Chapter Projects 597 1. Find the real solutions, if any, of the equation + = x x 3 1 4 . 2 2. Find an equation for the circle with center at the point ( ) −5, 1 and radius 3. Graph this circle. 3. Find the domain of the function ( ) = − − f x x x3 4 2 4. Graph the function: π( ) = y x 3sin 5. Graph the function: π ( ) = − − y x 2 cos 2 6. If θ = − tan 2 and π θ π < < 3 2 2 , find the exact value of: (a) θ sin (b) θ cos (c) θ ( ) sin 2 (d) θ ( ) cos 2 (e) θ ( ) sin 1 2 (f) θ ( ) cos 1 2 7. Graph each of the following functions on the interval [ ] 0, 4 : (a) = y ex (b) = y x sin (c) = y e x sin x (d) = + y x x 2 sin 8. Graph each the following functions: (a) = y x (b) = y x2 (c) = y x (d) = y x3 (e) = y ex (f) = y x ln (g) = y x sin (h) = y x cos (i) = y x tan 9. Solve the triangle for which side a is 20, side c is 15, and angle C is ° 40 . 10. In the complex number system, solve the equation − + − + − = x x x x x 3 10 21 42 36 8 0 5 4 3 2 11. Graph the rational function ( ) = − − + − R x x x x x 2 7 4 2 15 2 2 12. Solve = 3 12. x Round your answer to two decimal places. 13. Solve: ( ) + + = x x log 8 log 2 3 3 14. Suppose that ( ) = + f x x4 5 and ( ) = + − g x x x5 24. 2 (a) Solve ( ) = f x 0. (b) Solve ( ) = f x 13. (c) Solve ( ) ( ) = f x g x . (d) Solve ( ) > f x 0. (e) Solve ( ) ≤ g x 0. (f) Graph ( ) = y f x . (g) Graph ( ) = y g x . Cumulative Review O A P B b a c Q C 2. Use the Law of Cosines with triangles OPQ and CPQ to find two expressions for the length of PQ. 3. Subtract the expressions in part (2) from each other. Solve for the term containing cos c. 4. Use the Pythagorean Theorem to find another value for − OQ CQ 2 2 and − OP CP . 2 2 Now solve for cos c. Chapter Project I. Spherical Trigonometry When the distance between two locations on the surface of Earth is small, we can treat Earth as a plane and compute the distance in statutory miles. Using this assumption, we can use the Law of Sines and the Law of Cosines to approximate distances and angles. However, the Earth is a sphere, so as the distance between two points on its surface increases, the linear distance is less accurate. Under this circumstance, we need to take into account the curvature of Earth when using the Law of Sines and the Law of Cosines. 1. See the figure. The points A, B, and C are the vertices of a spherical triangle with sides a, b, and c, a three-sided figure drawn on the surface of a sphere with center at the point O. Connect each vertex by a radius to the center O of the sphere. Now draw tangent lines to the sides a and b of the triangle that go through C. Extend the lines OA and OB to intersect the tangent lines at P and Q, respectively. List the plane right triangles. Find the measures of the central angles. Credit: Michael Warwick/ Shutterstock
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