# User:Melikamp/alg-trig

## Real Numbers

Recall that ${\displaystyle \mathbb {N} =\{1,2,3,\ldots \}}$.

1. List 4 smallest elements of the set ${\displaystyle \{y\mid y=2x{\mbox{ and }}x\in \mathbb {N} \}}$

2, 4, 6, 8

2. List 4 smallest elements of the set ${\displaystyle \{z\mid z=|x+2|{\mbox{ and }}x{\mbox{ is a negative integer}}\}}$

0, 1, 2, 3

3. If ${\displaystyle A=\{0,1,2,3,4,5,6\}}$, ${\displaystyle B=\{1,3,5,7,9,11\}}$, and ${\displaystyle C=\{4,5,6,7,8\}}$, find
1. ${\displaystyle A\cup B}$
2. ${\displaystyle A\cap B}$
3. ${\displaystyle (A\cup B)\cap C}$
4. ${\displaystyle (A\cap B)\cup (B\cap C)}$
1. ${\displaystyle A\cup B=\{0,1,2,3,4,5,6,7,9,11\}}$
2. ${\displaystyle A\cap B=\{1,3,5\}}$
3. ${\displaystyle (A\cup B)\cap C=\{4,5,6,7\}}$
4. ${\displaystyle (A\cap B)\cup (B\cap C)=\{1,3,5,7\}}$
4. Graph and write in a set builder notation the set ${\displaystyle [0,1]\cup (2,3)}$

${\displaystyle \{x\mid 0\leq x\leq 1{\mbox{ or }}2

5. Graph and write in a set builder notation the set ${\displaystyle (-\infty ,-1]\cup (1,\infty )}$

${\displaystyle \{x\mid x\leq -1{\mbox{ or }}x>1\}}$

6. Graph and write in a set builder notation the set ${\displaystyle (-\infty ,1]\cap (-1,\infty )}$

${\displaystyle \{x\mid -1

7. Evaluate given expressions if ${\displaystyle x=1}$, ${\displaystyle y=-2}$, and ${\displaystyle z=3}$
1. ${\displaystyle 2x^{3}-y^{3}}$
2. ${\displaystyle \displaystyle {\frac {1}{x+y}}(x+z)}$
3. ${\displaystyle (5x+3y+z)^{3}}$
4. ${\displaystyle \displaystyle {\frac {x^{2}+y^{2}}{z-y}}}$
1. ${\displaystyle 10}$
2. ${\displaystyle -4}$
3. ${\displaystyle 8}$
4. ${\displaystyle 1}$

## Rational Exponents

10. Evaluate ${\displaystyle \displaystyle 9^{-2}}$

${\displaystyle 1/81}$

11. Evaluate ${\displaystyle \displaystyle {\frac {3}{2^{-3}}}}$

${\displaystyle 24}$

12. Simplify ${\displaystyle (-3ab^{3})(a^{-1}b^{-1})}$

${\displaystyle -3b^{2}}$

13. Simplify ${\displaystyle \displaystyle {\frac {10a^{3}b^{-2}}{5a^{2}b^{-3}}}}$

${\displaystyle 2ab}$

14. Simplify ${\displaystyle 16a^{3}b^{7}(2a^{-1}b^{2})^{-2}}$

${\displaystyle 4a^{5}b^{3}}$

15. Evaluate ${\displaystyle 125^{2/3}}$

${\displaystyle 25}$

16. Evaluate ${\displaystyle (27/8)^{-1/3}}$

${\displaystyle 2/3}$

17. Rationalize the denominator and simplify: ${\displaystyle \displaystyle {\frac {1-3{\sqrt {2}}}{{\sqrt {2}}+4}}}$

${\displaystyle \displaystyle {\frac {13{\sqrt {2}}-10}{14}}}$

## Polynomials

30. Rewrite the expression ${\displaystyle (x^{2}-1)(2x+3)}$ as a polynomial in the standard form and state the degree of the polynomial.

${\displaystyle 2x^{3}+3x^{2}-2x-3}$, degree 3

31. Rewrite the expression ${\displaystyle 5x^{17}(4x^{-3}-2x^{-2})}$ as a polynomial in the standard form and state the degree of the polynomial.

${\displaystyle -10x^{15}+20x^{14}}$, degree 15

32. Rewrite the expression ${\displaystyle 6(y^{3}-y)^{2}-2y^{4}(3y^{2}-6)}$ as a polynomial in the standard form and state the degree of the polynomial.

