2.3.2: 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 a conditional equation and an identity?
    2. To prove that an equation is an identity, is it enough to show that it is true for several different values of the variable? Why or why not?
    3. State the Ratio Identity for \(\tan(\theta)\).
    4. For which angles is the identity for \(\tan(\theta)\) not defined?
    5. State the three Reciprocal Identities that relate the six trigonometric functions.
    6. What is the Fundamental Pythagorean Identity?
    7. If you know that \(\sin(\theta) = \frac{7}{25}\), why is it also necessary to know the quadrant of \(\theta\) to find the exact value of \(\cos(\theta)\)?
    8. What are the two Pythagorean Identities that involve the tangent, cotangent, secant, and cosecant functions?
    9. Are \(\csc(\theta)\) and \(\sin(\theta)\) multiplicative inverses or functional inverses?
    10. How can you use the identity \(\cos^2(\theta) + \sin^2(\theta) = 1\) to derive the identity involving \(\tan^2(\theta)\) and \(\sec^2(\theta)\)?

    Skills Refresher

    Review the following skills you will need for this section.

    Skills Refresher

    For the following exercises, write the expression as a single fraction in simplest form.

    1. \(\dfrac{1}{x} + \dfrac{1}{y}\)

    2. \(1 - \dfrac{1}{x}\)

    3. \(\dfrac{1}{x-1} + \dfrac{1}{x+1}\)

    4. \(x - \dfrac{x}{x+1}\)

    5. \(\dfrac{\frac{y}{x}-\frac{x}{y}}{\frac{y}{x} + 1}\)

    6. \(\dfrac{1-\frac{x^2}{y^2}}{1+\frac{x^2}{y^2}}\)

    7. \(\dfrac{\frac{2a}{x}}{1-\frac{a^2}{x^2}}\)

    8. \(\dfrac{\frac{a}{x}+\frac{b}{y}}{1-\frac{ab}{xy}}\)

    Answers

    1. \(\dfrac{x+y}{xy}\)

    2. \(\dfrac{x-1}{x}\)

    3. \(\dfrac{2x}{x^2-1}\)

    4. \(\dfrac{x^2}{x+1}\)

    5. \(\dfrac{y-x}{y}\)

    6. \(\dfrac{y^2-x^2}{y^2+x^2}\)

    7. \(\dfrac{2ax}{x^2-a^2}\)

    8. \(\dfrac{ay+bx}{xy-ab}\)

    Homework

    Vocabulary Check

    1. A ___ equation is true only for certain variable values and false for others.

    2. An equation that is true for all legitimate values of the variables is called a(n) ___.

    3. The expressions on either side of the equal sign in an identity are called ___ because they have the same value for all values of the variable.

    Concept Check

    1. What is the difference between a conditional equation and an identity? Give an example of each.

    2. What happens when you try to "solve" an identity?

    3. Nhat claims that since \(\cos^2 \left(\theta\right) + \sin^2 \left(\theta\right) = 1\), we can take the square root of both sides to get the simpler form \(\cos \left(\theta\right) + \sin \left(\theta\right) = 1\). Is she correct? Why or why not?

    4. If \(\frac{a}{b} = \frac{2}{5}\), is it necessarily true that \(a = 2\) and \(b = 5\)? Explain.

    5. If \(\tan \left(\theta\right) = \frac{3}{11}\), is it true that \(\sin \left(\theta\right) = 3\) and \(\cos \left(\theta\right) = 11\)? Explain.

    6. Explain how you would arrive at the identity\[ 1 + \cot^2\left( \theta \right) = \csc^2\left( \theta \right) \nonumber \]given that you know \( \cos^2\left( \theta \right) + \sin^2\left( \theta \right) = 1. \)

    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. The reciprocal of the secant is the cosecant.

    2. For any angle \(\theta\),\[\cos ^2 \left(\theta\right)+\sin ^2 \left(\theta\right)=1.\nonumber \]

    3. For any angle \(\theta\),\[\cos \left(\theta\right)+\sin \left(\theta\right)=1.\nonumber \]

    4. For any angle \( \theta \),\[\tan \left(\theta\right)=\dfrac{\sin \left(\theta\right)}{\cos \left(\theta\right)}.\nonumber \]

    5. To show that an equation is an identity, we just need to find one value of the unknown that makes the statement true.

    Basic Skills

    For the following exercises, decide which of the following equations are identities. Explain your reasoning.

