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Pearson Malaysia Sdn Bhd Form 4 Chapter 9: Trigonometry II

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Pearson Malaysia Sdn Bhd Quadrants and angles in the unit circle 0°0° 90° 180° 270° 360° Quadrant I Quadrant II Quadrant III Quadrant IV Cartesian plane can be divided into four parts called quadrants. Quadrants are named in the anticlockwise direction. y x

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Pearson Malaysia Sdn Bhd Quadrants and angles in the unit circle O Angle is measured by rotating the line OP in the anticlockwise direction from the positive x -axis at the origin, O. y x P

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Pearson Malaysia Sdn Bhd Verify sin = y -coordinate in quadrant I of the unit circle O sin = = = y y x P ( x, y ) 1 y x Q sin = y -coordinate

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Pearson Malaysia Sdn Bhd Verify cos = x -coordinate in quadrant I of the unit circle O cos = = = x P y x ( x, y ) 1 y x Q cos = x -coordinate

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Pearson Malaysia Sdn Bhd Verify tan = in quadrant I of the unit circle O tan = = P y x ( x, y ) 1 y x Q tan =

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Pearson Malaysia Sdn Bhd y x III III IV A ll s in t an c os Determine whether the value is positive or negative Quadrant I = All positive Quadrant II = sin positive Quadrant III = tan positive Quadrant IV = cos positive

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Pearson Malaysia Sdn Bhd Example 1: sin 213° y x Determine whether the value is positive or negative The angle 213° lies in quadrant III. Therefore, the value of sin 213° is negative. 213° O Sin is positive in quadrant II. Not quadrant II

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Pearson Malaysia Sdn Bhd Example 2: cos 321° y x Determine whether the value is positive or negative The angle 321° lies in quadrant IV. Therefore, the value of cos 321° is positive. 321° O Cos is positive in quadrant IV. It is quadrant IV.

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Pearson Malaysia Sdn Bhd Example 3: tan 123° y x Determine whether the value is positive or negative The angle 123° lies in quadrant II. Therefore, the value of tan 123° is negative. 123° O Tan is positive in quadrant III. Not quadrant III

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Pearson Malaysia Sdn Bhd Example 4: sin 32° y x Determine whether the value is positive or negative The angle 32° lies in quadrant I. Therefore, the value of sin 32° is positive. 32° O It is quadrant I. All positive in quadrant I.

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Pearson Malaysia Sdn Bhd Determine the values of sine, cosine and tangent for special angles 45° 1 1 sin 45° =cos 45° = tan 45° = 1

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Pearson Malaysia Sdn Bhd Determine the values of sine, cosine and tangent for special angles 30° 60° 1 2 sin 30° =cos 30° =tan 30° =

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Pearson Malaysia Sdn Bhd Determine the values of sine, cosine and tangent for special angles 30° 60° 1 2 cos 60° =sin 60° =tan 60° =

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Pearson Malaysia Sdn Bhd Determine the values of sine, cosine and tangent for special angles y x O (1, 0) (0, 1) (–1, 0) (0, –1) 0°0°90°180°270°360° sin 010–10 cos 10–101 tan 000

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Pearson Malaysia Sdn Bhd Summary: Determine the values of sine, cosine and tangent for special angles 0°0°30°45°60°90°180°270°360° sin 010–10 cos 10–101 tan 0100

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Pearson Malaysia Sdn Bhd Determine the values of sine, cosine and tangent for special angles Question 1: Calculate the values of the following: 7 sin 90° + 4 cos 180 ° Solution: 7 sin 90° + 4 cos 180 ° = 7 × (1) + 4 × (–1) = 7 – 4 = 3

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Pearson Malaysia Sdn Bhd Values of angles in quadrant II y x O between x -axis and line O P = corresponding angle in quadrant I P

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Pearson Malaysia Sdn Bhd Values of angles in quadrant II y x O P where = 180° – sin = + sin cos = – cos tan = – tan = – 90° X

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Pearson Malaysia Sdn Bhd Values of angles in quadrant III y x O P where = – 180° sin = – sin cos = – cos tan = + tan = 270° – X

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Pearson Malaysia Sdn Bhd Values of angles in quadrant IV y x O P where = 360° – sin = – sin cos = + cos tan = – tan = – 270° X

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Pearson Malaysia Sdn Bhd Solution: = – sin 51° Finding the value of an angle Question 1: Find the value of sin 231°. 231° y x O P quadrant III sin 231° negative sin 231° = – sin (231° – 180°) = – 0.7771

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Pearson Malaysia Sdn Bhd Solution: = cos 56° 43' Finding the value of an angle Question 2: Find the value of cos 303° 17‘. 303° 17' quadrant IV cos 303° 17' positive cos 303° 17' = cos (360° – 303° 17') = 0.5488 303° 17' x y O P

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Pearson Malaysia Sdn Bhd Solution: = – tan 62° 47' Finding the value of an angle 117° 13' quadrant II tan 117° 13' negative tan 117° 13' = – tan (180° – 117° 13') = – 1.945 Question 3: Find the value of tan 117° 13'. 117° 13' x y O P

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Pearson Malaysia Sdn Bhd Solution: Finding angles between 0° and 360° 0.9511 positive Therefore, the acute angle is in quadrant I or II. Question 1: For sin x = 0.9511 where 0° ≤ x ≤ 360°, find the value of x. x O 72° P Quadrant I: x = Quadrant II: x = 72° x P x y and x y Corresponding acute angle, x = 72° 72° 180° – 72° = 108°

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Pearson Malaysia Sdn Bhd Solution: Finding angles between 0° and 360° – 1.746 negative Therefore, the acute angle is in quadrant II or IV. Question 2: For tan x = – 1.746 where 0° ≤ x ≤ 360°, find the value of x. 60° 12' O P x x y Quadrant IV: x = Quadrant II: x = 60° 12' P x and x y Corresponding acute angle, x = 60° 12' 180° – 60° 12' = 119° 48' 360° – 60° 12' = 299° 48'

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Pearson Malaysia Sdn Bhd Solution: Finding angles between 0° and 360° 0.5 positive Therefore, the acute angle is in quadrant I or IV. Question 3: For cos x = 0.5 where 0° ≤ x ≤ 360°, find the value of x. 60° O P x x y Quadrant IV: x = Quadrant I: x = 60° P x and x y Corresponding acute angle, x = 60° 60° 360° – 60° = 300°

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Pearson Malaysia Sdn Bhd Solve problems involving sine, cosine and tangent Question:In the diagram below, HMS and JHN are straight lines. H is the midpoint of JN. Given that HM = 12 cm, MN = 13 cm and FJ = 4 cm, calculate: (a)the length of HN, (b)the value of cos x °, (c)the value of tan y °. N HM S JF y°y° x°x°

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Pearson Malaysia Sdn Bhd Solution: Solve problems involving sine, cosine and tangent N HM S JF y°y° x°x° HN 2 = (a) = 169 – 144 = 25 HN = 5 cm Pythagoras’ theorem 13 2 – 12 2 12 cm 13 cm 4 cm

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Pearson Malaysia Sdn Bhd Solve problems involving sine, cosine and tangent N HM S JF y°y° x°x° Solution: x° x° = 180° – HMN (b) cos x ° = HMS is a straight line = – cos HMN

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Pearson Malaysia Sdn Bhd Solve problems involving sine, cosine and tangent N HM S JF y°y° x°x° Solution: y° y° = 180° – FHJ (c) tan y ° = JHN is a straight line = – tan FHJ

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Pearson Malaysia Sdn Bhd

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