Mathematics Trigonometry: Unit Circle

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1 a place of mind F A C U L T Y O F E D U C A T I O N Department of Curriculum and Pedagog Mathematics Trigonometr: Unit Circle Science and Mathematics Education Research Group Supported b UBC Teaching and Learning Enhancement Fund 0-0

2 The Unit Circle θ

3 The Unit Circle I P(, ) P(, ) θ A circle with radius is drawn with its center through the origin of a coordinate plane. Consider an arbitrar point P on the circle. What are the coordinates of P in terms of the angle θ? A. P(, ) Press for hint θ B. C. P(sin, P(cos, cos ) sin ) D. P(sin, cos ) E. P(cos, sin )

4 Solution Answer: C Justification: Draw a right triangle b connecting the origin to point P, and drawing a perpendicular line from P to the - ais. This triangle has side lengths,, and hpotenuse. θ P(, ) The trigonometric ratios sine and cosine for this triangle are: cos( ) cos( ) sin( ) sin( ) Therefore, the point P has the coordinates (cos θ, sin θ).

5 The Unit Circle II The line segment OP makes a 0 angle with the -ais. What are the coordinates of P? Hint: What are the lengths of the sides of the triangle? P(, ) A. B. P (, ) P (, ) C. P (, ) O 0 D. E. P ( P (,, ) )

6 Solution Answer: E Justification: The sides of the triangle give the distance from P to the -ais ( ) and the distance from P to the -ais ( ). P O 0 Therefore, the coordinates of P are: (, ) (, )

7 The Unit Circle III What are the eact values of sin(0 ) and cos(0 )? P (, ) A. B. C. sin(0) sin(0) sin(0),,, cos(0) cos(0) cos(0) D. E. sin(0), cos(0) Cannot be done without a calculator O 0

8 Solution Answer: A Justification: From question we learned that the -coordinate of P is cos(θ) and the -coordinate is sin(θ). In question, we found the and coordinates of P when θ = 0 using the triangle. Therefore, we have two equivalent epressions for the coordinates of P: P (cos0, sin 0 sin 0) (From question ) (From question ), P ( cos 0 You should now also be able to find the eact values of sin(60 ) and cos(60 ) using the triangle and the unit circle. If not, review the previous questions., )

9 The Unit Circle IV P (, ) The coordinates of point P are shown in the diagram. Determine the angle θ. A. 0 B. 45 C. 60 D. E. 75 Cannot be done without a calculator θ O

10 Solution Answer: C Justification: The coordinates of P are given, so we can draw the following triangle: The ratio between the side lengths of the triangle are the same as a triangle. This shows that θ = 60. θ

11 The Unit Circle V O θ Consider an arbitrar point P on the unit circle. The line segment OP makes an angle θ with the -ais. What is the value of tan(θ)? A. tan( ) P (, ) B. C. D. E. tan( ) tan( ) tan( ) Cannot be determined

12 Solution Answer: D Justification: Recall that: P(, ) θ cos( ), sin( ) Since the tangent ratio of a right triangle can be found b dividing the opposite side b the adjacent side, the diagram shows: tan( ) Using the formulas for and shown above, we can also define tangent as: sin( ) tan( ) cos( )

13 Summar P (, ) P (, ) The diagram shows the points on the unit circle with θ = 0, 45, and 60, as well as their coordinate values. P (, )

14 Summar The following table summarizes the results from the previous questions. sin(θ) cos(θ) tan(θ) θ = 0 θ = 45 θ = 60

15 The Unit Circle VI O P (, ) Consider the points where the unit circle intersects the positive -ais and the positive -ais. What are the coordinates of P and P? A. B. C. D. E. P (, ) P P P P (, 0), (0,), (,), (0, 0), None of P P P P the above (0,) (, 0) (0, 0) (,)

16 Solution Answer: A Justification: An point on the -ais has a -coordinate of 0. P is on the -ais as well as the unit circle which has radius, so it must have coordinates (,0). P = (0, ) Similarl, an point is on the -ais when the -coordinate is 0. P has coordinates (0, ). P = (, 0)

17 The Unit Circle VII What are values of sin(0 ) and sin(90 )? O P (0,) Hint: Recall what sinθ represents on the graph A. sin(0) 0, sin(90) 0 B. sin(0), sin(90) 0 C. sin(0) 0, sin(90) D. sin(0), sin(90) E. None of the above P (0,)

18 Solution Answer: C Justification: Point P makes a 0 angle with the -ais. The value of sin(0 ) is the -coordinate of P, so sin(0 ) = 0. P = (0, ) Point P makes a 90 angle with the -ais. The value of sin(90 ) is the -coordinate of P, so sin(90 ) =. P = (, 0) Tr finding cos(0 ) and cos(90 ).

19 The Unit Circle VIII For what values of θ is sin(θ) positive? θ A. B. C. D. E None of the above

20 Solution θ Answer: B Justification: When the coordinate of the points on the unit circle are positive (the points highlighted red) the value of sin(θ) is positive. Verif that cos(θ) is positive when θ is in quadrant I (0 <θ<90 ) and quadrant IV (70 < θ<60 ), which is when the -coordinate of the points is positive.

21 The Unit Circle IX What is the value of sin(0 )? Remember to check if sine is positive or negative. 0 A. B. C. D. E. None of the above

22 Solution Answer: B Justification: The point P is below the -ais so its -coordinate is negative, which means sin(θ) is negative. The angle between P and the -ais is 0 80 = 0. 0 sin(0) sin(0) P (, )

23 The Unit Circle X P = (, ) What is the value of tan(90 )? A B. C. D. E. None of the above

24 Solution P = (0,) 90 Answer: E Justification: Recall that the tangent of an angle is defined as: tan sin cos When θ=90, sin 90 tan90 cos90 0 Since we cannot divide b zero, tan90 is undefined.

25 The Unit Circle XI In which quadrants will tan(θ) be negative? Quadrant II Quadrant III Quadrant I Quadrant IV A. II B. III C. IV D. I and III E. II and IV

26 Solution Answer: B Quadrant II Quadrant III Quadrant I Quadrant IV Justification: In order for tan(θ) to be negative, sin(θ) and cos(θ) must have opposite signs. In the nd quadrant, sine is positive while cosine is negative. In the 4 th quadrant, sine is negative while cosine is positive. Therefore, tan(θ) is negative in the nd and 4 th quadrants.

27 Summar The following table summarizes the results from the previous questions. θ = 0 θ = 90 θ = 80 θ = 70 sin(θ) cos(θ) 0-0 tan(θ) 0 undefined 0 undefined

28 Summar Quadrant II sin(θ) > 0 cos(θ) < 0 tan(θ) < 0 Quadrant I sin(θ) > 0 cos(θ) > 0 tan(θ) > 0 sin(θ) < 0 cos(θ) < 0 tan(θ) > 0 Quadrant III sin(θ) < 0 cos(θ) > 0 tan(θ) < 0 Quadrant IV