${\displaystyle 6y^{2}}$, degree 2

33. Evaluate the polynomial expression ${\displaystyle 4x^{2}+3x-2}$ at ${\displaystyle x=-5}$

${\displaystyle 113}$

## Factoring

We are only interested in factors that have integer coefficients.

40. Factor ${\displaystyle 60x^{2}y+20xy-12xy^{3}}$

${\displaystyle 4xy(15x+5-3y^{2})}$

41. Factor ${\displaystyle x^{2}+x-20}$

${\displaystyle (x+5)(x-4)}$

42. Show that the polynomial ${\displaystyle 3x^{2}+5x+3}$ is irreducible (that is, cannot be factored using real coefficients)

${\displaystyle b^{2}-4ac=5^{2}-4\cdot 3\cdot 3=-11<0}$

43. Factor ${\displaystyle 6x^{2}-5x-4}$

${\displaystyle (3x-4)(2x+1)}$

44. Factor ${\displaystyle 9x^{6}-16x^{4}}$

${\displaystyle x^{4}(3x+4)(3x-4)}$

45. Factor ${\displaystyle 9x^{2}y^{2}-12xyz+4z^{2}}$

${\displaystyle (3xy-2z)^{2}}$

46. Factor ${\displaystyle -2z^{3}+z^{2}-6z+3}$

${\displaystyle (z^{2}+3)(1-2z)}$

## Linear Equations

50. You want to install hardwood floor tile. The delivery fee is $80, and the installation costs$6 per square foot (parts and labor). How many square feet can you tile on a $1100 budget? 170 square feet 51. Your distant relative's will stipulates that her money is to be divided between you, your sibling, and your child in such a way that your sibling gets twice as much as you do, and your child gets half as much as you do. How much money will you get if the total inheritance is seven hundred thousand dollars?$200000

52. Solve the equation ${\displaystyle 3-2x=x+9}$.

${\displaystyle x=-2}$

53. Solve the equation ${\displaystyle 2(x+7)=-3(x+1-2(x-1))}$.

${\displaystyle x=23}$

54. Determine whether the equation ${\displaystyle 2(x-1)=-2(1-x)}$ is conditional, a contradiction, or an identity.

Identity

55. Solve the equation ${\displaystyle \displaystyle \left|{\frac {4-x}{2}}\right|=3}$.

${\displaystyle \{-2,10\}}$

56. Solve the equation ${\displaystyle |1-2(3-x)|=15}$.

${\displaystyle \{-5,10\}}$

## Formulas

60. Solve the formula ${\displaystyle E=mc^{2}}$ for ${\displaystyle m}$ (Mass-energy equivalence).

${\displaystyle m=Ec^{-2}}$

61. Solve the formula ${\displaystyle \displaystyle C={\frac {5}{9}}(F-32)}$ for ${\displaystyle F}$ (Fahrenheit to Celsius conversion).

${\displaystyle F={\frac {9}{5}}C+32}$

70. Solve the equation ${\displaystyle x^{2}-6x+5=0}$ by factoring.

${\displaystyle \{1,5\}}$

71. Solve the equation ${\displaystyle 3(x+1)^{2}-48=0}$ using the square root procedure.

${\displaystyle \{-5,3\}}$

72. Solve the equation ${\displaystyle x^{2}-6x=16}$ by completing the square.

${\displaystyle \{-2,8\}}$

73. Solve the equation ${\displaystyle 3x^{2}+2x=1}$ by using the quadratic formula.

${\displaystyle \left\{-1,{\frac {1}{3}}\right\}}$

## Other Types of Equations

80. Solve the equation ${\displaystyle x^{3}+3x^{2}-4x-12=0}$.

${\displaystyle \{-3,-2,2\}}$

81. Solve the equation ${\displaystyle \displaystyle {\frac {2x+1}{x+4}}+3={\frac {-2}{x+4}}}$.

${\displaystyle x=-3}$

82. Solve the equation ${\displaystyle x={\sqrt {2x+3}}}$.

${\displaystyle x=3}$

83. Solve the equation ${\displaystyle 4x^{4/5}-54=270}$.

${\displaystyle x=\pm 243}$

84. Solve the equation ${\displaystyle 6x^{6}+x^{3}-15=0}$.

${\displaystyle \left\{{\sqrt[{3}]{\frac {3}{2}}},{\sqrt[{3}]{-{\frac {5}{3}}}}\right\}}$

