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Subsection 5.3 Solving Trigonometric Equations Using Identities Exercises
Solve each equation on the interval
\([0, 2\pi)\text{.}\)
\(\displaystyle -2 \sin \theta=\sqrt{3}\)
\(\displaystyle 2 \cos y + 1 = \cos y\)
\(\displaystyle \sin^2 \alpha - 1 = 0\)
\(\displaystyle 2\tan^2 \beta = 2\)
\(\displaystyle \sin^2 y + \cos y + 1 = 0\)
\(\displaystyle \sec \theta + \tan \theta = 0\)
\(\displaystyle -\sqrt{3} \tan \frac{t}{2}=1\)
\(\displaystyle \sec x = \sqrt{2}\)
\(\displaystyle \sin \theta + 3 \sin 2 \theta = 2 \sin 2 \theta\)
\(\displaystyle \tan^2 x +2 \tan x - 3 = 0\)
\(\displaystyle 2\sin^2 x - \sin x - 3 = 0\)
Refer to the functions
\(f(x)=2x+3\) and
\(g(x)=10-5x\text{.}\)
Find the solution to
\(f(x) \leq g(x)\text{.}\)
Write a function equation for the inverse,
\(f^{-1}(x)\text{.}\)
Let
\(h(x)=2f(x)\text{.}\) Write a function equation for
\(h(x)\text{.}\)
Let
\(k(x)=f(x)-g(x)\text{.}\) Graph
\(k(x)\) on the domain
\(0 \leq x \leq 6\text{.}\)
Let
\(m(x)=\frac{f(x)}{g(x)}\text{.}\) What is the set of values for which
\(m(x)\) is defined?
Graph
\(m(x)\) and indicate the important features of the graph.
Write parametric equations for a point traveling along the line
\(y=\frac{1}{2}x+5\text{,}\) such that at
\(t=0\) the point is at the
\(x\) -intercept, and at
\(t=3\) the point is at the
\(y\) -intercept.
The average price of a home in Cincinnati, OH, in 2015 was $111,200, and growing at about 3% per year.
If the current growth trend continues, write an exponential model for the average price of a home in Cincinnati
\(t\) years after 2015.
How long will it take for the average price to reach $130,000?
Let
\(g(x)=x^2+18x+35\) and
\(B(x)=-x^2+2x+5\text{.}\)
Use algebra to solve
\(g(x)=B(x)\text{.}\)
Use Desmos to check your answer. Sketch the graph with important points labeled.
Use algebra to solve
\(g(x) < B(x)\text{.}\)
A carousel (merry-go-round) is on a pier that juts out westward from a straight shoreline that runs north-south. The carousel has a radius of 20 feet and takes 25 seconds to complete one counterclockwise revolution. The center of the carousel is 40 feet from the shoreline. Jenny is riding one of the carousel horses, and starts at the point farthest from the shoreline. Let
\(t=0\) denote this starting point.
Draw a diagram with the point where the pier meets the shore as the origin, and the negative
\(x\) -axis stretching along the pier through the center of the carousel.
Graph the distance from Jenny to the shoreline as a function of time for the interval from
\(t=0\) to
\(t=25\) seconds.
Find a function of the form
\(f(t)=A\cos(B(t-h))+k\) that represents Jenny’s
\(x\) -coordinate as a function of time.
Write parametric equations
\((f(t),y(t))\) describing Jenny’s location at time
\(t\text{.}\)
How far does Jenny travel in the first 10 seconds of her ride?
What are Jenny’s coordinates at
\(t=10\) seconds?
Find two times when Jenny is 35 feet from the shoreline.
