Precalculus Final Exam Review – Step-by-Step Guidance

Q1. Classify each conic section by putting the conic section in standard form by completing the square.

Background

Topic: Conic Sections & Completing the Square

This question tests your ability to rewrite equations of conic sections (parabolas, ellipses, hyperbolas, circles) in standard form by completing the square, and then classify the type of conic.

Key Terms and Formulas

• Standard forms for conics (e.g., circle: $(x-h)^2 + (y-k)^2 = r^2$, ellipse: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$, hyperbola: $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$)

• Completing the square: $x^2 + bx \rightarrow (x + \frac{b}{2})^2 - (\frac{b}{2})^2$

Step-by-Step Guidance (Part a)

1. Start by rewriting the equation $2x^2 - 16x + y^2 = 32$ so that all variable terms are on one side and constants on the other.

2. Group $x$ terms together and $y$ terms together: $2x^2 - 16x + y^2 = 32$.

3. Factor out the coefficient of $x^2$ from the $x$ terms: $2(x^2 - 8x) + y^2 = 32$.

4. To complete the square for $x$, take half of $-8$, square it, and add/subtract inside the parentheses. Remember to balance the equation by adjusting the other side accordingly.

5. Rewrite the equation in standard form and identify the conic section based on the resulting equation.

Step-by-Step Guidance (Part b)

1. Start with $4x^2 + 4y^2 - 8y = 2$ and group like terms: $4x^2 + 4(y^2 - 2y) = 2$.

2. Factor out the $4$from the$y$terms: $4x^2 + 4(y^2 - 2y) = 2$.

3. Divide both sides by $4$to simplify:$x^2 + (y^2 - 2y) = \frac{1}{2}$.

4. Complete the square for $y^2 - 2y$ and adjust the equation accordingly.

5. Express the equation in standard form and classify the conic section.

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Q2. For part (b) above, give the ordered pair of the center and the major and minor axis vertices. Show your substitution into the formulas and simplify.

Background

Topic: Ellipses – Center, Vertices, and Axes

This question asks you to identify the center and vertices of an ellipse (or other conic) from its standard form, and to use the formulas for vertices based on the values of $a$ and $b$.

Key Terms and Formulas

• Standard form for ellipse: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

• Center: $(h, k)$

• Vertices: Move $a$ units from the center along the major axis, $b$ units along the minor axis

Step-by-Step Guidance

1. Identify $h$ and $k$ from the standard form you found in Q1(b).

2. Determine which denominator ($a^2$ or $b^2$) is larger to find the major axis direction.

3. Find the coordinates of the vertices by adding and subtracting $a$ and $b$ from the center in the appropriate direction.

4. Show your substitution into the vertex formulas and simplify, but do not compute the final coordinates yet.

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Q3. A plane takes a photograph of a building. The photograph is made when the angle of elevation of the sun is 43°. The shadow of the building is 65 feet long. How tall is the building? Round to the nearest whole number. Use right triangle trig to solve and show your set up of the equation you use.

Background

Topic: Right Triangle Trigonometry – Applications

This question tests your ability to apply right triangle trigonometry (specifically tangent) to solve for the height of an object given an angle and a side length.

Key Terms and Formulas

• Tangent ratio: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

• Here, $\theta = 43^\circ$, adjacent = 65 ft, opposite = height of building

Step-by-Step Guidance

1. Draw a right triangle representing the situation, labeling the angle of elevation, the shadow (adjacent), and the height (opposite).

2. Set up the equation using the tangent function: $\tan(43^\circ) = \frac{h}{65}$.

3. Rearrange the equation to solve for $h$: $h = 65 \cdot \tan(43^\circ)$.

4. Prepare to use your calculator to evaluate $\tan(43^\circ)$ and multiply by 65, but do not compute the final value yet.

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Q4. Solve the equation $\cot \theta = -5.2198$ for $\theta$ in $[0, 360^\circ)$ using reference angles and the calculator. List quadrants. Show what you type into the calculator, how you find your reference angle, and how you use your reference angle to find the answers. Round answers to the hundredths place.