## Functions and Graphs

90. Find the distance between points ${\displaystyle (3,5)}$ and ${\displaystyle (-2,17)}$.

13

91. Find the midpoint of the line segment connecting points ${\displaystyle (-4,7)}$ and ${\displaystyle (2,-1)}$.

${\displaystyle (-1,3)}$

92. Find the intercepts of the graph of the equation ${\displaystyle 3y=12-4x}$.

x-intercept at ${\displaystyle (3,0)}$, y-intercept at ${\displaystyle (0,4)}$

93. Sketch the graph of the function ${\displaystyle y=x^{2}-x-2}$ and find its intercepts.

94. Find the center and the radius of the circle ${\displaystyle x^{2}+y^{2}+2x-10y=16}$ by rewriting the equation in the standard form.

Center at ${\displaystyle (-1,5)}$, radius ${\displaystyle {\sqrt {42}}}$

## Functions

100. Given that ${\displaystyle A(x)={\sqrt {x^{2}+x}}}$, find ${\displaystyle A(2)}$ and ${\displaystyle A(r+1)}$.

${\displaystyle A(2)={\sqrt {6}}}$ and ${\displaystyle A(r+1)={\sqrt {r^{2}+3r+2}}}$.

101. Find the domain of ${\displaystyle f(x)=x+{\sqrt {x-4}}}$.

${\displaystyle x\geq 4}$

102. Find the domain of ${\displaystyle \displaystyle f(x)={\frac {1}{x+1}}+{\frac {1}{x-2}}}$.

${\displaystyle x\neq -1}$ and ${\displaystyle x\neq 2}$.

103. Find the domain of ${\displaystyle \displaystyle A(x)={\frac {\sqrt {x}}{x-1}}}$.

${\displaystyle [0,1)\cup (1,\infty )}$

104. Find the zeroes of ${\displaystyle f(x)=(x-1)^{2}-4}$.

${\displaystyle \{-1,3\}}$

105. Find the zeroes of ${\displaystyle g(x)=x^{2}+1}$.

${\displaystyle \emptyset }$

106. Find the zeroes of ${\displaystyle h(r)=x-1/x}$.

${\displaystyle \{-1,1\}}$

## Linear Functions

110. Find the slope of the line that passes through the points ${\displaystyle (-1,3)}$ and ${\displaystyle (2,-4)}$.

${\displaystyle -7/3}$

111. Graph the function ${\displaystyle f(x)=-3x-2}$ by finding the slope and the ${\displaystyle y}$-intercept.

112. Graph the function given by ${\displaystyle 3y+2x=2}$ by finding the slope and the ${\displaystyle y}$-intercept.

In the next four exercises, state the answer in the form ${\displaystyle y=mx+b}$.

113. Find an equation for the line with the slope ${\displaystyle m=2}$ and containing the point ${\displaystyle (2,1)}$.

${\displaystyle y=2x-3}$

114. Find an equation for the line containing the points ${\displaystyle (-1,4)}$ and ${\displaystyle (2,1)}$.

${\displaystyle y=-x+3}$

115. Find an equation of the line parallel to the line ${\displaystyle y=-2x-1}$ and containing the point ${\displaystyle (2,2)}$.

${\displaystyle y=-2x+6}$

116. Find an equation of the line perpendicular to the line ${\displaystyle y=-2x-1}$ and containing the point ${\displaystyle (-1,-1)}$.

${\displaystyle y=x/2-1/2}$

120. Complete the square to find the standard form of the quadratic function ${\displaystyle f(x)=x^{2}+6x-1}$ and use it to sketch its graph.

121. Complete the square to find the standard form of the quadratic function ${\displaystyle f(x)=-2x^{2}-4x+5}$ and use it to sketch its graph.

122. Use the vertex formula to find the vertex of the graph of the quadratic function ${\displaystyle f(x)=x^{2}-6x}$, and write the function in the standard form.

123. Use the vertex formula to find the vertex of the graph of the quadratic function ${\displaystyle f(x)=-5x^{2}-6x+3}$, and write the function in the standard form.

124. Find the maximum or minimum value of the function ${\displaystyle f(x)=2x^{2}+6x-5}$ and state the range of the function.

125. Find the maximum or minimum value of the function ${\displaystyle f(x)=-x^{2}-6x-2}$ and state the range of the function.

