2.2.2: Homework

Math 373: Trigonometry for Calculus · LibreTexts

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    From LibreTexts, under not declared, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    Reading Questions

    1. What is the difference between an algebraic expression and an equation?
    2. Explain the difference between evaluating \(( \cos(30^\circ) )^2\) and \(\cos( (30^\circ)^2 )\).
    3. What does the notation \(\sin^2(\theta)\) mean?
    4. Why is the notation \(\cos^{-1}(\theta)\) not used to represent \(\frac{1}{\cos(\theta)}\)?
    5. When multiplying expressions like \(\cos(t)(3\cos(t) - 2)\), what algebraic property is used?
    6. What is the purpose of using a substitution, like letting \(x = \sin(\theta)\), when factoring a complex trigonometric expression?
    7. What is the formula for factoring a difference of squares, \(F^2 - L^2\)?
    8. Is an expression like \(\sin^2(\theta) + 9\) factorable over the real numbers? Why or why not?
    9. What is the formula for factoring a sum of cubes, \(F^3 + L^3\)?
    10. What is the formula for factoring a difference of cubes, \(F^3 - L^3\)?

    Skills Refresher

    Review the following skills you will need for this section.

    Skills Refresher

    For the following exercises, factor completely.

    1. \(2x^2 + 5x - 12\)

    2. \(3x^2 - 2x - 8\)

    3. \(12x^2 + x - 1\)

    4. \(6x^2 - 13x - 15\)

    5. \(6x^2 - x - 12\)

    6. \(24x^2 + 10x - 21\)

    7. \( 9y^2 - 16 \)

    8. \( 12m^2 - 75n^2 \)

    9. \( 81x^4 - 1 \)

    10. \( x^8 - 100y^4 \)

    11. \( x^3 + 27 \)

    12. \( x^6 - 64y^3 \)

    13. \( y^9 - 1 \)

    14. \( 16m^3 - 54y^3 \)

    Answers

    1. \((2x-3)(x+4)\)

    2. \((3x+4)(x-2)\)

    3. \((4x-1)(3x+1)\)

    4. \((6x+5)(x-3)\)

    5. \((2x-3)(3x+4)\)

    6. \((6x+7)(4x-3)\)

    7. \( \left( 3y - 4 \right)\left( 3y+4 \right) \)

    8. \( 3\left( 2m - 5n \right)\left( 2m+5n \right) \)

    9. \( \left( 3x - 1 \right)\left( 3x + 1 \right)\left( 9x^2 + 1 \right) \)

    10. \( \left( x^4 - 10y^2 \right)\left( x^4 + 10y^2 \right) \)

    11. \( \left( x + 3 \right)\left( x^2 - 3x + 9 \right) \)

    12. \( \left( x^2 - 4y \right)\left( x^4 + 4x^2 y + 16y^2 \right) \)

    13. \( \left( y-1 \right)\left( y^2 +y +1 \right)\left( y^6 + y^3 + 1 \right) \)

    14. \( 2\left( 2m - 3y \right)\left( 4m^2 + 6my + 9y^2 \right) \)

    Homework

    Concept Check

    1. To evaluate \(\cos ^2 \left(30^{\circ}\right)\), Wyatt used the keystrokes\[\mathrm{COS} \quad 30 \quad \wedge \quad 2 \quad \text{ENTER}\nonumber \]and got the answer \(-1\). Were his keystrokes correct? Why or why not?

    2. Make up an example to show that \(\tan \left(\theta+\phi\right) \neq \tan \left(\theta\right)+\tan \left(\phi\right)\).

    True or False? For the following exercises, determine if the statement is true or false. If true, cite the definition or theorem stated in the text supporting your claim. If false, explain why it is false and, if possible, correct the statement.

    1. Expressions containing trigonometric functions can be simplified or evaluated like other algebraic expressions.

    2. We can solve expressions like \( 3 \sin\left( x \right) + 2 \sin\left( x \right) \).

    3. \(\sin (X+Y) = \sin \left(X\right)+\sin \left(Y\right)\)

    4. \(\cos ^2 \left(\theta\right) = \left(\cos \left(\theta\right)\right)^2\)

    5. \(\sin^{-1} \left(\theta\right) = \dfrac{1}{\sin \left(\theta\right)}\)

    6. \( 64x^2 - 9 = \left( 8x - 3 \right)\left( 8x + 3 \right) \)

    7. \( 64x^2 + 9 = \left( 8x + 3 \right)\left( 8x + 3 \right) = \left( 8x+3 \right)^2\)

    Basic Skills

    For the following exercises, evaluate the expressions, using exact values for the trigonometric ratios.

