Je bent je ingevulde velden bij deze pagina aan het verwijderen. Ben je zeker dat je dit wilt doen?
You are erasing your filled-in fields on this page. Are you sure that is what you want?
Nieuwe Versie BeschikbaarNew Version Available
Er is een update van deze pagina. Als je update naar de meest recente versie, verlies je mogelijk je huidige antwoorden voor deze pagina. Hoe wil je verdergaan ?
There is an updated version of this page. If you update to the most recent version, then your current progress on this page will be erased. Regardless, your record of completion will remain. How would you like to proceed?
In Exercises sumdifffirst - sumdifflast, use the Sum and Difference Identities to find the exact value. You may have need of the Quotient, Reciprocal
or Even / Odd Identities as well.
If \(\alpha \) is a Quadrant IV angle with \(\cos (\alpha ) = \frac {\sqrt {5}}{5}\), and \(\sin (\beta ) = \frac {\sqrt {10}}{10}\), where \(\frac {\pi }{2} < \beta < \pi \), find
If \(\sec (\alpha ) = -\frac {5}{3}\), where \(\frac {\pi }{2} < \alpha < \pi \), and \(\tan (\beta ) = \frac {24}{7}\), where \(\pi < \beta < \frac {3\pi }{2}\), find
\(\csc (\alpha - \beta )\)
\(\csc (\alpha - \beta ) = -\frac {5}{4}\)
\(\sec (\alpha + \beta )\)
\(\sec (\alpha + \beta ) = \frac {125}{117}\)
\(\cot (\alpha + \beta )\)
\(\cot (\alpha + \beta ) = \frac {117}{44}\)
In Exercises expandedsinusoidexerfirst - expandedsinusoidexerlast, use Example expandedsinusoidex1 as a guide to show that the function is a sinusoid by rewriting it in the forms \(C(t) = A \cos (\omega t + \phi ) + B\) and \(S(t) = A \sin (\omega t + \phi ) + B\) for \(\omega > 0\) and
\(0 \leq \phi < 2\pi \).
In Exercises expandedsinusoidexerfirst - expandedsinusoidexerlast, you should have noticed a relationship between the phases \(\phi \) for the \(S(t)\) and \(C(t)\). Show that if \(f(t) = A \sin (\omega t + \alpha ) + B\), then \(f(t) = A \cos (\omega t + \beta ) + B\) where \(\beta = \alpha - \frac {\pi }{2}\).
Let \(\phi \) be an angle measured in radians and let \(P(a,b)\) be a point on the terminal side of \(\phi \) when it is drawn in standard
position. Use Theorem cosinesinecircle and the sum identity for sine in Theorem sinesumdifference to show that \(f(t) = a \, \sin (\omega t) + b\, \cos (\omega t) + B\) (with \(\omega > 0\)) can be rewritten as \(f(t) = \sqrt {a^{2} + b^{2}}\sin (\omega t + \phi ) + B\).
In Example sinusoidsunlight in Section GraphsofSineandCosine, we developed two (seemingly) different formulas to model the hours of daylight, \(H(t)\): \(H_{1}(t) = 9.25 \sin \left (\frac {\pi }{6} t - \frac {\pi }{2}\right ) + 12.55\) and \(H_{2}(t) = -8.13 \sin \left (\frac {\pi }{6} t - 4.70\right )+ 12.5\). Use the
difference identities for sine to expand \(H_{1}(t)\) and \(H_{2}(t)\). How different are they?
Find the equation of the tangent line to the graph of \(y = \cos (t)\) at \((0,1)\), \(\left (\frac {\pi }{2}, 0\right )\) and \(\left (-\frac {\pi }{2}, 0\right )\).
Check your answers graphically.
at \((0,1)\): \(y = 1\); at \(\left (\frac {\pi }{2}, 0\right )\): \(y = -x + \frac {\pi }{2}\) at \(\left (-\frac {\pi }{2}, 0\right )\): \(y = x + \frac {\pi }{2}\)
Use the fact (See Exercise sintovertexercise3 in Section TheOtherCircularFunctions.) that \(\lim _{\theta \rightarrow 0}\) \(\frac {\sin (\theta )}{\theta } = 1\) to prove \(\lim _{\theta \rightarrow 0}\) \(\frac {\tan (\theta )}{\theta } = 1\).
Use part tanthetaoverthetalimit to show \(f'(t) = \lim _{h \rightarrow 0} \frac {f(t+h) - f(t)}{h} = \sec ^{2}(t)\)
Find the equation of the tangent line to the graph of \(y = \tan (t)\) at \((0,0)\), \(\left (\frac {\pi }{4}, 1 \right )\) and \(\left (-\frac {\pi }{4}, -1\right )\).
In Exercises idenhalfanglefirst - idenhalfanglelast, use the Half Angle Formulas to find the exact value. You may have need of the Quotient, Reciprocal or Even
/ Odd Identities as well.
\(\cos (75^{\circ })\) (compare with Exercise cos75)
Suppose \(\theta \) is a Quadrant I angle with \(\sin (\theta ) = x\). Verify the following formulas
\(\cos (\theta ) = \sqrt {1-x^2}\)
\(\sin (2\theta ) = 2x\sqrt {1-x^2}\)
\(\cos (2\theta ) = 1 - 2x^2\)
Discuss with your classmates how each of the formulas, if any, in Exercise preludetoarctrigsine change if we change assume \(\theta \) is a Quadrant II,
III, or IV angle.