Background

Topic: Trigonometric Equations – Cotangent, Reference Angles, Calculator Use

This question tests your ability to solve a trigonometric equation for $\theta$ using the cotangent function, reference angles, and knowledge of quadrants.

Key Terms and Formulas

• $\cot \theta = \frac{1}{\tan \theta}$

• Reference angle: The acute angle formed by the terminal side of $\theta$ and the x-axis

• Quadrants where cotangent is negative: Quadrants II and IV

Step-by-Step Guidance

1. Rewrite $\cot \theta = -5.2198$ as $\tan \theta = \frac{1}{-5.2198}$.

2. Use your calculator to find the reference angle: $\theta_{ref} = \arctan\left(\left|\frac{1}{-5.2198}\right|\right)$.

3. Determine which quadrants $\theta$ can be in (where cotangent is negative: II and IV).

4. Find the two possible solutions for $\theta$ in $[0, 360^\circ)$ by using the reference angle in the appropriate quadrants, but do not compute the final values yet.

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Q5. Find the exact value of $\sin(2\theta)$ and $\cos(2\theta)$ given that $\sin \theta = -\frac{7}{5}$ and $\theta$ is in quadrant III. Use XYR or triangles.

Background

Topic: Double Angle Formulas, Reference Triangles

This question tests your ability to use double angle identities and reference triangles to find exact trigonometric values.

Key Terms and Formulas

• Double angle formulas: $\sin(2\theta) = 2\sin\theta\cos\theta$, $\cos(2\theta) = \cos^2\theta - \sin^2\theta$

• Reference triangle: Use Pythagorean identity to find $\cos \theta$

• Pythagorean identity: $\sin^2\theta + \cos^2\theta = 1$

Step-by-Step Guidance

1. Given $\sin \theta = -\frac{7}{5}$, recognize that this is not possible for a real triangle (since $|\sin \theta| \leq 1$), so check for a typo or clarify with your instructor. If you proceed, use the given value as is for practice.

2. Use the Pythagorean identity to solve for $\cos \theta$ (remember the sign in quadrant III).

3. Plug $\sin \theta$ and $\cos \theta$ into the double angle formulas for $\sin(2\theta)$ and $\cos(2\theta)$.

4. Simplify the expressions as much as possible, but do not compute the final values yet.

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Q6. Solve each equation by the given method over the given interval. Show all work as done in notes and videos.

Background

Topic: Solving Trigonometric Equations

These questions test your ability to solve trigonometric equations using factoring, linear methods, and general solutions.

Key Terms and Formulas

• Factoring trigonometric equations

• Linear trigonometric equations

• General solutions for sine and cosine equations

Step-by-Step Guidance (a)

1. Start with $2 - 6\cos x + 7\cos^2 x = 0$ for $x$ in $[0, 360^\circ)$.

2. Rewrite in standard quadratic form: $7\cos^2 x - 6\cos x + 2 = 0$.

3. Factor the quadratic equation in terms of $\cos x$.

4. Solve for $\cos x$ and set up equations to find $x$ in the given interval, but do not solve for $x$ yet.

Step-by-Step Guidance (b)

1. Start with $8\sin \theta + 1 = 6\sin \theta$ for $\theta$ in $[0, 360^\circ)$.

2. Isolate $\sin \theta$ on one side: $8\sin \theta - 6\sin \theta = -1$.

3. Solve for $\sin \theta$.

4. Set up equations to find $\theta$ in the given interval, but do not solve for $\theta$ yet.

Step-by-Step Guidance (c)

1. Start with $\sin 2x = -2$ for $x$ in $[0, 2\pi)$.

2. Recall the range of the sine function and discuss if a solution is possible.

3. If possible, write the general solution for $2x$ and then for $x$ over the interval, but do not solve for $x$ yet.

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Q7. Find the magnitude and direction angle of the vector $\langle -2, -6 \rangle$. Give the magnitude as a reduced radical and the direction angle rounded to the tenths place.