## Polynomial Long Division

130. Use polynomial long division to divide ${\displaystyle P(x)=-x^{4}+2x^{3}-3x^{2}-1}$ by ${\displaystyle D(x)=x+1}$.

${\displaystyle -x^{3}+3x^{2}-6x+6+{\frac {-7}{x+1}}}$

131. Use polynomial long division to divide ${\displaystyle P(x)=2x^{5}+x^{4}+x-3}$ by ${\displaystyle D(x)=x^{2}-x+2}$.

${\displaystyle 2x^{3}+3x^{2}-x-7+{\frac {-4x+11}{x^{2}-x+2}}}$

## Polynomial Functions of Higher Degrees

140. Use the Intermediate Value Theorem to show that the polynomial function $\displaystyle p(x) = x^6 − 3x^5 + 6x^2 − 1$ has a zero in the interval ${\displaystyle [-1,1]}$.

${\displaystyle p(0)<0

141. Given that the function $\displaystyle \displaystyle f(x) = \frac{x^6 − 10}{x^5+10x^2−x}$ is continuous on the interval ${\displaystyle [1,3]}$, prove that it has a zero in that interval.

${\displaystyle f(1)<0

## Zeros of Polynomial Functions

150. Find all zeroes of a polynomial function ${\displaystyle p(x)=(x+4)^{3}(x^{2}-9)^{2}}$ and state the multiplicity of each zero.

-4 (3), -3 (2), 3 (2)

151. Use the Descarte's Rule of Signs to state the possible numbers of positive and negative zeroes of the polynomial function ${\displaystyle p(x)=x^{4}-3x^{3}+x^{2}-1}$.

3 or 1 positive zeroes, 1 negative zero.

152. Find the zeroes of the polynomial function ${\displaystyle p(x)=x^{3}-8x^{2}+8x+24}$ and state their multiplicity.

${\displaystyle 6}$ (1), ${\displaystyle 1-{\sqrt {5}}}$ (1), ${\displaystyle 1+{\sqrt {5}}}$ (1)

153. Find the zeroes of the polynomial function ${\displaystyle p(x)=2x^{4}-9x^{3}-2x^{2}+27x-12}$ and state their multiplicity.

${\displaystyle 1/2}$ (1), ${\displaystyle 4}$ (1), ${\displaystyle -{\sqrt {3}}}$ (1), ${\displaystyle {\sqrt {3}}}$ (1)

## Fundamental Theorem of Algebra

160. Find a polynomial function of the lowest degree with roots ${\displaystyle 1,\ -1,\ 2+3i,\ 2-3i}$.

${\displaystyle p(x)=x^{4}-4x^{3}+12x^{2}+4x-13}$

161. Find all zeroes of a polynomial function ${\displaystyle p(x)=4x^{4}-4x^{3}+13x^{2}-12x+3}$, given that it has a root ${\displaystyle 1/2}$ of multiplicity ${\displaystyle 2}$. State the answer by rewriting the polynomial as a product of linear factors.

${\displaystyle p(x)=(x-1/2)^{2}(x+i{\sqrt {3}})(x-i{\sqrt {3}})}$

## Inverse Functions

170. Use composition of functions to determine whether ${\displaystyle \displaystyle f(x)={\frac {x}{2}}-{\frac {3}{2}}}$ and ${\displaystyle g(x)=2x+3}$ are inverses of each other.

Yes

171. Find the inverse of the function ${\displaystyle f=\{(-2,4),(-1,1),(0,0),(1,1),(2,4)\}}$ or prove that it does not exist.

This function is not injective, since ${\displaystyle f(-1)=f(1)=1}$

172. Given ${\displaystyle \displaystyle f(x)={\frac {x}{x-2}}}$ find the inverse of ${\displaystyle f}$, and state the domains and the ranges for both ${\displaystyle f}$ and ${\displaystyle f^{-1}}$.

${\displaystyle \displaystyle f^{-1}(x)={\frac {2x}{x-1}}}$, ${\displaystyle \mathop {\mathrm {dom} } (f)=\mathop {\mathrm {ran} } (f^{-1})=\mathbb {R} -\{2\}}$, ${\displaystyle \mathop {\mathrm {ran} } (f)=\mathop {\mathrm {dom} } (f^{-1})=\mathbb {R} -\{1\}}$

173. Given ${\displaystyle \displaystyle f(x)={\sqrt {4-x}}}$ find the inverse of ${\displaystyle f}$, and state the domains and the ranges for both ${\displaystyle f}$ and ${\displaystyle f^{-1}}$.

