Trigonometry Review - The Questions - 1,001 Calculus Practice Problems

1,001 Calculus Practice Problems

Part I

The Questions

Chapter 2

Trigonometry Review

In addition to having a strong algebra background, you need a strong trigonometry skill set for calculus. You want to know the graphs of the trigonometric functions and to be able to evaluate trigonometric functions quickly. Many calculus problems require one or more trigonometric identities, so make sure you have more than a few of them memorized or at least can derive them quickly.

The Problems You'll Work On

In this chapter, you solve a variety of fundamental trigonometric problems that cover topics such as the following:

· Understanding the trigonometric functions in relation to right triangles

· Finding degree and radian measure

· Finding angles on the unit circle

· Proving identities

· Finding the amplitude, period, and phase shift of a periodic function

· Working with inverse trigonometric functions

· Solving trigonometric equations with and without using inverses

What to Watch Out For

Remember the following when working on the trigonometry review questions:

· Being able to evaluate the trigonometric functions at common angles is very important since they appear often in problems. Having them memorized will be extremely useful!

· Watch out when solving equations using inverse trigonometric functions. Calculators give only a single solution to the equation, but the equation may have many more (sometimes infinitely many solutions), depending on the given interval. Thinking about solutions on the unit circle is often a good way to visualize the other solutions.

· Although you may be most familiar with using degrees to measure angles, radians are used almost exclusively in calculus, so learn to love radian measure.

· Memorizing many trigonometric identities is a good idea because they appear often in calculus problems.

Basic Trigonometry

103–104 Evaluate 9781118496718-eq02001.eps, 9781118496718-eq02002.eps, and 9781118496718-eq02003.eps for the given right triangle. Remember to rationalize denominators that contain radicals.

103. 9781118496718-un0201.tif

104. 9781118496718-un0202.tif

105–108 Evaluate the trig function. Remember to rationalize denominators that contain radicals.

105. Given 9781118496718-eq02004.eps, where 9781118496718-eq02005.eps, find 9781118496718-eq02006.eps.

106. Given 9781118496718-eq02007.eps, where 9781118496718-eq02008.eps, find 9781118496718-eq02009.eps.

107. Given 9781118496718-eq02010.eps, where 9781118496718-eq02011.eps and 9781118496718-eq02012.eps, find 9781118496718-eq02013.eps.

108. Given 9781118496718-eq02014.eps, where 9781118496718-eq02015.eps, find 9781118496718-eq02016.eps.

Converting Degree Measure to Radian Measure

109–112 Convert the given degree measure to radian measure.

109. 135 °

110. –280°

111. 36 °

112. –315°

Converting Radian Measure to Degree Measure

113–116 Convert the given radian measure to degree measure.

113. 9781118496718-eq02017.eps rad

114. 9781118496718-eq02018.eps rad

115. 9781118496718-eq02019.eps rad

116. 9781118496718-eq02020.eps rad

Finding Angles in the Coordinate Plane

117–119 Choose the angle that most closely resembles the angle in the given diagram.

117. Using the diagram, find the angle measure that most closely resembles the angle 9781118496718-eq02021.eps.

9781118496718-un0203.tif

(A) 9781118496718-eq02022.eps

(B) 9781118496718-eq02023.eps

(C) 9781118496718-eq02024.eps

(D) 9781118496718-eq02025.eps

(E) 9781118496718-eq02026.eps

118. Using the diagram, find the angle measure that most closely resembles the angle 9781118496718-eq02027.eps.

9781118496718-un0204.tif

(A) 9781118496718-eq02028.eps

(B) 9781118496718-eq02029.eps

(C) 9781118496718-eq02030.eps

(D) 9781118496718-eq02031.eps

(E) 9781118496718-eq02032.eps

119. Using the diagram, find the angle measure that most closely resembles the angle 9781118496718-eq02033.eps.

9781118496718-un0205.tif

(A) 9781118496718-eq02034.eps

(B) 9781118496718-eq02035.eps

(C) 9781118496718-eq02036.eps

(D) 9781118496718-eq02037.eps

(E) 9781118496718-eq02038.eps

Finding Common Trigonometric Values

120–124 Find 9781118496718-eq02039.eps, 9781118496718-eq02040.eps, and 9781118496718-eq02041.eps for the given angle measure. Remember to rationalize denominators that contain radicals.

120. 9781118496718-eq02042.eps

121. 9781118496718-eq02043.eps

122. 9781118496718-eq02044.eps

123. 9781118496718-eq02045.eps

124. 9781118496718-eq02046.eps

Simplifying Trigonometric Expressions

125–132 Determine which expression is equivalent to the given one.