    1. \((\sqrt{a}+\sqrt{b})^2=a+b\)

    2. \(\sqrt{a^2-b^2}=a-b\)

    3. \(\dfrac{1}{a+b}=\dfrac{1}{a}+\dfrac{1}{b}\)

    4. \(\dfrac{a+b}{a}=b\)

    5. \(\tan (\alpha+\beta)=\dfrac{\sin (\alpha+\beta)}{\cos (\alpha+\beta)}\)

    6. \(\dfrac{1}{\tan \theta}=\dfrac{\cos \theta}{\sin \theta}\)

    7. \((1+\tan \theta)^2=1+\tan ^2 \theta\)

    8. \(\sqrt{1-\sin ^2 \phi}=1-\sin \phi\)

    For the following exercises, evaluate (without using a calculator).

    1. \(3 \cos ^2 1.7^{\circ}+3 \sin ^2 1.7^{\circ}\)

    2. \(4-\cos ^2 338^{\circ}-\sin ^2 338^{\circ}\)

    3. \(\left(\cos ^2 20^{\circ}+\sin ^2 20^{\circ}\right)^4\)

    4. \(\dfrac{18}{\cos ^2 17^{\circ}+\sin ^2 17^{\circ}}\)

    5. \(\dfrac{6}{\cos ^2 53^{\circ}}-6 \tan ^2 53^{\circ}\)

    6. \(\dfrac{1}{\sin ^2 102^{\circ}}-\dfrac{\cos ^2 102^{\circ}}{\sin ^2 102^{\circ}}\)

    For the following exercises, find exact values for the other trigonometric functions of the given angle.

    1. \(\cos A=\dfrac{12}{13} \quad\) and \(270^{\circ}<A<360^{\circ}\)

    2. \(\sin B=\dfrac{-3}{5} \quad\) and \(180^{\circ}<B<270^{\circ}\)

    3. \(\sin \phi=\dfrac{1}{7} \quad\) and \(90^{\circ}<\phi<180^{\circ}\)

    4. \(\cos t=\dfrac{-2}{3} \quad\) and \(180^{\circ}<t<270^{\circ}\)

    5. \(\tan \theta=-\dfrac{1}{2} \quad\) and \(270^{\circ}<\theta<360^{\circ}\)

    6. \(\tan \theta=2\) and \(180^{\circ}<\theta<270^{\circ}\)

    7. \(\tan \theta=\dfrac{3}{4} \quad\) and \(180^{\circ}<\theta<270^{\circ}\)

    8. \(\tan \theta=-3\) and \(90^{\circ}<\theta<180^{\circ}\)

    9. \(2 \cos A+9=8\) and \(90^{\circ}<A<180^{\circ}\)

    10. \(25 \sin B+8=-12\) and \(180^{\circ}<B<270^{\circ}\)

    11. \(8 \tan \beta+5=-11\) and \(90^{\circ}<\beta<180^{\circ}\)

    12. \(6(\tan \beta-4)=-24\) and \(90^{\circ}<\beta<270^{\circ}\)

    13. \(\tan ^2 C - 3 =0 \quad\) and \(90^{\circ}<C<180^{\circ}\) or \(270^{\circ}<C<360^{\circ}\)

    14. \(4 \cos ^2 A-\cos A=0\) and \(0^{\circ}<A<180^{\circ}\)

    Synthesis Questions

    1. Proving the Pythagorean Identity. Prove the Pythagorean Identity \(\cos ^2 \theta+\sin ^2 \theta=1\) by carrying out the following steps. Sketch an angle \(\theta\) in standard position, and label a point \((x, y)\) on the terminal side, at a distance \(r\) from the vertex.

      1. Begin with the equation \(\sqrt{x^2+y^2}=r\), and square both sides.

      2. Divide both sides of your equation from part (a) by \(r^2\).

      3. Write the left side of the equation as the sum of the squares of two fractions.

      4. Substitute the appropriate trigonometric ratio for each fraction.

    2. Proving the Ratio Identity. Prove the Ratio Identity \(\tan \theta=\frac{\sin \theta}{\cos \theta}\) by carrying out the following steps. Sketch an angle \(\theta\) in standard position, and label a point \((x, y)\) on the terminal side, at a distance \(r\) from the vertex.

      1. Write \(\sin \theta\) in terms of \(y\) and \(r\), and solve for \(y\).

      2. Write \(\cos \theta\) in terms of \(x\) and \(r\), and solve for \(x\).

      3. Write \(\tan \theta\) in terms of \(x\) and \(y\), then substitute your results from parts (a) and (b).

      4. Simplify your fraction in part (c).

    2.3.2: Homework is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.