Practice Test 8 - Precalculus

20 questions. Answer them in any order, then submit to see your score and every explanation.

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1. An angle of $\theta=\frac{9\pi}{4}$ is drawn in standard position on the unit circle (one full rotation plus $\frac{\pi}{4}$). Using the fact that points on the unit circle satisfy $(\cos\theta,\sin\theta)$, which coordinates represent the terminal point of this angle on the unit circle?
2. Using the Pythagorean identity $\sin^2(\theta)+\cos^2(\theta)=1$, if $\tan(\theta)=\frac{4}{3}$ and $\theta$ is in Quadrant III, what is $\sin(\theta)$?
3. Point $A$ is located at $(0, 3)$ and point $B$ is located at $(5, 3)$. For the vector described, what is the component form of $\overrightarrow{AB}$?
4. A linear system has the matrix representation: $\begin{pmatrix} 1 & -2 & 3 \\ 0 & 5 & -1 \\ 4 & 0 & 2 \end{pmatrix} \begin{pmatrix} p \\ q \\ r \end{pmatrix} = \begin{pmatrix} 8 \\ -3 \\ 12 \end{pmatrix}$ What is the second equation in this system when written in standard form?
5. Point $A$ is located at $(1, 6)$ and point $B$ is located at $(-3, 2)$. For the vector described, what is the component form of $\overrightarrow{AB}$?
6. Given the matrix equation $\begin{bmatrix}4&-2\\-1&5\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}8\\-3\end{bmatrix},$ which system of equations does it represent?
7. Two different linear systems are represented by: System 1: $A\vec{x} = \vec{b}$ where $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ System 2: $C\vec{x} = \vec{d}$ where $C = \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}$ What is the relationship between these coefficient matrices?
8. On the unit circle, point $P$ is located at coordinates $\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right)$. An angle $\theta$ in standard position has its terminal side passing through $P$, so that $\cos(\theta)=x$ and $\sin(\theta)=y$. Based on the unit circle, what is the value of $\sin(\theta)$?
9. The number of daylight hours varies periodically throughout the year. The maximum is $15$ hours on day 172, and the minimum is $9$ hours on day 355. Assume the pattern repeats every 365 days. For the phenomenon described, which function models the daylight hours $D(d)$ as a function of day number $d$ using a cosine model?
10. Based on the unit circle, examine the graph shown. The graph shows the relationship between angle measures and the corresponding y-coordinates of points on the unit circle. What can be concluded about the symmetry of this function?
11. Using the Pythagorean identity $\sin^2(\theta)+\cos^2(\theta)=1$, if $\sin(\theta)=\frac{1}{2}$ and $\theta$ is in Quadrant IV, what is $\cos^2(\theta)$?
12. In right triangle $\triangle MNO$, $\angle O = 90^\circ$, the hypotenuse is $MN = 12$ cm, and $\angle M = 30^\circ$. Using the given information, what is the exact length of side $NO$ (the side opposite $\angle M$)?
13. Using the end-to-end (head-to-tail) method: vector $\overrightarrow{AB}=\langle 2,1\rangle$ and vector $\overrightarrow{BC}=\langle 3,4\rangle$. What is $\overrightarrow{AC}$ in component form?
14. Using the cofunction identity for complementary angles, which equation correctly expresses the complementary angle relationship for any angle $\theta$?
15. Circle $O$ has radius $4$, and an external point $P$ satisfies $OP=5$. Tangent segments from $P$ touch the circle at $T$ and $T'$. Using the given circle and point, what is the length of the tangent segment $PT$?
16. A small café tracks how many sandwiches and salads it has in stock at two locations. Rows represent items (row 1 = Sandwiches, row 2 = Salads) and columns represent locations (column 1 = Downtown, column 2 = Uptown). Inventory matrix $A$ (in units) is $A=\begin{bmatrix}20 & 30\\ 15 & 25\end{bmatrix}$ Units sold during the day are in matrix $B$ (in units): $B=\begin{bmatrix}5 & 10\\ 10 & 15\end{bmatrix}$ Based on the matrices, what does the matrix $A-B$ represent in this context?​
17. An object's displacement over two steps is represented by vectors $\vec v=\langle 6,0\rangle$ and $\vec w=\langle 1,0\rangle$ drawn from the same starting point. The difference $\vec v-\vec w$ can be found as $\vec v+(-\vec w)$. What are the components of $\vec v-\vec w$?
18. In right triangle $\triangle STU$, $\angle U = 90^\circ$, $SU = 12$ (leg adjacent to $\angle S$), and $TU = 5$ (leg opposite $\angle S$). Using the given information, what is the value of $\tan(\angle S)$?
19. A ladder leans against a wall, making an angle $\alpha$ with the ground. The ladder's length is 20 feet, and it reaches 16 feet up the wall. Using the relationship between sine and cosine of complementary angles, what is $\cos(\alpha) + \sin(90° - \alpha)$?
20. On a coordinate plane, vector $\mathbf{v}$ is the arrow from the origin to $(3,4)$. Which statement correctly describes the visual effect of scalar multiplication by $2$ (that is, $2\mathbf{v}$) compared to $\mathbf{v}$?

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