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CMV6120 Foundation Mathematics Unit 11: Reduction Principle and Simple Trigonometric Equations Objectives Students should be able to state the reduction principle use the reduction principle to find the trigonometric ratios of angles related to special angles such as 30, 45 and 60 solve simple trigonometric equations Unit 11: Trigonometric Equations page 1 of 5 CMV6120 Foundation Mathematics Reduction principle and simple trigonometric equations 1. Reduction Principle 1.1 Reference Angle The reference angle for an angle is the positive acute angle formed by the terminal side of and the x-axis. Given sin = 4/5, cos = 3/5 and tan = 4/3, you can easily fill in the table below. Angles = Quadrant I = (180-) II = (180+) III = (360-) IV sin 4 5 4 5 4 5 4 5 sin 4 5 4 5 4 5 4 5 cos 3 5 3 5 3 5 3 5 cos 3 5 3 5 3 5 3 5 tan 4 3 4 3 4 3 4 3 tan 4 3 4 3 4 3 4 3 The trigonometric ratios of an angle and its reference angle have the same numerical value, i.e. sin sin cos cos tan tan where the choice of sign (+ or -) depends on the quadrant in which lies. 1.2 The Reduction Principle In the first quadrant, sin sin cos cos tan tan Unit 11: Trigonometric Equations = , we have page 2 of 5 CMV6120 Foundation Mathematics In the second quadrant, = (180-), we have sin sin( 180 ) sin cos cos(180 ) cos tan tan(180 ) tan In the third quadrant, = (180+), we have sin sin( 180 ) sin cos cos(180 ) cos tan tan(180 ) tan In the fourth quadrant, = (360-), we have sin sin( 360 ) sin cos cos(360 ) cos tan tan( 360 ) tan N.B. It is also interesting to note that sin(90+) = cos; cos(90+) = -sin; tan(90+) = -cot; sin2 + cos2 = 1 sin(-) = -sin cos(-) = cos tan(-) = -tan 1.3 Trigonometric ratios of some special angles The trigonometric ratios of some special angles are listed below for easy reference. These could be found by using Pythagoras’ theorem. Ratio\θ 0o sinθ 0 1 2 ) 4 2 2 cosθ 1 tanθ 0 3 2 1 2 2 1 30o( ) 6 45o( ) 3 3 2 60o( 1 2 3 90o( ) 2 1 0 ∞ 3 Unit 11: Trigonometric Equations page 3 of 5 CMV6120 Foundation Mathematics Example 1 Find the values of the following without using a calculator. (i) (ii) cos150 tan(- ) 3 Solution (i) cos 150 = cos (180o – 30o) = -cos30o =- (ii) tan(- 3 2 )=tan(-60o) 3 = -tan 60o = _________ Example 2 sin( 90 ) Simplify sin( 90 ) Solution sin( 90 ) cos 1 sin( 90 ) cos Example 3 Simplify sin( 180 ) cos(180 ) Solution sin( 180 ) = cos(180 ) 2. Simple Trigonometric Equations We shall demonstrate some methods in solving simple equations involving trigonometric ratios of angle. Unit 11: Trigonometric Equations page 4 of 5 CMV6120 Foundation Mathematics Example 4 Solve the equation 2 sin 1 0 , 0 360 Solution sin 1 , θ = 30o, 150o 2 Example 5 4 cos 2 3 0 , 0 2 Solve the equation Solution Factorizing, (cos when (cos 3 3 3 )(cos ) 0 or simply cos 2 2 2 3 ) 0, 2 5 7 , 6 6 3 ) 0, 2 θ= ___________,__________ when (cos Example 6 Solve the equation 5 sin 2 cos 2 1 0 , 0 360 Solution Since sin2θ + cos2θ=1, the equation becomes 5sinθ –2(1- sin2θ) –1 = 0 (2sinθ– 1)(sinθ +3) = 0 gives sinθ=1/2 whence θ= _________,________ [Note: (sinθ+3) = 0 has no solution.] Unit 11: Trigonometric Equations page 5 of 5