Grade 12 Mathematics 1 1 Trigonometric Identities Mathematics

Grade 12 Mathematics (1 -1) Trigonometric Identities ﺍﻟﻤﺘﻄﺎﺑﻘﺎﺕ ﺍﻟﻤﺜﻠﺜﻴﺔ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 4. 1 Right-angled triangles A right-angled triangle contains a right angle. The longest side opposite the right angle is called the hypotenuse. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 4. 1 The opposite and adjacent sides The two shorter sides of a right-angled triangle are named with respect to one of the acute angles. The side opposite the marked angle is called the opposite side. The side between the marked angle and the right angle is called the adjacent side. x ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Similar right-angled triangles PA 4. 1 If two right-angled triangles have an acute angle of the same size they must be similar. For example, two triangles with an acute angle of 37° are similar. 3 cm 5 cm 37° 4 cm 10 cm 6 cm 37° 8 cm The ratio of the side lengths in each triangle is the same. opp 3 6 = = adj 4 8 opp 3 6 = = hyp 5 10 adj 4 8 = = hyp 5 10 ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 4. 1 Similar right-angled triangles ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

The sine ratio PA 4. 1 the length of the opposite side is the sine ratio. the length of the hypotenuse The ratio of The value of the sine ratio depends on the size of the angles in the triangle. O P P O S I T E H Y P O T E N U S θ E opposite sin θ = hypotenuse ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

The cosine ratio PA 4. 1 the length of the adjacent side The ratio of is the cosine ratio. the length of the hypotenuse The value of the cosine ratio depends on the size of the angles in the triangle. H Y P O T E N U S E adjacent cos θ = hypotenuse θ ADJACENT ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

The tangent ratio PA 4. 1 the length of the opposite side The ratio of is the tangent ratio. the length of the adjacent side The value of the tangent ratio depends on the size of the angles in the triangle. O P P O S I T E tan θ = opposite adjacent θ ADJACENT ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 4. 1 O P P O S I T E H Y P O T E N U S E The three trigonometric ratios Opposite Sin θ = Hypotenuse SOH Adjacent Cos θ = Hypotenuse CAH Opposite Tan θ = Adjacent TOA θ ADJACENT ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 4. 1 Finding angles 8 cm Find θ to 2 decimal places. 5 cm θ We are given the lengths of the sides opposite and adjacent to the angle, so we use: opposite tan θ = adjacent 8 tan θ = 5 θ = tan– 1 (8 ÷ 5) = 57. 99° (to 2 d. p. ) ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Finding side lengths PA 4. 1 If we are given one side and one acute angle in a rightangled triangle, we can use one of the three trigonometric ratios to find the lengths of other sides. Find x to 2 decimal places. To find the length of the side opposite the angle, given the hypotenuse, use: 12 cm 56° x opposite sin θ = hypotenuse x sin 56° = 12 x = 12 × sin 56° = 9. 95 cm ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

Finding side lengths PA 4. 1 A 5 m ladder is resting against a wall. It makes an angle of 70° with the ground. What is the distance between the base of the ladder and the wall? To find the length of the side adjacent to the angle, given the hypotenuse, use: 5 m 70° x adjacent cos θ = hypotenuse x cos 70° = 5 x = 5 × cos 70° = 1. 71 m (to 2 d. p. ) ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 4. 1 Finding angles ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

