 tan π 4 tan π (d) tan π (e) tan π (f) tan π (g) tan 5π 6 (h) tan π 4 (i) tan π (j) tan 7π 6 (m) tan π 6 (k) tan 5π 4 (n) tan 7π 4 (l) tan 4π (o) tan 5π R I can sketch or identify a basic trig graph under a single transformation.. Write down the equation of each of the graphs (d)

5 . Write down the equation of each of the graphs (d). Write down the equation of each of the graphs (d)

6 4. Write down the equation of each of the graphs 5. Sketch each graph showing clearly the coordinates of the maximum and minimum values and where each graph cuts the axes. y = cos x for 0 x 60 y = sin x + for 0 x 60 y = cos x for 0 x π (d) y = sin x for 0 x π (e) y = cos x for 0 x 60 (f) y = tan(x 45) for 0 x 60 (g) y = cos (x π ) for 0 x π (h) y = sin x for 0 x π R4 I can sketch or identify a basic trig graph under combined transformations.. Write down the equation of each of the graphs

7 (d) (e) (f) (g) (h)

8 (i) (j). Sketch each graph showing clearly the coordinates of the maximum and minimum values and where each graph cuts the axes. y = cos x for 0 x 60 y = cos x for 0 x 60 (d) y = cos (x π ) for 0 x π y = sin (x π ) + for 0 x π 6 (e) y = 4 cos x for 0 x 60 (f) y = sin(x 0) for 0 x 60 (g) y = cos x for 0 x π (h) y = sin x for 0 x π R5 I can use the addition and double angle formulae.. Expand and use exact values to simplify sin (x + π 6 ) sin(x 60) cos (x π 4 ) (d) cos(x + 45) (e) cos (x + π ) (f) sin(x + 60) (g) sin(x 90) (h) sin(x + π) (i) cos(x + 80)

9 . Use an appropriate substitution (such as 45 0 = 5) then expand to find the exact values of sin 5 sin 75 cos 05. Given that sin x = and cos x = 4, find the exact values of: 5 5 sin x cos x sin x (Hint x = x + x) 4. Given that sin x = 5 and cos x =, find the exact values of: sin x cos x sin 4x (Hint 4x = (x)) 5. Given that sin x = and cos x =, find the exact values of: 5 5 sin x cos x cos x 6. Given that sin x = and cos x =, find the exact values of: sin x cos x cos 4x

10 R6 I can convert acosx + bsinx to kcos(x ± α) or ksin(x ± α), where α is in any quadrant k > 0.. A function f is defined as f(x) = 5 cos x sin x. Express f(x) in the form k cos(x + a) where k > 0 and 0 a < 60.. Express sin x cos x in the form k sin(x α) where k > 0 and 0 α < π.. A function g is defined as g(x) = cos x + sin x. Express g(x) in the form k sin(x + α) where k > 0 and 0 α < Express sin x + cos x in the form r cos(x a) where r > 0 and 0 a < π. 5. A function Q is defined as Q(x) = cos x sin x. Express Q(x) in the form k cos(x + a) where k > 0 and 0 a < Express sin x 4 cos x in the form a sin(x b) where a > 0 and 0 b < π. 7. A function f is defined as f(x) = cos x sin x. Express f(x) in the form k sin(x a) where k > 0 and 0 a < Express sin x cos x in the form k cos(x + a) where k > 0 and 0 a < π. 9. A function f is defined as f(x) = 5 cos x + sin x. Express f(x) in the form k cos(x + a) where k > 0 and 0 a < 60.

11 R7 I have revised solving basic trigonometric equations in degrees and radians.. Solve the equations: 5tanx 6 =, 0 x 60. 7sinx + = 5, 0 x 60. 4cosx + = 0, 0 x 60. (d) tanx + = 7, (e) 4sinx =, 0 x π. 0 x π. (f) 9cosx 5 = 0, 0 x π.. Solve the equations: 9tanx 5 =, 0 x 80. 4sinx + =, 0 x 60. cosx + = 0, 0 x 60.. Solve the equations: tan(x + 0) =, 0 x 60. 5sin(x + 0) + =, 0 x 60. 4cos(x + 6) + = 0, 0 x 60. (d) tan (x + π 5 ) + = 0, (e) 6sin(x + ) =, 0 x π. 0 x π. (f) cos (x + π ) + = 0, 0 x π. 6

12 Section B This section is designed to provide examples which develop Course Assessment level skills NR I can apply Trig Formulae to Mathematical Problems (excluding where trig equations have to be solved but including exact values). y. On the coordinate diagram shown, P is the point (8, 6) and Q is the point (5, ). P(8, 6) Angle POR = a and angle ROQ = b. O a b R x Find the exact value of sin(a b). Find the exact value of cos a. Q(5, ). In triangle ABC, show that: The exact value of sin p = A p q C The exact value of cos(p + q) = 5 B. For the shape shown, find the exact value of cos(ab C) C A 5 x B x

