Practice Test 3 - Precalculus

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

Start practice test

1. A circular track has radius $r=6$ meters. A runner travels along the edge through a central angle of $\theta=\frac{\pi}{3}$ radians. For the circle described, what is the length of the arc intercepted by the angle?​
2. On the unit circle, an arc from point $A$ to point $B$ has length $\frac{7\pi}{6}$. If point $A$ corresponds to an angle of $\frac{\pi}{4}$ radians in standard position, what angle in standard position corresponds to point $B$?
3. A circle has radius $r=5$ cm and a central angle of $\theta=\frac{2\pi}{3}$ radians. If the radius is doubled while the angle remains constant, how does the arc length change?
4. Using the periodicity property of sine, $\sin(\theta+2\pi)=\sin(\theta)$ for all $\theta$. What is $\sin\left(\frac{9\pi}{4}\right)$?​
5. In the unit circle, if an arc of length $k$ subtends an angle of $k$ radians, and another arc of length $2k$ is drawn on a circle of radius $3$, what is the ratio of the angle subtended by the second arc to the angle subtended by the first arc?
6. Circle $O$ has center $O$ and radius $5$. Point $P$ is an external point (so $OP>5$). Using the given circle and point, how many tangent lines can be drawn from point $P$ to circle $O$?
7. Triangle $GHI$ has sides $GH=9\text{ cm}$ and $GI=16\text{ cm}$ with included angle $\angle HGI=30^\circ$. For the triangle described, what is the area of the triangle?
8. Vector $\vec{v}$ is represented in component form as $\vec{v}=\langle 3,4\rangle$ on the coordinate plane (units in meters). Based on the vector described, what is the magnitude of vector $\vec{v}$?
9. In triangle $XYZ$, the median from vertex $X$ to side $YZ$ has length $m$ and makes an angle of $\beta$ with side $XY$. If $XY = a$ and the median divides the triangle into two triangles of equal area, what is the total area of triangle $XYZ$?
10. The directed line segment from $M(2, 2)$ to $N(2, -4)$ represents a vector. Based on the coordinates, what is the component form of $\overrightarrow{MN}$?
11. In right triangle $\triangle ABC$, $\angle C = 90^\circ$. If $AC = 8$ and $AB = 17$ (hypotenuse), what is the exact value of $\sin(\angle A)$?
12. Given the vector $\vec v$ points from the origin to $(2,1)$ on a coordinate plane, in what direction does $-\vec v$ point compared to $\vec v$?
13. A force vector $\mathbf{v}$ has magnitude $60\ \text{N}$ directed due north. What are the magnitude and compass direction of $-\tfrac{1}{2}\mathbf{v}$?
14. Given the velocities, an airplane has airspeed $300$ km/h due east, and the wind blows $50$ km/h due south. What is the magnitude of the resultant ground velocity $|\vec v_{ground}|$?
15. A ladder is $10$ ft long and leans against a vertical wall. The bottom of the ladder is $6$ ft from the wall, forming a right triangle with the ground and the wall. What is the height (in feet) the ladder reaches up the wall?
16. Using the Pythagorean identity, what is the value of $\sin^2(\theta)+\cos^2(\theta)$ for any angle $\theta$?​
17. A point $\langle x,y\rangle$ is transformed by the matrix $S=\begin{bmatrix}3&0\\0&1\end{bmatrix}.$ Where does the point $\langle -2,4\rangle$ map to under this transformation (i.e., what is $S\begin{bmatrix}-2\\4\end{bmatrix}$)?
18. If $\sin\left(\frac{\pi}{6}\right)=\frac{1}{2}$, then based on sine being an odd function, what is $\sin\left(-\frac{\pi}{6}\right)$?
19. A triangular garden has two sides of lengths $15\text{ m}$ and $20\text{ m}$ that meet at an included angle of $45^\circ$. Based on the area formula $A=\tfrac12 ab\sin(C)$, what is the area of the garden?​
20. Vector $\mathbf{v}$ goes from the origin to $(4,6)$ on a coordinate plane. Where does $1.5\mathbf{v}$ terminate?

Practice tests