${\displaystyle \displaystyle f^{-1}(x)=4-x^{2}}$, ${\displaystyle \mathop {\mathrm {dom} } (f)=\mathop {\mathrm {ran} } (f^{-1})=(-\infty ,4]}$, ${\displaystyle \mathop {\mathrm {ran} } (f)=\mathop {\mathrm {dom} } (f^{-1})=[0,\infty )}$

## Exponential Functions

180. Evaluate ${\displaystyle f(x)=(2/5)^{x}}$ for ${\displaystyle x=2}$ and ${\displaystyle x=-3}$.

${\displaystyle 4/25}$ and ${\displaystyle 125/8}$

181. Use a computer to estimate the value of ${\displaystyle f(x)=\pi ^{x}}$ for ${\displaystyle x={\sqrt {2}}}$ and ${\displaystyle x=\pi }$.

${\displaystyle 5.047497}$ and ${\displaystyle 36.46216}$

182. Sketch the graph of the function ${\displaystyle f(x)=3^{x}}$.

183. Sketch the graph of the function ${\displaystyle f(x)=2^{-x}+1}$.

## Logarithmic Functions

190. Rewrite the equation ${\displaystyle \log _{4}(x+1)=3y}$ in exponential form.

${\displaystyle 4^{3y}=x+1}$

191. Rewrite the equation ${\displaystyle (x-1)^{2z-3}=0.5}$

${\displaystyle \log _{(x-1)}(0.5)=2z-3}$

192. Evaluate ${\displaystyle \log _{0.3}(100/9)}$ without using a computer.

${\displaystyle -2}$

193. Find the domain of the function ${\displaystyle f(x)=\ln(x+2)}$.

${\displaystyle (-2,\infty )}$

194. Find the domain of the function ${\displaystyle f(x)=\ln(x^{2}-9)}$.

${\displaystyle |x|>3}$

## Properties of Logarithms

200. Use a computer to estimate ${\displaystyle \log _{\pi }(1.618)}$.

${\displaystyle 0.4203532}$

201. Rewrite the expression ${\displaystyle \displaystyle \log _{5}\left({\frac {z^{4}{\sqrt {x}}}{125y}}\right)}$ in a way that leaves the arguments of ${\displaystyle \log }$ function as simple as possible.

${\displaystyle \displaystyle 4\log _{5}(z)+{\frac {1}{2}}\log _{5}(x)-3-\log _{5}(y)}$

202. Rewrite the expression ${\displaystyle \displaystyle {\frac {1}{2}}\log _{3}(x)-\log _{3}(y^{2})+2\log _{3}(x+2)}$ as a single logarithm with coefficient ${\displaystyle 1}$.

${\displaystyle \displaystyle \log _{3}\left({\frac {(x+2)^{2}{\sqrt {x}}}{y}}\right)}$

## Exponetial and Logarithmic Equations

210. Solve the equation ${\displaystyle 10^{7-x}=1000}$.

${\displaystyle x=4}$

211. Solve the equation ${\displaystyle \log _{10}(x^{2}+19)=2}$.

${\displaystyle x=\pm 9}$

212. Solve the equation ${\displaystyle \log _{3}(x)+\log _{3}(x+6)=3}$.

${\displaystyle \{3,\ -9\}}$

213. Solve the equation ${\displaystyle 5^{3x}=3^{x+4}}$.

${\displaystyle \displaystyle x={\frac {4\log _{5}(3)}{3-\log _{5}(3)}}}$

## Angles and Arcs

220. Find the complement and the supplement of the angle ${\displaystyle 15^{\circ }}$.

Complement is ${\displaystyle 75^{\circ }}$ and supplement is ${\displaystyle 165^{\circ }}$

221. Find the complement and the supplement of the angle ${\displaystyle 0.7}$.

Complement is ${\displaystyle \pi /2-0.7}$ and supplement is ${\displaystyle \pi -0.7}$

222. Convert the degree measure ${\displaystyle 18^{\circ }}$ into the exact radian measure.

${\displaystyle \pi /10}$

223. Convert the radian measure ${\displaystyle \displaystyle {\frac {5\pi }{6}}}$ into the exact degree measure.

${\displaystyle 150^{\circ }}$

## Right Triangle Trigonometry

230. Find the values of ${\displaystyle \sin(\theta )}$, ${\displaystyle \cos(\theta )}$, ${\displaystyle \tan(\theta )}$, ${\displaystyle \cot(\theta )}$, ${\displaystyle \sec(\theta )}$, and ${\displaystyle \csc(\theta )}$ if ${\displaystyle \theta =\angle A}$ in the right triangle ${\displaystyle ABC}$ with ${\displaystyle \angle B=\pi /2}$ and side lengths ${\displaystyle AB=5}$ and ${\displaystyle AC=13}$.

