BackPrecalculus Final Study Guide – Step-by-Step Guidance
Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
Q1. Determine the exact value of tan-1(tan(3\pi/4)) without a calculator.
Background
Topic: Inverse Trigonometric Functions
This question tests your understanding of the inverse tangent function and its principal values, as well as how to evaluate composite trigonometric expressions.
Key Terms and Formulas:
Principal value of $\tan^{-1}(x)$ is in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$
$\tan(\theta)$ is periodic with period $\pi$
Step-by-Step Guidance
Evaluate $\tan(3\pi/4)$. Recall that $3\pi/4$ is in the second quadrant.
Find the value of $\tan(3\pi/4)$ using the unit circle or reference angles.
Apply the inverse tangent function $\tan^{-1}$ to the result from step 2. Remember, $\tan^{-1}$ returns a value in its principal range $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.
Determine which answer choice matches the principal value you found.
Try solving on your own before revealing the answer!
Final Answer: b. -\frac{\pi}{4}
$\tan(3\pi/4) = -1$, so $\tan^{-1}(-1) = -\frac{\pi}{4}$, which is in the principal range of the inverse tangent function.
Q2. What is the value of $\cos^{-1}(-1)$?
Background
Topic: Inverse Trigonometric Functions
This question tests your knowledge of the range and values of the inverse cosine function.
Key Terms and Formulas:
Principal value of $\cos^{-1}(x)$ is in $[0, \pi]$
$\cos(\pi) = -1$
Step-by-Step Guidance
Recall the definition: $\cos^{-1}(x)$ gives the angle in $[0, \pi]$ whose cosine is $x$.
Set up the equation: $\cos(\theta) = -1$ and solve for $\theta$ in the interval $[0, \pi]$.
Identify which angle in this interval has a cosine of $-1$.
Try solving on your own before revealing the answer!
Final Answer: a. $\pi$
$\cos(\pi) = -1$, so $\cos^{-1}(-1) = \pi$.
Q3. If $\sin(\theta) = \frac{\sqrt{2}}{2}$, what are all possible solutions for $\theta$?
Background
Topic: Solving Trigonometric Equations
This question tests your ability to find all solutions to a basic sine equation, including using periodicity.
Key Terms and Formulas:
General solution for $\sin(\theta) = k$ is $\theta = \arcsin(k) + 2\pi n$ and $\theta = \pi - \arcsin(k) + 2\pi n$, where $n$ is any integer.
$\sin(\pi/4) = \frac{\sqrt{2}}{2}$
Step-by-Step Guidance
Identify the reference angle whose sine is $\frac{\sqrt{2}}{2}$.
List all angles in $[0, 2\pi)$ where sine has this value.
Write the general solution using periodicity: $\theta = \text{angle} + 2\pi n$ for each solution.
Try solving on your own before revealing the answer!
Final Answer: a. $\theta = \frac{\pi}{4} + 2\pi n,\ \frac{3\pi}{4} + 2\pi n$, where $n$ is an integer
The sine function is positive in the first and second quadrants, so both $\frac{\pi}{4}$ and $\frac{3\pi}{4}$ are solutions.
Q4. Is the given equation an identity? $\cot x \cos x = \cot x + \frac{2}{\sin x} - 1$
Background
Topic: Trigonometric Identities
This question tests your ability to verify or refute a trigonometric identity by simplifying both sides.
Key Terms and Formulas:
$\cot x = \frac{\cos x}{\sin x}$
Trigonometric identities and algebraic manipulation
Step-by-Step Guidance
Rewrite $\cot x$ and $\cos x$ in terms of sine and cosine on both sides of the equation.
Simplify the left side: $\cot x \cos x$.
Simplify the right side: $\cot x + \frac{2}{\sin x} - 1$.
Compare the simplified forms to see if they are always equal for all $x$ in the domain.
Try solving on your own before revealing the answer!
Final Answer: b. No
After simplifying both sides, they are not always equal for all $x$ in the domain, so the equation is not an identity.
Q5. What is the exact value of $\sin(75^\circ)$ using sum and difference identities?
Background
Topic: Sum and Difference Identities
This question tests your ability to use the sine sum identity to find the exact value of a non-standard angle.
Key Terms and Formulas:
Sum identity: $\sin(A + B) = \sin A \cos B + \cos A \sin B$
Common angles: $45^\circ, 30^\circ$
Step-by-Step Guidance
Express $75^\circ$ as a sum of two common angles (e.g., $45^\circ + 30^\circ$).
Apply the sum identity: $\sin(75^\circ) = \sin(45^\circ + 30^\circ)$.
Substitute the exact values for $\sin(45^\circ)$, $\cos(30^\circ)$, $\cos(45^\circ)$, and $\sin(30^\circ)$.