13 4. The diagram shows the right angled triangle PQR, with dimensions given. P Find the exact value of sin a. By expressing sin a as sin(a + a), find the exact value of sin a. Q a R 5. If cos x = and 0 < x < π, find the exact values of cosx and sin x Using the fact that π 4 + π 6 = 5π, find the exact value of cos (5π ). 7. It is given that cos a = 5 and sin b =. Find the exact value of sin(a + b) and cos(a + b). Hence find the exact value of tan(a + b). P 8. Triangles PSQ and RSQ are right angled with dimensions as shown in the diagram. Show that cos(a + b) is 85. Calculate the value of sin(a + b). S Q Hence calculate the value of tan(a + b). R

14 NR I have experience of using wave functions to find the maximum and minimum values.. A function f is defined as f(x) = 5 cos x sin x. Express f(x) in the form k cos(x + a) where k > 0 and 0 a < 60. Part of the graph of y = f(x) is shown in the diagram. y 0 60 y = f(x) x Find the coordinates of the minimum turning point A. A. Express sin x cos x in the form k sin(x a) where k > 0 and 0 a < π. Hence state the maximum and minimum values of sin x cos x and determine the values of x, in the interval 0 x < π, at which these maximum and minimum values occur.. Express sin x + 5 cos x in the form k sin(x + a) where k > 0 and 0 a < π. Hence state the maximum value of 4 + sin x + 5 cos x and determine the value of x, in the interval 0 x < π, at which the maximum occurs. 4. A function f is defined as f(x) = 4 cos x + sin x. Find the maximum and minimum values of f(x) and the values of, in the range 0 x < 60, at which the maximum and minimum values occur. 5. A function g is defined as g(x) = cos x sin x. Find the maximum and minimum values of g(x) and the values of, in the range 0 x < π, at which the maximum and minimum values occur.

15 NR I can solve trigonometric equations in the context of a problem.. Solve the equation sin x cos x = 0, in the interval 0 x < 80.. Solve the equation sin x sin x = 0, in the interval 0 x < 60.. Solve the equation cos x + 0 cos x = 0, in the interval 0 x < π. 4. Solve the equation cos x + sin x = sin x, in the interval 0 x < Solve the equation cos x 5 cos x 4 = 0, in the interval 0 x < π. 6. Solve the equation tan x =, in the interval 0 x < π. 7. Solve the equation sin θ = 4 cos θ, in the interval 0 x < π. 8. Express sin x + 4 cos x in the form k sin(x + a) where k > 0 and 0 a < π. Hence solve the equation sin x + 4 cos x = 0 in the interval 0 x < π. 9. Express 5sin x + cos x in the form k cos(x a) where k > 0 and 0 a < 60. Hence solve the equation 5sin x + cos x = 4 in the interval 0 x < Two curves have equations y = 6 cos x and y = sin x. Find the coordinates of the points of intersection in the range 0 x < 60.

16 . Two curves have equations y = cos x and y = cos x +. Find the coordinates of the points of intersection in the range 0 x < 80.. A curves has the equation = cos x cos x +. Find the coordinates of the points where the curve cuts the x-axis in the range 0 x < 60.. A curves has the equation y = sin x + cos x. Find the coordinates of the points where the curve cuts the x-axis in the range 0 x < The graph shows two curves which have equations y = cos x and y = sin x in the range 0 x < 80. y A O B 8O x Find the coordinates of A and B, the points of intersection between the two curves.

17 NR4 I have experience of cross topic exam standard questions. Trigonometry and integration. The expression sin x 5 cos x can be written in the form k sin(x + α) where k > 0 and 0 α < π. Calculate the values of k and α. Hence find the value of p for which 0 ( sin x 5 cos x)dx = p.. A curve for which dy = 5 cos x passes through the point ( π, ). dx Find y in terms of x.. Find the area enclosed by y = cos x and y = 4 cos x +. y y = cos x x y = 4 cos x +

18 Trigonometry and differentiation. Two functions are defined as f(x) = 7 cos x and g(x) = sin x. Write f(x) + g(x) in the form k cos(x α) where k > 0 and 0 α < π. Hence find f (x) + g (x) as a single trigonometric expression.. A curve has equation y = 7 cos x 4 sin x. Write 7 cos x 4 sin x in the form k sin(x α) where k > 0 and 0 α < π. Hence find, in the interval 0 x π, the x-coordinate of the point on the curve where the gradient is.. Find the equation of the tangent to the curve y = cos (x π ) at the point 6 where x = π Trigonometry, functions and graphs. A function f is defined as f(x) = cos x + sin x. Express f(x) in the form k cos(x a) where k > 0 and 0 a < 60. Sketch the graph of y = f(x) between 0 x < 60, showing clearly the coordinates of the maximum and minimum turning points.. Express sin x + 4 cos x in the form k sin(x + a) where k > 0 and 0 a < 60. Sketch the graph of y = sin x + 4 cos x + between 0 x < 60, showing clearly the coordinates of the maximum and minimum turning points and where the curve cuts the axes.