125. 9781118496718-eq02047.eps

(A) 9781118496718-eq02048.eps

(B) 9781118496718-eq02049.eps

(C) 9781118496718-eq02050.eps

(D) 9781118496718-eq02051.eps

(E) 9781118496718-eq02052.eps

126. sec x – cos x

(A) 1

(B) sin x

(C) tan x

(D) cos x cot x

(E) sin x tan x

127. (sin x + cos x)2

(A) 2 + sin 2x

(B) 2 + cos 2x

(C) 1 + sec 2x

(D) 1 + sin 2x

(E) 1 + cos 2x

128. sin(π – x)

(A) cos x

(B) sin x

(C) csc x

(D) sec x

(E) tan x

129. sin x sin 2x + cos x cos 2x

(A) cos x

(B) sin x

(C) csc x

(D) sec x

(E) tan x

130. 9781118496718-eq02053.eps

(A) 9781118496718-eq02054.eps

(B) 9781118496718-eq02055.eps

(C) 9781118496718-eq02056.eps

(D) 9781118496718-eq02057.eps

(E) 9781118496718-eq02058.eps

131. 9781118496718-eq02059.eps

(A) csc x + cot x

(B) sec x + cot x

(C) csc x – cot x

(D) sec x – tan x

(E) csc x – tan x

132. 9781118496718-eq02060.eps

(A) 5 cos3θ – 3 cos θ

(B) 2 cos3θ – 3 cos θ

(C) 4 cos3θ – 3 cos θ

(D) 4 cos3θ + 3 cos θ

(E) 2 cos3θ + 5 cos θ

Solving Trigonometric Equations

133–144 Solve the given trigonometric equations. Find all solutions in the interval [0, 2π].

133. 2 sin x – 1 = 0

134. sin x = tan x

135. 2 cos2 x + cos x – 1 = 0

136. 9781118496718-eq02061.eps

137. 2 sin2 x – 5 sin x – 3 = 0

138. cos x = cot x

139. 9781118496718-eq02062.eps

140. sin 2x = cos x

141. 2 cos x + sin 2x = 0

142. 2 + cos 2x = –3 cos x

143. tan(3x) = –1

144. cos(2x) = cot(2x)

Amplitude, Period, Phase Shift, and Midline

145–148 Determine the amplitude, the period, the phase shift, and the midline of the function.

145. 9781118496718-eq02063.eps

146. 9781118496718-eq02064.eps

147. f (x) = 2 – 3 cos(πx – 6)

148. 9781118496718-eq02065.eps

Equations of Periodic Functions

149–154 Choose the equation that describes the given periodic function.

149. 9781118496718-un0206.tif

(A) f (x) = 2 sin(2x)

(B) f (x) = –2 sin(2x)

(C) f (x) = 2 sin(x)

(D) f (x) = 2 sin(πx)

(E) 9781118496718-eq02066.eps

150. 9781118496718-un0207.tif

(A) f (x) = 2 cos(x)

(B) f (x) = 2 cos(2x)

(C) f (x) = 2 cos(πx)

(D) f (x) = –2 cos(2x)

(E) 9781118496718-eq02067.eps

151. 9781118496718-un0208.tif

(A) f (x) = 2 cos(2x) + 1

(B) f (x) = –2 cos(2x) + 2

(C) f (x) = 2 cos(2x)

(D) 9781118496718-eq02068.eps

(E) f (x) = 2 cos(πx)

152. 9781118496718-un0209.tif

(A) f (x) = –2 cos(2x)

(B) f (x) = –2 cos(2x) + 2

(C) f (x) = 2 cos(2x) + 1

(D) f (x) = 2 cos(πx) + 1

(E) 9781118496718-eq02069.eps

153. 9781118496718-un0210.tif

(A) 9781118496718-eq02070.eps

(B) 9781118496718-eq02071.eps

(C) 9781118496718-eq02072.eps

(D) 9781118496718-eq02073.eps

(E) 9781118496718-eq02074.eps

154. 9781118496718-un0211.tif

(A) 9781118496718-eq02075.eps

(B) 9781118496718-eq02076.eps

(C) 9781118496718-eq02077.eps

(D) 9781118496718-eq02078.eps

(E) 9781118496718-eq02079.eps

Inverse Trigonometric Function Basics

155–160 Evaluate the inverse trigonometric function for the given value.

155. Find the value of 9781118496718-eq02080.eps.

156. Find the value of arctan(–1).

157. Find the value of 9781118496718-eq02081.eps.

158. Find the value of 9781118496718-eq02082.eps.

159. Find the value of 9781118496718-eq02083.eps.

160. Find the value of 9781118496718-eq02084.eps.

Solving Trigonometric Equations Using Inverses

161–166 Solve the given trigonometric equation using inverses. Find all solutions in the interval [0, 2π].

161. sin x = 0.4

162. cos x = –0.78

163. 5 sin(2x) + 1 = 4

164. 7 cos(3x) – 1 = 3

165. 2 sin2 x + 8 sin x + 5 = 0

166. 3 sec2 x + 4 tan x = 2