Simplify the expression to match one of the answer choices.
Try solving on your own before revealing the answer!
Final Answer: a. $\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}$
Using the sum identity and substituting the exact values gives this result.
Q6. Evaluate $\cos(2\theta)$ if $\theta = \frac{\pi}{4}$ using double angle identities.
Background
Topic: Double Angle Identities
This question tests your ability to use the double angle formula for cosine.
Key Terms and Formulas:
Double angle identity: $\cos(2\theta) = 2\cos^2\theta - 1$
Alternatively: $\cos(2\theta) = \cos^2\theta - \sin^2\theta$
Step-by-Step Guidance
Recall the double angle identity for cosine.
Find $\cos\left(\frac{\pi}{4}\right)$ and $\sin\left(\frac{\pi}{4}\right)$.
Substitute these values into the double angle formula.
Simplify the expression to match one of the answer choices.
Try solving on your own before revealing the answer!
Final Answer: c. -1
Substituting $\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$ into the double angle formula yields $-1$.
Q7. Which identity is used to express $\sin^2(\theta)$ in terms of $\cos(\theta)$?
Background
Topic: Pythagorean Identities
This question tests your knowledge of the fundamental Pythagorean identity and how to rearrange it to solve for $\sin^2(\theta)$.
Key Terms and Formulas:
Pythagorean identity: $\sin^2\theta + \cos^2\theta = 1$
Step-by-Step Guidance
Start with the Pythagorean identity.
Rearrange the equation to solve for $\sin^2\theta$ in terms of $\cos^2\theta$.
Identify which answer choice matches your rearranged equation.
Try solving on your own before revealing the answer!
Final Answer: d. $1 - \cos^2(\theta)$
Rearranging the Pythagorean identity gives $\sin^2\theta = 1 - \cos^2\theta$.
Q8. What is the Law of Sines?
Background
Topic: Law of Sines
This question tests your understanding of the Law of Sines and its application in solving triangles.
Key Terms and Formulas:
Law of Sines: $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$
Step-by-Step Guidance
Recall the Law of Sines formula and what it states about the relationship between angles and sides in a triangle.
Match the description in the answer choices to the Law of Sines formula.
Try solving on your own before revealing the answer!
Final Answer: c. The ratio of the sine of an angle to its opposite side is constant for all angles in a triangle.
This matches the Law of Sines formula.
Q9. Which formula represents the Law of Cosines for finding side c?
Background
Topic: Law of Cosines
This question tests your ability to recognize the Law of Cosines formula for solving triangles.
Key Terms and Formulas:
Law of Cosines: $c^2 = a^2 + b^2 - 2ab\cos(C)$
Step-by-Step Guidance
Recall the Law of Cosines formula for side $c$ in a triangle.
Compare each answer choice to the standard formula.
Try solving on your own before revealing the answer!
Final Answer: b. $c^2 = a^2 + b^2 - 2ab\cos(C)$
This is the standard Law of Cosines formula for side $c$.
Q10. Which formula represents the area of a triangle with sides $b$ and $c$ and angle $A$?
Background
Topic: Area of Triangles (SAS/ASA)
This question tests your knowledge of the area formula for a triangle given two sides and the included angle.
Key Terms and Formulas:
Area formula: $\text{Area} = \frac{1}{2}bc\sin(A)$
Step-by-Step Guidance
Recall the area formula for a triangle using two sides and the included angle.
Identify which answer choice matches this formula.
Try solving on your own before revealing the answer!
Final Answer: d. $\frac{1}{2} b c \sin(A)$
This is the correct formula for the area of a triangle given two sides and the included angle.
Q11. A vector makes a 45-degree angle with the east direction. What is the direction of the vector?
Background
Topic: Vector Directions
This question tests your understanding of vector directions and standard compass bearings.
Key Terms and Formulas:
East is the positive x-axis; 45 degrees from east is northeast.
Step-by-Step Guidance
Recall the standard compass directions and their corresponding angles.
Determine which direction is 45 degrees from east (the positive x-axis).
Try solving on your own before revealing the answer!
Final Answer: b. Northeast
45 degrees from east points directly northeast.
Q12. What is the component form of a vector with initial point (0, 0) and terminal point (5, 5)?
Background
Topic: Vectors in Component Form
This question tests your ability to find the component form of a vector given its initial and terminal points.
Key Terms and Formulas:
Component form: $\langle x_2 - x_1, y_2 - y_1 \rangle$
Step-by-Step Guidance
Subtract the coordinates of the initial point from the terminal point: $(x_2 - x_1, y_2 - y_1)$.
Apply this to the given points: $(0, 0)$ and $(5, 5)$.
Try solving on your own before revealing the answer!