    1. \(5 \tan 135^{\circ}+6 \cos 60^{\circ}\)

    2. \(3 \tan 240^{\circ}+8 \sin 300^{\circ}\)

    3. \(\sin \left(15^{\circ}+30^{\circ}\right)\)

    4. \(\cos \left(2 \cdot 75^{\circ}\right)\)

    5. \(8 \cos ^2\left(30^{\circ}\right)\)

    6. \(12 \sin ^2\left(315^{\circ}\right)\)

    7. \(3 \tan ^2\left(150^{\circ}\right)-\sin ^2\left(45^{\circ}\right)\)

    8. \(1+\tan ^2\left(120^{\circ}\right)\)

    For the following exercises, evaluate the expressions for \(x=30^{\circ}, y=45^{\circ}\), and \(z=60^{\circ}\). Give exact values for your answers.

    1. \(3 \sin x+5 \cos y\)

    2. \(4 \tan y+6 \cos y\)

    3. \(-2 \tan 3 y\)

    4. \(\sin (3 x-2 x)\)

    5. \(\cos ^2 x+\sin ^2 x\)

    6. \(7 \sin ^2 y+7 \cos ^2 y\)

    7. \(\cos x \cos x-\sin x \sin z\)

    8. \(\tan \left(180^{\circ}-x\right) \tan x\)

    For the following exercises, combine like terms.

    1.  

      1. \(3x^2 - x - 5x^2\)

      2. \(3 \cos ^2 \theta-\cos \theta-5 \cos ^2 \theta\)

    2.  

      1. \(4 x+5 y-y\)

      2. \(4 \cos \theta+5 \sin \theta-\sin \theta\)

    3.  

      1. \(-3 S C+7 S C\)

      2. \(-3 \sin \theta \cos \theta+7 \sin \theta \cos \theta\)

    4.  

      1. \(4 S C+11 S C-17 S C\)

      2. \(4 \sin \theta \cos \theta+11 \sin \theta \cos \theta-17 \sin \theta \cos \theta\)

    5.  

      1. \(-C^2 S^3+6 C^2 S^3\)

      2. \(-\cos ^2 \theta \sin ^3 \theta+6 \cos ^2 \theta \sin ^3 \theta\)

    6.  

      1. \(7 C^2 S^2-(2 C S)^2\)

      2. \(7 \cos ^2 \theta \sin ^2 \theta-(2 \cos \theta \sin \theta)^2\)

    For the following exercises, simplify the expression, then evaluate.

    1. \(\cos t+2 \cos t \sin t-3 \cos t\), for \(t=30^{\circ}\)

    2. \(11 \tan w-4 \tan w+6 \tan w \cos w\), for \(w=60^{\circ}\)

    3. \(7 \tan \theta-4 \tan \phi+3 \tan \phi-6 \tan \theta\), for \(\theta=45^{\circ}, \phi=120^{\circ}\)

    4. \(5 \sin A+\sin B-6 \sin B+6 \sin A\), for \(A=210^{\circ}, B=135^{\circ}\)

    5. \(-\sin x \cos x-\sin (2 x)+3 \sin x \cos x\), for \(x=45^{\circ}\)

    6. \(4 \cos u \sin u+3 \sin 2 u-10 \sin u \cos u\), for \(u=270^{\circ}\)

    For the following exercises, multiply or expand.

    1.  

      1. \(x(2 x-1)\)

      2. \(\sin A(2 \sin A-1)\)

    2.  

      1. \(q(5 r-q)\)

      2. \(\cos \theta(5 \sin \theta-\cos \theta)\)

    3.  

      1. \(a(b-3 a)\)

      2. \(\tan A(\tan B-3 \tan A)\)

    4.  

      1. \(3 w(2 w-z)\)

      2. \(3 \sin \alpha(2 \sin \alpha-\sin \beta)\)

    5.  

      1. \((C+1)(2 C-1)\)

      2. \((\cos \phi+1)(2 \cos \phi-1)\)

    6.  

      1. \((3 S-2)(S+1)\)

      2. \((3 \sin B-2)(\sin B+1)\)

    7.  

      1. \((a+b)(a-b)\)

      2. \((\cos \theta+\cos \phi)(\cos \theta-\cos \phi)\)

    8.  

      1. \((t-w)(t+4 w)\)

      2. \((\tan \alpha-\tan \beta)(\tan \alpha+4 \tan \beta)\)

    9.  

      1. \((1-T)^2\)

      2. \((1-\tan \theta)^2\)

    10.  

      1. \((2+3 S)^2\)

      2. \((2+3 \sin \theta)^2\)

    11.  

      1. \(\left(T^2+2\right)\left(T^2-2\right)\)

      2. \(\left(\tan ^2 \theta+2\right)\left(\tan ^2 \theta-2\right)\)

    12.  

      1. \(\left(2 c^2-3\right)\left(2 c^2+3\right)\)

      2. \(\left(2 \cos ^2 \phi-3\right)\left(2 \cos ^2 \phi+3\right)\)

    For the following exercises, decide whether or not the expressions are equivalent. Justify your response by citing a theorem or providing a counterexample.

