Math Analysis Problem Set 03-01 1. Re-derive the sines, cosines, and tangents of 30° , 45° , and 60° angles using the special right triangles that you learned so long ago. 2. Sketch 30° , 150° , 210° , and 330° angles. Explain what’s “the same” about them. 3. Sketch 45° , 405° , and 765° angles. Explain what’s “the same” about them. 4. Given sin ( 2.22 ) ≈ 0.797 . a. Verify that on your calculator. b. Without a calculator, in what quadrant does 2.22 radians fall? c. Do your best to sketch 2.22 radians. d. What is the smallest positive radian measure for which sin ( x ) ≈ 0.797 is true? e. How about the largest negative radian measure for which it is true? 5. Without a calculator—using what you know of the unit circle—solve, over the real numbers, the inequality sin ( x ) ≥ 0.5 . 6. Order these by size, smallest to largest: cos (15° ) , cos ( 85° ) , cos ( 90° ) , cos (133° ) , cos (180° ) , cos ( 220° ) , cos ( 277° ) , cos ( 320° ) , cos ( 359° ) . This is a non-calculator question; obviously you should check your answer on your calculator.
MA Notes 03 Problem Sets
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Math Analysis Problem Set 03-02 1. Find the exact and approximate area of the figure.
2. Is it possible for cos (φ ) = −
2 1 and sin (φ ) = for an angle φ ? Explain. 3 3
1 and cot ( x ) = − 15 . 4 a. In what quadrant does the angle x terminate? b. Find the other trig functions of x. c. Find the exact and approximate values of angle x.
3. Given that sin ( x ) = −
4. Compare the values by putting >, <, or = between them. These are not calculator problems. Use your understanding of the Unit Circle, among other things. sin ( 85° ) cos ( 352° )
sin ( 232° )
sin ( 311° )
sec ( 87° )
csc (175° )
cos (113° )
MA Notes 03 Problem Sets
sin ( 205° )
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Math Analysis Problem Set 03-04 1. Find the indicated trig function(s) for the given angle in standard position passing through the given point. If possible find an easier point to use. a. For the angle through the point ( −12,20 ) , find csc (θ ) and cot (θ ) .
(
)
b. For the angle through the point 3 2,− 4 , find cos (θ ) and tan (θ ) . c. For the angle through the point ( −6, 3) , find sin (θ ) and sec (θ ) . 2. Find the angle of inclination of the line passing through the points ( 6, 3) and ( −5,9 ) . 3. Write each of the following in the indicated way. Show intermediate steps. a. cos ( 9,954° ) in terms of the cosine of a QI angle. b. sin ( 6,532° ) in terms of the sine of a QI angle. c. cos ( 2,253° ) in terms of the sine of a QII angle. d. sin ( 5,543° ) in terms of the sine of a QIII angle. e. sec (193, 453° ) in terms of the cosecant of a QIV angle. f. csc ( 998,877° ) in terms of the secant of a QIV angle. g. tan ( 632° ) in terms of the tangent of a QII angle. h. tan ( 4, 775° ) in terms of the cotangent of a QII angle. 4. Find all values of x satisfying the given information. b. tan ( x ) = − 3 3 a. sin ( x ) = − 2 c. csc ( x ) = 2 d. sec ( x ) = 1 5. Write the equation of the circle in standard form. Identify the center and radius. a. x 2 + y 2 + 8x − 4y = 3 b. x 2 + y 2 − 9x + 4y = 0
MA Notes 03 Problem Sets
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Math Analysis Problem Set 03-05 1. Write sin ( 554,231,807° ) in terms of the cosine of a QIII angle. 2. Find all six trig functions at the angle in standard position passing through the point ( 280,−525 ) . State where the angle intersects the unit circle. 3. Find the exact length of CD and the exact value of angle BDC.
4. Given that cot ( x ) =
15 and that x ∈QIII , find the value of 4
a. sin 2 ( x ) + cos 2 ( x ) b. 1+ tan 2 ( x ) c. sec 2 ( x )
5. Find all x such that tan ( x ) = −
3 . 3
6. Find all x such that sin ( x ) = −
3 . 2
1 ( x − 1) intersects the Unit Circle. 2 a. Find the intersection point(s). b. Find the exact value of the angle(s) in standard position that pass through the intersection point(s). c. If there is more than one intersection point, find the area of the triangle with vertices at the points of intersection and the origin.
7. The line y =
8. Find the length of a tunnel dug between two cities on the same meridian on Earth, radius 6,378 km, whose latitudes are 15°25' 32'' and 17°24 '18'' .
MA Notes 03 Problem Sets
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Math Analysis Problem Set 03-06 1. Use the rectangle below to find the sine and cosine of 15° and 75° .
2. From memory, rapidly and completely fill in a Unit Circle. 3. Write cos ( 5, 322,543° ) in terms of the: a. Sine of a QI angle c. Sine of a QIV angle
b. Cosine of a QIII angle d. Cosine of a QII angle
4. Based solely on the Unit Circle, state the intervals on which the function f ( x ) = sin ( x ) is an increasing function. 5. Using a Unit Circle and a calculator, state all angles such that csc (θ ) = 2.583 .
MA Notes 03 Problem Sets
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