Final Answer: b. (5, 5)
The component form is found by subtracting the initial point from the terminal point.
Q13. A vector has components (3, -4). In which quadrant is the vector located?
Background
Topic: Vector Quadrants
This question tests your understanding of the signs of vector components and their corresponding quadrants.
Key Terms and Formulas:
First quadrant: (+, +)
Second quadrant: (-, +)
Third quadrant: (-, -)
Fourth quadrant: (+, -)
Step-by-Step Guidance
Identify the signs of the vector components.
Match the signs to the correct quadrant.
Try solving on your own before revealing the answer!
Final Answer: a. Fourth quadrant
Positive x and negative y components place the vector in the fourth quadrant.
Q14. Is the vector $0.5\mathbf{i} + 0.5\mathbf{j}$ a unit vector?
Background
Topic: Unit Vectors
This question tests your ability to determine if a vector has a magnitude of 1 (unit vector).
Key Terms and Formulas:
Magnitude: $\sqrt{a^2 + b^2}$ for vector $a\mathbf{i} + b\mathbf{j}$
Unit vector: magnitude equals 1
Step-by-Step Guidance
Calculate the magnitude: $\sqrt{(0.5)^2 + (0.5)^2}$.
Determine if the magnitude equals 1.
Try solving on your own before revealing the answer!
Final Answer: d. No, because its magnitude is not 1
The magnitude is less than 1, so it is not a unit vector.
Q15. Find $\mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}$ for $\mathbf{a} = 9\mathbf{i} - 4\mathbf{j}$, $\mathbf{b} = \mathbf{i} + 5\mathbf{j}$, $\mathbf{c} = 8\mathbf{i} + 3\mathbf{j}$.
Background
Topic: Dot Product of Vectors
This question tests your ability to compute the dot product and use vector addition properties.
Key Terms and Formulas:
Dot product: $\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2$
Distributive property: $\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}$
Step-by-Step Guidance
Compute $\mathbf{a} \cdot \mathbf{b}$ using the formula.
Compute $\mathbf{a} \cdot \mathbf{c}$ using the formula.
Add the two results together.
Try solving on your own before revealing the answer!
Final Answer: b. 88
After calculating both dot products and adding, you get 88.
Q16. Calculate the cross product of $\mathbf{u} = (1, 2, 3)$ and $\mathbf{v} = (4, 5, 6)$ using the matrix method.
Background
Topic: Cross Product of Vectors
This question tests your ability to compute the cross product using the determinant of a 3x3 matrix.
Key Terms and Formulas:
Cross product: $\mathbf{u} \times \mathbf{v} = (u_2v_3 - u_3v_2,\ u_3v_1 - u_1v_3,\ u_1v_2 - u_2v_1)$
Step-by-Step Guidance
Set up the determinant for the cross product.
Calculate each component using the formula.
Write the resulting vector in component form.
Try solving on your own before revealing the answer!
Final Answer: b. (-3, 6, -3)
Using the cross product formula, you get (-3, 6, -3).
Q17. What is the pole in the polar coordinate system?
Background
Topic: Polar Coordinates
This question tests your understanding of the terminology used in the polar coordinate system.
Key Terms and Formulas:
Pole: The origin in polar coordinates, equivalent to (0, 0) in Cartesian coordinates.
Step-by-Step Guidance
Recall the definition of the pole in polar coordinates.
Match the definition to the answer choices.
Try solving on your own before revealing the answer!
Final Answer: a. The origin point, equivalent to (0,0) in Cartesian coordinates.
The pole is the origin in the polar coordinate system.
Q18. What are the rectangular coordinates of the polar point (0, 0)?
Background
Topic: Converting Polar to Rectangular Coordinates
This question tests your ability to convert a polar point to rectangular (Cartesian) coordinates.
Key Terms and Formulas:
Conversion: $x = r\cos\theta$, $y = r\sin\theta$
Step-by-Step Guidance
Substitute $r = 0$ and $\theta = 0$ into the conversion formulas.
Calculate $x$ and $y$.
Try solving on your own before revealing the answer!
Final Answer: d. (0, 0)
The origin in polar coordinates is also the origin in rectangular coordinates.
Q19. What is the graph shape of the equation $r = 6\cos\theta$ after converting to rectangular form?
Background
Topic: Converting Polar Equations to Rectangular Form
This question tests your ability to recognize the shape of a graph after converting from polar to rectangular form.
Key Terms and Formulas:
$x = r\cos\theta$
Circle equation: $(x - h)^2 + (y - k)^2 = r^2$
Step-by-Step Guidance
Multiply both sides by $r$ to get $r = 6\cos\theta$ into rectangular form using $x = r\cos\theta$.
Rearrange the equation to match the standard form of a circle.