The sine, cosine and tangent of any angle PA 4. 1 These definitions are limited because a right-angled triangle cannot contain any angles greater than 90°. To extend the three trigonometric ratios to include angles greater than 90° and less than 0° we consider the rotation of a straight line OP of fixed length r about the origin O of a coordinate grid. Angles are then measured anticlockwise from the positive x-axis. y P(x, y) r θ α O x For any angle θ there is an associated acute angle α between the line OP and the x-axis. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 4. 1 The sine, cosine and tangent of any angle The three trigonometric ratios are then given by: The x and y coordinates can be positive or negative, while r is always positive. This means that the sign of the required ratio will depend on the sign of the x-coordinate and the ycoordinate of the point P. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 4. 1 The sine, cosine and tangent of any angle If we take r to be 1 unit long then these ratios can be written as: The relationship between θ measured from the positive xaxis and the associated acute angle α depends on the quadrant that θ falls into. For example, if θ is between 90° and 180° it will fall into the second quadrant and α will be equal to (180 – θ)°. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 4. 1 The sine of any angle If the point P is taken to revolve about a unit circle then sin θ is given by the y-coordinate of the point P. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 4. 1 The cosine of any angle If the point P is taken to revolve about a unit circle then cos θ is given by the x-coordinate of the point P. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 4. 1 The tangent of any angle Tan θ is given by the y-coordinate of the point P divided by the x-coordinate. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 4. 1 The tangent of any angle Tan θ can also be given by the length of tangent from the point P to the x-axis. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 4. 1 Remember CAST We can use CAST to remember in which quadrant each of the three ratios are positive. 2 nd quadrant 1 st quadrant S Sine is positive A All are positive 3 rd quadrant 4 th quadrant T Tangent is positive C Cosine is positive ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 4. 1 The sine, cosine and tangent of any angle The sin, cos and tan of angles in the first quadrant are positive. In the second quadrant: sin θ = sin α cos θ = –cos α tan θ = –tan α In the third quadrant: sin θ = –sin α cos θ = –cos α tan θ = tan α In the fourth quadrant: sin θ = –sin α cos θ = cos α tan θ = –tan α where α is the associated acute angle. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 4. 1 The sine, cosine and tangent of any angle The value of the associated acute angle α can be found using a sketch of the four quadrants. For angles between 0° and 360° it is worth remembering that: when 0° < θ < 90°, α=θ when 90° < θ < 180°, α = 180° – θ when 180° < θ < 270°, α = θ – 180° when 270° < θ < 360°, α = 360° – θ For example, if θ = 230° we have: α = 230° – 180° = 50° 230° is in the third quadrant where only tan is positive and so: sin 230° = –sin 50° cos 230° = –cos 50° tan 230° = tan 50° ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 4. 1 The graphs of sin θ, cos θ and tan θ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. 25 Mathematicsof 47 Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation © Boardworks Ltd 2005

PA 4. 1 The graph of y = sin θ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 4. 1 The graph of y = sin θ The graph of sine is said to be periodic since it repeats itself every 360°. We can say that the period of the graph y = sin θ is 360°. Other important features of the graph y = sin θ include the fact that: It passes through the origin, since sin 0° = 0. The maximum value is 1 and the minimum value is – 1. Therefore, the amplitude of y = sin θ is 1. It has rotational symmetry about the origin. In other words, it is an odd function and so sin (–θ) = –sin θ. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 4. 1 The graph of y = cos θ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 4. 1 The graph of y = cos θ Like the graph of y = sin θ the graph of y = cos θ is periodic since it repeats itself every 360°. We can say that the period of the graph y = cos θ is 360°. Other important features of the graph y = cos θ include the fact that: It passes through the point (0, 1), since cos 0° = 1. The maximum value is 1 and the minimum value is – 1. Therefore, the amplitude of y = cos θ is 1. It is symmetrical about the y-axis. In other words, it is an even function and so cos (–θ) = cos θ. It the same as the graph of y = sin θ translated left 90°. In other words, cos θ = sin (90° – θ). ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 4. 1 The graph of y = tan θ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 4. 1 The graph of y = tan θ You have seen that the graph of y = tan θ has a different shape to the graphs of y = sin θ and y = cos θ. Important features of the graph y = tan θ include the fact that: It is periodic with a period of 180°. It passes through the point (0, 0), since tan 0° = 0. Its amplitude is not defined, since it ranges from +∞ to –∞. It is symmetrical about the origin. In other words, it is an odd function and so tan (–θ) = –tan θ is not defined for θ = ± 90°, ± 270°, ± 450°, … that is, for odd multiples of 90°. The graph y = tan θ therefore contains asymptotes at these points. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 4. 1 The reciprocal trigonometric functions ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. 32 Mathematicsof 35 Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation © Boardworks Ltd 2006

PA 4. 1 The reciprocal trigonometric functions are cosecant, secant and cotangent. They are related to the three main trigonometric ratios as follows: cosec x = This is short for cosecant. sec x = This is short for secant. cot x = This is short for cotangent. Notice that the first letter of sin, cos and tan happens to be the same as the third letter of the corresponding reciprocal functions cosec, sec and cot. ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 4. 1 The graph of sec x ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 4. 1 The graph of cosec x ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 4. 1 The graph of cot x ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation

PA 4. 1 Problems involving reciprocal trig functions Given that x is an acute angle and tan x = exact values of cot x, sec x and cosec x. find the Using the following right-angled triangle: 5 The length of the hypotenuse is 3 x 5 4 So tan x = cos x = sin x = sec x = cosec x = Therefore cot x = ﺍﻟﺮﻳﺎﺿﻴﺎﺕ – ﺗﻔﻜﻴﺮ ﺳﻠﻴﻢ – ﺩﻗﺔ ﻭﺗﻌﺎﻭﻥ – ﺻﺒﺮ ﻭﻧﻈﺎﻡ – ﺗﺬﻭﻕ ﺍﻟﺠﻤﺎﻝ ﺍﻟﻌﻠﻤﻲ. Mathematics- Proper Thinking- Accuracy and Cooperation- Patience and Discipline- Science Beauty sensation