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Mathematics Grade 10 GRADE 10 MATHEMATICS www.learnxtra.co.za SESSION 4: DEFINING TRIG RATIOS ON THE CARTESIAN PLANE SESSION 7: DEFINING TRIG RATIOS ON THE CARTESIAN PLANE KEY CONCEPTS: Reciprocal of Trig Ratios The Cartesian plane CAST diagram Special angles 0o and 90o X-PLANATION Definition of Trig Ratios for triangles: Defining the Reciprocal Ratios Brought to you by Page 1 Mathematics Grade 10 www.learnxtra.co.za For triangle ABC we define the reciprocal ratios as follows: Trigonometry on the Cartesian plane A line, OP, drawn from the origin, will form an angle A with the x-axis. We measure angle A from the x-axis in an anti-clockwise direction. X We can construct a right and triangle OPX, with OP as the hypothenuse. Since the line OP can rotate about O, we think of it as a radius and call its length r. The x coordinates of P, gives the size of the side adjacent to A and the y co-ordinates of P, gives the size of the side opposite to A. Now we can re-define the trigonometric ratios on the Cartesian plane as follows: Brought to you by Page 2 Mathematics Grade 10 www.learnxtra.co.za The CAST diagram In the Cartesian plane, we can use our definitions of the trig ratios to deal with angles greater than 900. However, the sign of the ratios will change in Quadrants II, III and IV. The CAST diagram is a memory tool to help us remember which ratios are positive in the different quadrants. Special Angles We expand our table of special angles to include 00 and 900 Brought to you by Page 3 Mathematics Grade 10 www.learnxtra.co.za Brought to you by Page 4 Mathematics Grade 10 www.learnxtra.co.za X-AMPLE QUESTIONS: Question 1: P(3; 4) is a point on the Cartesian plane. XOP = θ, where X is a positive point on the x-axis. Without using a calculator, determine the value of: a.) cos θ b.) 3 tan θ c.) ½ cosec θ Question 2: XOK = θ is an angle in the third quadrant and K is the point (-5; y). OK is 13 units. Determine without using a calculator. a.) b.) The value of y Prove that tan2 θ + 1 = sec2 θ Question 3: If sin θ = 0,4 and θ is an obtuse angle, determine: a.) b.) cos θ √21 tan θ Question 4: Given tan θ =t /2 , where 0o ≤ θ ≤ 90o. Determine the following in terms of t: a.) b.) c.) d.) sec θ cot θ cos2 θ tan2 θ - sec2 θ Brought to you by Page 5