Try solving on your own before revealing the answer!
Final Answer: b. Circle
The equation represents a circle in rectangular coordinates.
Q20. How many petals does the rose curve $r = 3\sin 4\theta$ have?
Background
Topic: Graphing Polar Equations (Rose Curves)
This question tests your knowledge of rose curves and how the coefficient of $\theta$ affects the number of petals.
Key Terms and Formulas:
Rose curve: $r = a\sin(n\theta)$ or $r = a\cos(n\theta)$
If $n$ is even, number of petals is $2n$; if $n$ is odd, number of petals is $n$.
Step-by-Step Guidance
Identify the value of $n$ in the equation $r = 3\sin 4\theta$.
Determine if $n$ is even or odd.
Apply the rule for the number of petals based on $n$.
Try solving on your own before revealing the answer!
Final Answer: d. 8 petals
Since $n = 4$ (even), the rose curve has $2n = 8$ petals.
Q21. How do parametric equations differ from standard two-variable equations?
Background
Topic: Parametric Equations
This question tests your understanding of the structure and purpose of parametric equations.
Key Terms and Formulas:
Parametric equations: $x = f(t)$, $y = g(t)$, where $t$ is a parameter
Step-by-Step Guidance
Recall that parametric equations introduce a third variable (the parameter).
Compare this to standard equations involving only $x$ and $y$.
Try solving on your own before revealing the answer!
Final Answer: d. They involve a third variable, typically t.
Parametric equations use a parameter to define $x$ and $y$.
Q22. For the parametric equations $x = 2\cos t$ and $y = 2\sin t$, use a Pythagorean identity to eliminate the parameter and find the rectangular equation.
Background
Topic: Eliminating the Parameter
This question tests your ability to convert parametric equations to a rectangular equation using trigonometric identities.
Key Terms and Formulas:
Pythagorean identity: $\cos^2 t + \sin^2 t = 1$
Step-by-Step Guidance
Solve each parametric equation for $\cos t$ and $\sin t$ in terms of $x$ and $y$.
Substitute these expressions into the Pythagorean identity.
Simplify to obtain the rectangular equation.
Try solving on your own before revealing the answer!
Final Answer: d. $x^2 + y^2 = 4$
Substituting and simplifying gives the equation of a circle with radius 2.
Q23. Why is it generally advised to avoid choosing $t$ as an even power of $x$ when parameterizing equations?
Background
Topic: Parameterization
This question tests your understanding of domain restrictions and the implications of parameter choices.
Key Terms and Formulas:
Even powers can restrict the domain to non-negative values, possibly resulting in imaginary numbers for negative $x$.
Step-by-Step Guidance
Consider what happens when $t = x^2$ and $x$ is negative.
Think about the domain of the parameterization and whether all real values are allowed.
Try solving on your own before revealing the answer!
Final Answer: b. Even powers of $x$ can lead to domain restrictions, such as negative values resulting in imaginary numbers.
Choosing $t$ as an even power of $x$ can restrict the domain and cause issues with negative values.
Q24. If you plot the complex number $-2 + 5i$, where will the point be located on the complex plane?
Background
Topic: Graphing Complex Numbers
This question tests your ability to interpret and plot complex numbers on the complex plane.
Key Terms and Formulas:
Complex number: $a + bi$ is plotted at $(a, b)$
Real axis: horizontal; Imaginary axis: vertical
Step-by-Step Guidance
Identify the real part ($-2$) and the imaginary part ($5$).
Plot the point: move left 2 units (real axis), up 5 units (imaginary axis).
Match this location to the answer choices.
Try solving on your own before revealing the answer!
Final Answer: d. 2 units left on the real axis and 5 units up on the imaginary axis.
The point $-2 + 5i$ is plotted at $(-2, 5)$.
Q25. Which of the following is the correct polar form of the complex number $-2 - 2i$?
Background
Topic: Polar Form of Complex Numbers
This question tests your ability to convert a complex number from rectangular to polar form.
Key Terms and Formulas:
Polar form: $r(\cos\theta + i\sin\theta)$
$r = \sqrt{a^2 + b^2}$, $\theta = \arctan\left(\frac{b}{a}\right)$ (adjust for quadrant)
Step-by-Step Guidance
Calculate the modulus: $r = \sqrt{(-2)^2 + (-2)^2}$.
Find the argument $\theta$ using $\arctan\left(\frac{-2}{-2}\right)$ and adjust for the correct quadrant (third quadrant).
Write the polar form using $r$ and $\theta$.
Try solving on your own before revealing the answer!
Final Answer: d. $2\sqrt{2}(\cos(225^\circ) + i\sin(225^\circ))$
The modulus is $2\sqrt{2}$ and the argument is $225^\circ$ (third quadrant).