${\displaystyle \sin(\theta )=12/13}$, ${\displaystyle \cos(\theta )=5/13}$, ${\displaystyle \tan(\theta )12/5}$, ${\displaystyle \cot(\theta )=5/12}$, ${\displaystyle \sec(\theta )=13/12}$, and ${\displaystyle \csc(\theta )=13/5}$

231. Let ${\displaystyle \theta }$ be the acute angle in a right triangle and ${\displaystyle \sin(\theta )=0.3}$. Find ${\displaystyle \cos(\theta )}$ and ${\displaystyle \tan(\theta )}$.

${\displaystyle \cos(\theta )={\sqrt {0.91}}}$ and ${\displaystyle \displaystyle \tan(\theta )={\frac {0.3}{\sqrt {0.91}}}}$

## Trigonometric Functions of any Angle

240. Find the value of ${\displaystyle \sin(x)}$, ${\displaystyle \cos(x)}$, ${\displaystyle \tan(x)}$, ${\displaystyle \csc(x)}$, ${\displaystyle \sec(x)}$, and ${\displaystyle \cot(x)}$ for the angle, in standard position, whose terminal side passes through the point ${\displaystyle (-3,-10)}$.

${\displaystyle \sin(x)=-10/{\sqrt {109}}}$, ${\displaystyle \cos(x)=-3/{\sqrt {109}}}$, ${\displaystyle \tan(x)=10/3}$, ${\displaystyle \csc(x)=-{\sqrt {109}}/10}$, ${\displaystyle \sec(x)=-{\sqrt {109}}/3}$, and ${\displaystyle \cot(x)=0.3}$

241. Find ${\displaystyle \cos(\theta )}$ if ${\displaystyle \cot(\theta )=-1}$ and ${\displaystyle \pi /2<\theta <\pi }$.

${\displaystyle -1/{\sqrt {2}}}$

## Trigonometric Functions of Real Numbers

250. Find the exact value of ${\displaystyle \cot(2\pi /3)}$.

${\displaystyle -{\sqrt {3}}}$

251. Find the exact value of ${\displaystyle \sec(-5\pi /6)}$.

${\displaystyle -0.5}$

252. Find the exact value of ${\displaystyle \tan(12\pi )}$.

${\displaystyle 0}$

253. Rewrite ${\displaystyle \cot(t)\sin(t)}$ in terms of a single trigonometric function or a constant.

${\displaystyle \cos(t)}$

254. Rewrite ${\displaystyle \displaystyle {\frac {1-\sin ^{2}(t)}{\cot ^{2}(t)}}}$ in terms of a single trigonometric function or a constant.

${\displaystyle \sin ^{2}(t)}$

255. Rewrite ${\displaystyle \displaystyle {\frac {1}{1-\sin(t)}}+{\frac {1}{1+\sin(t)}}}$ in terms of a single trigonometric function or a constant.

${\displaystyle 2\sec ^{2}(t)}$

## Graphs of Sine and Cosine

260. Find the amplitude and the period of the function ${\displaystyle y=-2\sin(x)}$. Plot one full period.

Amplitude ${\displaystyle 2}$ and period ${\displaystyle 2\pi }$

261. Find the amplitude and the period of the function ${\displaystyle y=\pi \sin(3x)}$. Plot one full period.

Amplitude ${\displaystyle \pi }$ and period ${\displaystyle 2\pi /3}$

262. Find the amplitude and the period of the function ${\displaystyle y=3\cos(x)+1}$. Plot one full period.

Amplitude ${\displaystyle 3}$ and period ${\displaystyle 2\pi }$

263. Find the amplitude and the period of the function ${\displaystyle y=2\cos(x+1)}$. Plot one full period.

Amplitude ${\displaystyle 2}$ and period ${\displaystyle 2\pi }$

## Verification of Trigonometric Identities

270. Verify the identity ${\displaystyle \tan(x)(1-\cot(x))=\tan(x)-1}$.

271. Verify the identity ${\displaystyle \sin ^{4}(x)-\cos ^{4}(x)=\sin ^{2}(x)-\cos ^{2}(x)}$.

272. Verify the identity ${\displaystyle (\tan(x)+1)^{2}=\sec ^{2}(x)+2\tan(x)}$.

273. Verify the identity ${\displaystyle \sec(x)\csc(x)=\cot(x)+\tan(x)}$.