    1. \(\cos (x+y), \qquad \cos x+\cos y\)

    2. \(\sin (\theta-\phi), \qquad \sin \theta-\sin \phi\)

    3. \(\tan (2 A), \qquad 2 \tan A\)

    4. \(\cos \left(\frac{1}{2} \beta\right), \qquad \frac{1}{2} \cos \beta\)

    5. \((\sin \alpha)^2, \qquad \sin ^2 \alpha\)

    6. \((\tan B)^2, \qquad \tan \left(B^2\right)\)

    7. \(\sin 3 t+\sin 5 t, \qquad \sin 8 t\)

    8. \(3 \cos 2 x+5 \cos 2 x, \qquad 8 \cos 2 x\)

    9. \(\tan \left(\theta+45^{\circ}\right)-\tan \theta, \qquad \tan 45^{\circ}\)

    10. \(\sin \left(30^{\circ}+z\right)+\sin \left(30^{\circ}-z\right), \qquad \sin 60^{\circ}\)

    1. Factor each expression, if possible.

      1. \(x^2-4 x\)

      2. \(x^2-4\)

      3. \(x^2+4\)

      4. \(x^2+4 x+4\)

      5. \(x^2-4 x+4\)

      6. \(x^2+4 x\)

      7. \(x^2-4 x-4\)

      8. \(-x^2+4\)

    For the following exercises, factor.

    1.  

      1. \(9 m+15 n\)

      2. \(9 \cos \alpha+15 \cos \beta\)

    2.  

      1. \(12 p-20 q\)

      2. \(12 \sin \theta-20 \sin \phi\)

    3.  

      1. \(5 r^2-10 q r\)

      2. \(5 \tan ^2 C-10 \tan B \tan C\)

    4.  

      1. \(2 x^2-6 x y\)

      2. \(2 \sin ^2 A -6 \sin A \cos A\)

    5.  

      1. \(9 C^2-1\)

      2. \(9 \cos ^2 \beta-1\)

    6.  

      1. \(25 S^2-16 S\)

      2. \(25 \sin ^2 \beta-16 \sin \beta\)

    7.  

      1. \(6 T^3-8 T^2\)

      2. \(6 \tan ^3 A-8 \tan ^2 A\)

    8.  

      1. \(9 T^2-15 T^3\)

      2. \(9 \tan ^2 B-15 \tan ^3 B\)

    9.  

      1. \(t^2-t-20\)

      2. \(\tan ^2 \theta-\tan \theta-20\)

    10.  

      1. \(s^2+5 s-6\)

      2. \(\sin ^2 A+5 \sin A-6\)

    11.  

      1. \(3 c^2+2 c-1\)

      2. \(3 \cos ^2 B+2 \cos B-1\)

    12.  

      1. \(8 S^2-6 S+1\)

      2. \(8 \sin ^2 \phi-6 \sin \phi+1\)

    13.  

      1. \(6 S^2-5 S-1\)

      2. \(6 \sin ^2 \alpha-5 \sin \alpha-1\)

    14.  

      1. \(T^2-4 T-12\)

      2. \(\tan ^2 \alpha-4 \tan \alpha-12\)

    15.  

      1. \( 9C^2 - 1 \)

      2. \( 9\cos^2\left( \theta \right) - 1\)

    16.  

      1. \( 16 - 9 S^2 \)

      2. \( 16 - 9 \sin^{ 2 }\left( \alpha \right) \)

    17.  

      1. \( 81T^2 - 16 C^2 \)

      2. \( 81\tan^2\left( A \right) - 16\cos^2\left( B \right) \)

    18.  

      1. \( 81T^4 - 16C^4 \)

      2. \( 81\tan^4\left( A \right) - 16\cos^4\left( B \right) \)

    19.  

      1. \( \dfrac{1}{9} S^2 - \dfrac{4}{25}C^2 \)

      2. \( \dfrac{1}{9} \sin^2\left( \phi \right) - \dfrac{4}{25}\cos^2\left( \lambda \right) \)

    20.  

      1. \( 27C^3 - 1 \)

      2. \( 27\cos^3\left( \theta \right) - 1\)

    21.  

      1. \( 64 + 27 S^3 \)

      2. \( 64 + 27 \sin^{ 3 }\left( \alpha \right) \)

    22.  

      1. \( 729T^3 + 64 C^3 \)

      2. \( 729\tan^3\left( A \right) + 64\cos^3\left( B \right) \)

    23.  

      1. \( 729T^6 - 64C^6 \)

      2. \( 729\tan^6\left( A \right) - 64\cos^6\left( B \right) \)

    24.  

      1. \( \dfrac{1}{27} S^3 - \dfrac{8}{125}C^3 \)

      2. \( \dfrac{1}{27} \sin^3\left( \phi \right) - \dfrac{8}{125}\cos^3\left( \lambda \right) \)

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