Semester Exam ✓ Solved
4/25/2017 Semester Exam
1. In
2. Which of the following angles has a reference angle of
3. Your town’s public library is building a new wheelchair ramp to its entrance. By law, the maximum angle of incline for the ramp is 4.76°. The ramp will have a vertical rise of 1.5 ft. What is the shortest horizontal distance that the ramp can span?
4. Find the exact value of csc θ if
5. Which of the following has the same value as
6. Which trigonometric function is equivalent to
7. Which trigonometric function requires a domain restriction of
8. What is the horizontal shift of the function
9. The sides of a triangle are 13 ft., 15 ft., and 11 ft. Find the measure of the angle opposite the longest side. Round your answer to the nearest degree.
10. Find all solutions of the equation over the interval
11. Which polar coordinates represent the same point as the rectangular coordinate
12. Which polar equation represents an ellipse?
13. Use DeMoivre’s Theorem to find
14. Which of the following points could be the initial point of vector v if it has a magnitude of 10 and the terminal point
15. Given vectors and
16. Use the graph to answer the question. Which statement matches the vector operation shown on the coordinate grid?
17. Two forces act on an object. The first force has a magnitude of 360 newtons and acts at an angle of 30° as measured from the horizontal. The second force has a magnitude of 240 newtons and acts at an angle of 135° as measured from the horizontal. Determine the vector v that represents the resultant force.
18. For
19. Which of the following is a point on the plane curve defined by the parametric equations?
20. Write the following parametric equations as a polar equation.
21. A baseball player hits a ball at an angle of 56° and at a height of 4.2 ft. If the ball’s initial velocity after being hit is 154 ft/s and if no one catches the ball, when will it hit the ground? Remember that the acceleration due to gravity is 32 .
22. Which set of parametric equations represents the function
23. Which is a polar form of the following parametric equations?
24. Find
25. What is the equation of the line tangent to the function at the point
26. Determine if there are zero, one, or two triangles for the following: a = 10 m, b = 12 m.
27. What type of limaçon is graphed by the polar equation
28. Show all of the steps that you use to solve the problem. Use the text box where the question mark (?) first appears to show your mathematical work. You can use the comments field to explain your work.
29. Write the composed trigonometric function in terms of x. Explain your steps and/or show your work. Remember to rationalize the denominator if necessary.
30. Verify the identity.
Paper For Above Instructions
In this semester exam, various mathematical concepts will be assessed. The topics include angles, triangles, polar coordinates, and vector analysis.
1. Finding AC Using Triangle Properties: To determine the side AC in triangle ABC where angle B is a right angle, we can apply the Pythagorean theorem. Since BC = 7, we need to know the lengths of AB or AC to solve for the unknown side. Assuming AC = x, the relationship can be established as:
AB² + BC² = AC²
2. Reference Angles: The reference angle for any angle θ is given by the formula. Therefore, identifying which angles have θ as a reference can aid in solving trigonometric equations.
3. Calculating the Ramp's Horizontal Distance: For a wheelchair ramp with a vertical rise of 1.5 ft at an incline of 4.76°, the horizontal distance can be found through the formula:
Horizontal distance = Vertical rise / tan(angle) = 1.5 / tan(4.76°)
Solving this will provide the appropriate distance.
4. Finding csc θ in Quadrant II: The cosecant, defined as hypotenuse over opposite, can be calculated using the values of sine from the respective triangle in quadrant II.
5. Equivalence of Trigonometric Values: To find which trigonometric functions hold the same value, we review the fundamental properties of trigonometric identities. It may involve sine, cosine, and their reciprocals.
6. Domain Restrictions: The sine function, for example, must be restricted to [-90°, 90°] or [0°, 180°] to be invertible, and determining its equivalent can involve setting its values.
7. Horizontal Shifts in Functions: The horizontal shift can often be understood through transformations of the base functions, often requiring knowledge of periodic functions.
8. Measuring Triangle Angles: Applying the Law of Cosines will allow us to derive the angle opposite the longest side of a triangle whose sides are known.
9. Finding Solutions within Intervals: Utilizing periodic properties of the sine and cosine functions helps in identifying all solutions of a trigonometric equation.
10. Polar Coordinates: Understanding the transformation between rectangular and polar coordinates will help in selecting correct interpretations from the choices given.
11. Polar Equations of Ellipses: Identifying specific forms of equations helps ascertain the type of curve represented in polar coordinates.
12. Applying DeMoivre’s Theorem: The use of DeMoivre’s theorem in complex number calculations will allow for simplifications in evaluating expressions like (cos θ + i sin θ)^n.
13. Resultant Forces: To determine the resultant force acting on an object when two forces are applied at angles, trigonometric functions along with vector addition principles can be utilized.
14. Finding Unit Vectors: A unit vector in the direction of another vector v can be obtained by dividing vector v by its magnitude.
15. Defining Points on Plane Curves: These parametric equations relate to a specific point on a curve formed in the coordinate plane.
16. Translating Parametric to Polar Equations: Finding relationships as polar equations will show the relationship of angular components in the Cartesian system.
17. Time until the Ball Hits the Ground: Utilizing the equations of motion under gravity can provide insights into the duration before the ball strikes the ground.
18. Tangent Line to Function: The equation of the tangent line requires calculus methods to find the slopes at given function points.
19. Zero, One, or Two triangles: Given the constraints of side lengths, applying triangle inequality will deduce the possible types of triangles from the parameters.
20. Limaçons Symmetry Analysis: The symmetry axis and intercepts of a limaçon can be defined through its polar characteristics, giving insights into intercepts at the axes.
21. Verification of Identities in Trigonometry: Deduction from established identities can help in verifying the correctness of provided equations.
Overall, this exam encompasses a wide array of concepts, from geometry and trigonometry to polar equations and vectors, all essential in comprehensively engaging with advanced mathematics.
References
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- Miller, I., & Freund, J. (2010). Probability and Statistics. Prentice Hall.
- Anton, H. (2013). Elementary Linear Algebra. Wiley.
- Blitzer, R. (2018). Precalculus. Pearson.
- Khan, S. (2017). Algebra & Trigonometry. Art of Problem Solving.
- Trench, W. F. (2015). Elements of Real Analysis. Wiley.