19 . Functions a(x) = sin x, b(x) = cos x and c(x) = x π are defined on a 4 suitable set of real numbers. Find expressions for; (i) a(c(x)); (ii) b(c(x)). (i) Show that a(c(x)) = sin x cos x. (ii) Find a similar expression for b(c(x)) and hence solve the equation a(c(x)) + b(c(x)) = for 0 x π. 4. Functions f and g are defined on suitable domains by f(x) = sin x and g(x) = x. Find expressions for; (i) f(g(x)); (ii) g(f(x)). Solve f(g(x)) = g(f(x)) for 0 x 60. Trigonometry and straight line. P is the point (6, 5). The line OP is inclined at an angle of a to the x-axis. y Find the exact values of sina and cosa. O a P(6, 5) x The line OQ is inclined at an angle of a to the x-axis. Write down the exact value of the gradient of OQ. y Q a O x

20 Answers R. π 6 π 4 π (d) π (e) π (f) π (g) π (h) π 4 (i) 5π 6 (j) 7π 6 (k) 5π 4 (l) 4π (m) π (n) 5π (o) 7π 4 (p) π 6 (q) π (r) 4π (d) 4 90 (e) 5 6 (f) (d) 90 (e) 70 (f) 450 (g) 60 (h) 0 (i) 40 (j) 00 (k) 40 (l) 45 (m) 5 (n) 5 (o) 5 (p) 0 (q) 50 (r) 0 (s) 0 (t) 5 (u) (d) 9 (e) 80 (f) 55 R. (d) (e) (f) (g) (h) 0 (i) (j) (k) (l) (m) (n) (o). (d) (e) (f) (g) 0 (h) (i) 0 (j) (k) (l) (m) (n) (o). (d) (e) (f) (g) Undefined (h) 0 (i) Undefined (j) (k) (l) (m) (n) (o) 4. (d) 0 (e) 0 (f)

21 (g) (h) (i) (j) (k) (l) (m) (n) (o) 5. (d) (e) (f) 0 (g) (h) (i) (j) (k) (l) (m) (n) (o) 6. (d) 0 (e) 0 (f) Undefined (g) (h) (i) (j) (k) (l) (m) (n) (o) R. y = sin x y = cos x y = tan x y = sin x. y = cos x y = 4 sin x y = sin x y = cos x. y = sin x y = sin x + y = cos x y = cos x y = sin(x + 0) y = tan(x 90) 5. Sketches can be check by using a graph drawing package R4. y = 4cos x y = sin x + y = cos x (d) y = sin(x + 45) or y = cos(x 45) (e) y = tan(x + 0) (f) y = sin(x 0) + (g) y = cos x + 5 (h) y = sin x (i) y = sin x (j) y = cos x. Sketches can be check by using a graph drawing package

22 R5. (d) sin x + cos x cos x sin x (e) sin x cos x cos x + sin x cos x sin x (f) sin x + cos x (g) cos x (h) sin x (i) cos x R6. 9 cos(x + 8). sin (x π ). 0 sin(x + 7.6) sin(x 0 46) 5. cos(x + 56 ) 6. 5 sin(x 0 9) 7. 5 sin(x 4 4) 8. cos (x + 7π ) 9. 4 cos(x + 07) 6 R7. x = 58, 8 x = 9, 0 x = 9, (d) x = 0 9, 4 (e) x = 4, 6 0 (f) x = 0 98, 5. x = 0 8, 0.8 x = 76, 0.8, 96.,.8, 6., 4.8 x = 65 9, 4., 45.9, 94.. x = 4 6,.6 x =, 97 x =.6, 95.4 (d) x = 0, 6 (e) x = 0 6, 4 8 (f) x = 7π, π

23 NR Proof Proof cos x = ± 40, sin x = ± sin(a + b) = , cos(a + b) = Proof NR. 9 cos(x +.8) (58, 9). sin (x π ) 4 min at x = 7π, max at x = π 4 4. sin(x ) max 7 at x =.8 4. min 5 at x = 6 9, max 5 at x = min at x = 55, max at x = 5 69 NR. x = 0, 90, 50. x = 0, 60, 80, 00. x =, x = 90, 99 5, x = 4, x = π, π 7. θ =, sin(x + 0 9) x = 57, cos(x 59) x =, x = 90, 70. x = 60, 9. x = 0, 60, 00. x = 90, 0, 70, 0 4. A(45, ) and B(90, 0) NR4 Trigonometry and integration 9 sin(x ) p = 4 7. y = 5 sin x 4. Area =.4 square units

24 Trigonometry and differentiation cos(x 0 65) sin(x 0 65). 5 sin(x.4) x = 89. y = (x π ) Trigonometry and functions and graphs cos(x 0) Sketches can be check by using a graph drawing package. 5 sin(x + 5 ) Sketches can be check by using a graph drawing package. (i) sin (x π 4 ) (ii) cos (x π 4 ) (i) Proof (ii) x = π 4, π 4 4. (i) sin x (ii) sin x x = 0, 70 5, 80, 89 5, 60 Trigonometry and the straight line sin a = 60, cos a = 6 6 tan a = 60