Suppose \(\theta \) is a Quadrant I angle with \(\tan (\theta ) = x\). Verify the following formulas
\(\cos (\theta ) = \frac {1}{\sqrt {x^2+1}}\)
\(\sin (\theta ) = \frac {x}{\sqrt {x^2+1}}\)
\(\sin (2\theta ) = \frac {2x}{x^2+1}\)
\(\cos (2\theta ) = \frac {1-x^2}{x^2+1}\)
Discuss with your classmates how each of the formulas, if any, in Exercise preludetoarctrigtan change if we change assume \(\theta \) is a Quadrant II,
III, or IV angle.
If \(\sin (t) = x\) for \(-\frac {\pi }{2} < t < \frac {\pi }{2}\), find an expression for \(\tan (t)\) in terms of \(x\).
\(\tan (t) = \frac {x}{\sqrt {1 - x^2}}\)
If \(\tan (\theta ) = x\) for \(-\frac {\pi }{2} < \theta < \frac {\pi }{2}\), find an expression for \(\sec (\theta )\) in terms of \(x\).
\(\sec (\theta ) = \sqrt {1+x^2}\)
If \(\sec (\theta ) = x\) where \(\theta \) is a Quadrant II angle, find an expression for \(\tan (\theta )\) in terms of \(x\).
\(\tan (\theta ) = \answer {-\sqrt {x^2-1}}\)
If \(\sin (t) = \frac {x}{2}\) for \(-\frac {\pi }{2} < t < \frac {\pi }{2}\), find an expression for \(\cos (2t)\) in terms of \(x\).
\(\cos (2t) = 1 - \frac {x^{2}}{2}\)
If \(\tan (\theta ) = \frac {x}{7}\) for \(-\frac {\pi }{2} < \theta < \frac {\pi }{2}\), find an expression for \(\sin (2\theta )\) in terms of \(x\).
\(\sin (2\theta ) = \frac {14x}{x^{2} + 49}\)
If \(\sec (t) = \frac {x}{4}\) for \(0 < t < \frac {\pi }{2}\), find an expression for \(\ln |\sec (t) + \tan (t)|\) in terms of \(x\).
Show that \(\cos ^{2}(\theta ) - \sin ^{2}(\theta ) = 2\cos ^{2}(\theta ) - 1 = 1 - 2\sin ^{2}(\theta )\) for all \(\theta \).
Let \(\theta \) be a Quadrant III angle with \(\cos (\theta ) = -\frac {1}{5}\). Show that this is not enough information to determine the sign of \(\sin \left (\frac {\theta }{2}\right )\) by first assuming \(3\pi < \theta < \frac {7\pi }{2}\) and
then assuming \(\pi < \theta < \frac {3\pi }{2}\) and computing \(\sin \left (\frac {\theta }{2}\right )\) in both cases.
Without using your calculator, show that \(\frac {\sqrt {2 + \sqrt {3}}}{2} = \frac {\sqrt {6} + \sqrt {2}}{4}\)
In part cosinepolynomial of Example doubleangleex, we wrote \(\cos (3\theta )\) as a polynomial in terms of \(\cos (\theta )\). In Exercise cosine4theta, we had you verify an identity which expresses \(\cos (4\theta )\) as
a polynomial in terms of \(\cos (\theta )\). Can you find a polynomial in terms of \(\cos (\theta )\) for \(\cos (5\theta )\)? \(\cos (6\theta )\)? Can you find a pattern so that \(\cos (n\theta )\) could be written as a
polynomial in cosine for any natural number \(n\)?
In Exercise sine3theta, we has you verify an identity which expresses \(\sin (3\theta )\) as a polynomial in terms of \(\sin (\theta )\). Can you do the same for \(\sin (5\theta )\)? What
about for \(\sin (4\theta )\)? If not, what goes wrong?
In Exercises idengraphfirst - idengraphlast, verify the identity by graphing the right and left hand using a graphing utility.
In Exercises prodsumfirst - prodsumlast, write the given product as a sum. Note: you may need to use an Even/Odd Identity to match the answer
provided.
\(\cos (3\theta )\cos (5\theta )\)
\(\frac {\cos (2\theta ) + \cos (8\theta )}{2}\)
\(\sin (2t)\sin (7t)\)
\(\frac {\cos (5t) - \cos (9t)}{2}\)
\(\sin (9x)\cos (x)\)
\(\answer {\frac {\sin (8x) + \sin (10x)}{2}}\)
\(\cos (2\theta ) \cos (6\theta )\)
\(\frac {\cos (4\theta ) + \cos (8\theta )}{2}\)
\(\sin (3t) \sin (2t)\)
\(\frac {\cos (t) - \cos (5t)}{2}\)
\(\cos (x) \sin (3x)\)
\(\frac {\sin (2x) + \sin (4x)}{2}\)
In Exercises sumprodfirst - sumprodlast, write the given sum as a product. Note: you may need to use an Even/Odd or Cofunction Identity to match
the answer provided.
In the remarks following Example expandedsinusoidex1, we rewrote the function \(f(t)\) in number beatsprodsumex from Example prodtosumtoprod using a sum to product identity as a
product of two sinusoids, one acting as a variable amplitude creating a ‘wave envelope.’ Using this discussion as a guide,
rewrite each of the functions \(f(t)\) in Exercises beatexfirst - beatexlast as a product of sinusoids. Identify the functions which create the
‘wave envelope.’ Check your answer by graphing the function along with the ‘wave-envelope’ using a graphing
utility.
PreCalculus