Solutions to Exercises

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Timothy Williamson
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1 Solutions to Exercises Exercise a) Determine the values of the trigonometric functions at the angles: with radian measure π. Solution: Since π = 0 π and the coordinates of the point on the unit circle determined by 0 rad are, 0), the coordinates of the point P determined by π are, 0). Of course,this observation follows directly by simply using a drawing to locate the point on the unit circle determined by the angle π rad and then observing that the coordinates of that point P are, 0). The table below list the desired values or a dash indicates that the function is undefined at π rad. π, 0) 0 0.

2 Solutions to Exercises 7 Exercise b) Determine the values of the trigonometric functions at the angles: with radian measure π/. Solution: Since π = π + π, the coordinates of the point P determined by the angle π rad are 0, ). The table below list the desired values or a dash indicates that the function is undefined at π/ rad. π/ 0, ) 0 0.

3 Solutions to Exercises 8 Exercise c) Determine the values of the trigonometric functions at the angles: with radian measure π/. Solution: Since π = π π, the coordinates of the point P determined by the angle π rad are 0, ). The table below list the desired values or a dash indicates that the function is undefined at π/ rad. Note that this answer is exactly the same as that for the angle π rad given in the previous part of this exercise. This is because the two angles are coterminal since they differ by π. π/ 0, ) 0 0

4 Solutions to Exercises 9 Exercise d) Determine the values of the trigonometric functions at the angles: with radian measure π/4. Solution: The coordinates of the point P determined by the angle π ) 4 rad are,. The table below list the desired values. ) π/4,

5 Solutions to Exercises 0 Exercise e) Determine the values of the trigonometric functions at the angles: with radian measure π/. Solution: Since π = π π, the coordinates of the point P determined by the angle π rad are, ). The table below list the desired values. π/, )

6 Solutions to Exercises Exercise f) Determine the values of the trigonometric functions at the angles: with radian measure 5π/6. Solution: Since 5π = π π, the coordinates of the point P determined by the angle 6 ) 6 5π rad are. The table below list the desired values. 6, ), 5π 6

7 Solutions to Exercises Exercise g) Determine the values of the trigonometric functions at the angles: with radian measure π/. Solution: The table below list the desired values. ) π/,

8 Solutions to Exercises Exercise h) Determine the values of the trigonometric functions at the angles: with radian measure π/4. Solution: The table below list the desired values. ), π/4

9 Solutions to Exercises 4 Exercise. Determine the values of the trigonometric functions at the angles with radian measure a) 4π b) 5π/ c) 7π/ d) 9π/4 e) 5π/ f) π/6 g) 4π/ h) 9π/4. Solution: The following table is derived using the values in Table.,the symmetry properties of the unit circle,and the periodic behavior of the trigonometric functions. a) 4π, 0) 0 0 b) 5π/ 0, ) 0 0 c) 7π/ 0, ) 0 0 ) d) 9π/4, ) e) 5π/, ) f) π/6 g) 4π/ h) 9π/4,, ), ) Exercise

10 Solutions to Exercises 5 Exercise. Determine the values of the trigonometric functions at the angles with radian measure a) 97π b) 5π/ c) π/ d) 9π/4 e) 9π/, f) π/6 g) 4π/ h) 9π/4. Solution: The following table is derived using the values in Table.,the symmetry properties of the unit circle,and the periodic behavior of the trigonometric functions. Notes a) 97π, 0) π = 48)π + π b) 5π/ 0, ) 0 0 5π = )π + π π c) π/ 0, ) 0 0 = 4π + ) π d) 9π/4, 9π = )π + π 4 4 e) 9π/, 0) 0 0 9π =π + π π f) π/6 0, ) 0 0 =π + π 6 g) 4π/, ) 4π = )π + π h) 9π/4 ), 9π = 4π + π 4 4 Exercise

11 Solutions to Exercises 6 Exercise 4. Find the coordinates of the point P x, y) on the unit circle determined by the angle π/6 rad. Solution: Consider the triangle in the figure below formed by the points ±x, ±y) and the origin. The two sides formed by radii of the unit circle ensure that this triangle is isosceles. Hence,this is an equilateral triangle since the angle at the origin has measure π/ rad and the remaining two angles must be equal and,consequently,have radian measure π/. O N O F $ F $ N O N O Since the triangle has one side of length one,all three sides must be one unit long.

12 Solutions to Exercises 7 Since the vertical side has length one we have y = or y =. Further,since x, y) in the figure is a point on the unit circle,we know that x +y =. Substituting y = into this equation gives ) x + == x = 4. Since x must be positive it follows that We have P = x =. ),. Exercise 4

13 Solutions to Exercises 8 Exercise 5. Show that the period of the tangent and cotangent function is π. Solution: We have already seen that tan t + π) = tan t for all numbers t in the domain of the tangent function. It remains to show that π is the smallest such number. Suppose there is a number c such that tan t + c) = tan t for all t in the domain of the tangent function. Then setting t = 0 yields tan 0 = tan c suggesting that c is an integer multiple of π. It follows that π is the smallest positive number for which tan t + π) = tan t for all numbers t in its domain. Since the cot t = it follows that the cotangent function also has period π. Exercise 5 tan t

Math 144 Activity #3 Coterminal Angles and Reference Angles

144 p 1 Math 144 Activity #3 Coterminal Angles and Reference Angles For this activity we will be referring to the unit circle. Using the unit circle below, explain how you can find the sine of any given