Final Exam Review s7

Final Exam Topic Guide MATH 1112A

Graphs of trigonometric functions (including amplitude, frequency/period, phase shifts) and inverse trigonometric function; Graphs of conic sections;

Recognizing equations with or without transformations: vertical shift, horizontal shifts, reflections across the x-axis, y-axis, and y=x line.

Solving trigonometric equations.

Finding exact values and coordinates using trigonometric identities.

Simplifying trigonometric expressions using identities.

DeMoivre’s Theorem

Conic Sections-recognizing type of graph with equation and total number of points of intersections

Final Exam Sample Questions

1. If sinq > 0 and secq < 0, in which quadrant does the terminal side of the angle lie? a) I b) II c) III d) IV

2. Expressed as a function of a positive acute angle, sin (-230°) is equal to: a) sin (50°) b) -sin (50°) c) cos (50°) d) -cos (50°)

3. In the accompanying diagram of a 3 1 unit circle, the ordered pair (- , - ) 2 2 represents the point where the terminal side of θ intersects the unit circle.

What is the m q ?

a) 210° b) 225° c) 233° d) 240° 4. The path traveled by a roller coaster is modeled by the equation: y=27sin(13 x ) + 30. What is the maximum altitude of the roller coaster? (hint: how is altitude is not the same as amplitude, but it is related. a) 13 b) 27 c) 30 d) 57

5. A dog has a 20-foot leash attached to the corner where a garage and a fence meet, as shown in the accompanying diagram. When the dog pulls the leas tight and walks from the fence to the garage, the arc the leash makes is 55.8 feet.

What is the m q between the garage and the fence, in radians? a) 0.36 b) 2.79 c) 3.14 d) 160

6. If θ is an angle in standard position and P(-3,4) is a point on the terminal side of θ, what is the value of sin θ?

3 3 4 4 a) b) - c) d) - 5 5 5 5

7. In physics class, Eva noticed the pattern shown in the accompanying diagram on an oscilloscope. Which equation best represents the pattern shown on the oscilloscope? (A)

1 a) y= sin( x ) + 1 2 b) y= 2sin x + 1 c) y= sin x + 1 1 d) y= 2sin( - x ) + 1 2

8. What is the radian measure of the angle formed by the hands of a clock at 2:00pm (B) p p p p a) b ) c ) d ) 2 3 4 6

5 4 9. If A and B are positive acute angles, sin A = , and cos B = , what is the value 13 5 of sin(A+ B )? (C) 16 33 56 63 a) - b ) c ) d ) 65 65 65 65

12 3 10. If sin x = , cos y = , and x and y are acute angles, what is the value of 13 5 cos(x- y ) ? (D) 33 14 21 63 a) - b ) - c ) d ) 65 65 65 65

5 11. If q is an acute angle such that sinq = , what is the value of sin 2q ? (D) 13 10 12 60 120 a) b ) c ) d ) 26 13 169 169

12. The expression (1+ cosx )(1 - cos x ) is equivalent to (B) a) 1 b ) sin2 x c ) sec 2 x d ) csc 2 x

2cosq 13. The expression is equivalent to (C) sin 2q a) sinq b ) sec q c) csc q d ) cot q

secq 14. The expression is equivalent to (C) cscq a) sinq b ) cos q c ) tan q d ) cot q

tanq 15. The expression is equivalent to (C) secq cos2 q sin q a) b ) c ) sinq d ) cos q sinq cos2 q 16. The expression cos 40° cos10 °+ sin 40 ° sin10 ° is equivalent to (A) a) cos30° b ) cos50 ° c ) sin30 ° d ) sin50 °

17. What value of x in the interval 0° x <180 ° satisfies the equation 2cos2 x+ cos x = 1? (C) a) - 30° b ) 30 ° c ) 60 ° d ) 150 °

18. The center of a circle represented by the equation (x- 2)2 + ( y + 3) 2 = 100 is located in Quadrant: (D) a) I b ) II c ) III d ) IV

19. What is the equation of a circle with center (-3,1) and radius 7? (D) a) ( x- 3)2 + ( y + 1) 2 = 7 b ) ( x - 3) 2 + ( y + 1) 2 = 49 c) ( x+ 3)2 + ( y - 1) 2 = 7 d ) ( x + 3) 2 + ( y - 1) 2 = 49

20. A circle has the equation (x+ 1)2 + ( y - 3) 2 = 16 . What are the coordinates of its center and the length of its radius? (A) a) (-1,3) and 4 b ) (1,-3) and 4 c ) (-1,3) and 16 d ) (1,-3) and 16

21. A commercial artist plans to include an ellipse in a design and wants the length of the horizontal axis to equal 10 and the length of the vertical axis to equal 6. Which equation could represent the ellipse? (A) a) 9x2 +25y 2 =225 b ) 9x 2 -25y 2 =225 c ) x 2 +y 2 =100 d ) 3y=20x 2

22. The accompanying diagram shows the construction of a model of an elliptical orbit of a planet travelling around a star. Point P and the center of the star represent the foci of the orbit.

Which equation could represent the relation shown? (B)

x2 y 2 x 2 y 2 x 2 y 2 x 2 y 2 a) + =1 b ) + =1 c ) + =1 d ) - = 1 81 225 225 81 15 9 15 9 23. Which equation, when graphed on a Cartesian coordinate plane, would best represent an elliptical racetrack? (A) a) 3x2 +10y 2 =288,000 b ) 3x 2 -10y 2 =288,000 c ) 3x+10y=288,000 d ) 30xy=288,000

24. An object orbiting a planet travels in a path represented by the equation 3(y+ 1)2 + 5( x + 4) 2 = 15 . In which type of pattern does the object travel? (C) a) parabola b ) circle c ) ellipse d ) hyperbola

25. Which equation represents a hyperbola? (D) a) y2 =16-x 2 b ) y=16-x 2 c ) y=16x 2 d ) y 2= 16 + x 2

Resource: New York Math B Regents