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Maths Quest Maths B Year 11 for Queensland WorkSHEET 6.2 Chapter 6 Trigonometric equations WorkSHEET 6.2 1 Trigonometric equations Name: ___________________________ 1 (a) State the Pythagorean identity. (a) sin2x + cos2x = 1 (b) Write the Pythagorean identity with sin2x (b) as the subject. (c) Write the Pythagorean identity with cos2x as the subject. sin2x = 1 cos2x (c) 2 If sin = 0.7, and 0o < < 90o, find, correct to (a) three decimal places: (a) cos (b) 3 cos2x = 1 sin2x cos 2 1 sin 2 4 1 (0.7) 2 1 0.49 tan 0.51 cos 0.51 0.714 (b) 3 Find all possible values of sin x if cos x = 0.25. sin cos 0 .7 0.714 0.980 tan sin2x = 1 cos2x = 1 (0.25)2 = 1 0.0625 = 0.9375 sin x = 0.9375 x = 0.968 or 0.968 3 Maths Quest Maths B Year 11 for Queensland 4 Find the exact value of sin x if cos x = Chapter 6 Trigonometric equations WorkSHEET 6.2 2 3 2 and x 5 is in the fourth quadrant. Third side of triangle = 21 21 From triangle sin x = but x is in the fourth 5 21 quadrant, so sin x = . 5 5 Find ao if 0o < < 90o and (a) sin ao = cos 67o (b) 6 (a) sin ao = cos 67o = sin (90 – 67)o = sin 23o ao = 23o (b) cos ao = sin 8o = cos (90 – 8)o = cos 82o ao = 82o cos ao = sin 8o If 0o a 90o and sin ao = 2 4 1 , find the exact 3 value of : (a) cos ao (b) tan ao (c) sin (90 – a)o Third side of triangle = (d) cos (180 + a)o (a) cos ao = (b) tan ao = (c) (d) 8 8 3 1 8 sin (90 – a)o = cos ao 8 = 3 o cos (180 + a) = cos ao 8 = 3 Maths Quest Maths B Year 11 for Queensland Chapter 6 Trigonometric equations WorkSHEET 6.2 3 7 Solve the equation 2 sin2x = sin x over the domain 0 x 2 . 2 sin2x = sin x 2 sin2x sin x = 0 sin x (2 sin x 1) = 0 sin x = 0 2 sin x 1 = 0 x = 0, , 2 2 sin x = 1 1 sin x = 2 5 x= , 6 6 5 Solution: x = 0, , , , 2 6 6 4 8 Solve the equation 2cos2x + the domain 0 x 2 . 2cos2x + 3 cos x = 0 cos x (2 cos x + 3 ) = 0 cos x = 0 2 cos x + 3 = 0 3 x= , 2 cos x = 2 2 4 3 cos x = 0 for cos x = x= Solution: x = 9 Solve 2 sin2x = 3 sin x 1 in the domain 0 x 2 . 3 3 2 5 7 , 6 6 5 7 3 , , or 2 6 2 6 2 sin2x = 3 sin x 1 2 sin2x 3 sin x + 1 = 0 (2 sin x 1)(sin x 1) = 0 2 sin x 1 = 0 sin x 1 = 0 2 sin x = 1 sin x = 1 3 1 sin x = x= , 2 2 2 5 x= , 6 6 5 3 Solution: x = , , or 2 6 2 6 4 Maths Quest Maths B Year 11 for Queensland 10 Solve 2sin2x = 2 0 x 2 . Chapter 6 Trigonometric equations WorkSHEET 6.2 3 cos x in the domain 2sin2x = 2 3 cos x 2 2(1 cos x) = 2 3 cos x 2cos2x 3 cos x = 0 cos x (2cos x 3 ) = 0 cos x = 0 2 cos x 3 = 0 3 x= , 2 cos x = 3 2 2 3 cos x = 2 11 x= , 6 6 3 11 Solution: x = , , or 6 6 2 2 4 4