Background

Topic: Vectors – Magnitude and Direction

This question tests your ability to find the magnitude and direction angle of a vector in the plane.

Key Terms and Formulas

• Magnitude: $|\vec{v}| = \sqrt{a^2 + b^2}$ for $\vec{v} = \langle a, b \rangle$

• Direction angle: $\theta = \arctan\left(\frac{b}{a}\right)$ (adjust for quadrant)

Step-by-Step Guidance

1. Calculate the magnitude: $|\vec{v}| = \sqrt{(-2)^2 + (-6)^2}$.

2. Simplify the radical as much as possible.

3. Find the direction angle: $\theta = \arctan\left(\frac{-6}{-2}\right)$.

4. Determine the correct quadrant for the angle (both components are negative, so the vector is in quadrant III).

5. Set up the calculation for the direction angle, but do not compute the final value yet.

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Q8. Find the following product and give exact value for reduced trig form and round a + bi form to hundredths place: $(2\cos 240^\circ + i\sin 240^\circ)(6\cos 140^\circ + i\sin 140^\circ)$

Background

Topic: Complex Numbers in Trigonometric Form (De Moivre's Theorem)

This question tests your ability to multiply complex numbers in trigonometric form and convert to $a + bi$ form.

Key Terms and Formulas

• Multiplication: $r_1\text{cis}\theta_1 \cdot r_2\text{cis}\theta_2 = (r_1r_2)\text{cis}(\theta_1 + \theta_2)$

• $\text{cis}\theta = \cos \theta + i\sin \theta$

Step-by-Step Guidance

1. Express each complex number in $r\text{cis}\theta$ form: $2\text{cis}240^\circ$ and $6\text{cis}140^\circ$.

2. Multiply the moduli: $2 \times 6$.

3. Add the angles: $240^\circ + 140^\circ$.

4. Write the product in $r\text{cis}\theta$ form and set up the conversion to $a + bi$ form, but do not compute the final values yet.

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Q9. Find the exact values of the six trig functions for the angle in standard position that passes through the point $(-3, 5)$. Use XYR or triangles.

Background

Topic: Trigonometric Functions from a Point

This question tests your ability to find all six trigonometric functions for an angle given a point on its terminal side.

Key Terms and Formulas

• Given $(x, y)$, $r = \sqrt{x^2 + y^2}$

• $\sin \theta = \frac{y}{r}$, $\cos \theta = \frac{x}{r}$, $\tan \theta = \frac{y}{x}$

• Cosecant, secant, cotangent are reciprocals

Step-by-Step Guidance

1. Calculate $r = \sqrt{(-3)^2 + 5^2}$.

2. Write expressions for $\sin \theta$, $\cos \theta$, $\tan \theta$ using $x$, $y$, and $r$.

3. Write expressions for $\csc \theta$, $\sec \theta$, $\cot \theta$ as reciprocals.

4. Simplify the expressions as much as possible, but do not compute the final values yet.

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Q10. Solve the following right triangle using right triangle trig. Show an equation for each part you need to find and put all solutions in a solution box. Don’t forget units with side lengths. Round sides to tenths place and give the other angle in degrees and minutes. $A = 29^ 13'$, $b = 14.7$ yd

Background

Topic: Right Triangle Trigonometry – Solving Triangles

This question tests your ability to solve a right triangle given one angle and one side, using trigonometric ratios.

Key Terms and Formulas

• Sum of angles in a triangle: $180^\circ$

• Sine, cosine, tangent ratios

Step-by-Step Guidance

1. Find the missing angle using $90^\circ$ (since it's a right triangle): $C = 90^\circ$, so $B = 90^\circ - A$.

2. Use the given side $b$ and angle $A$ to find another side using sine, cosine, or tangent as appropriate.

3. Set up equations for the unknown sides and solve for them, but do not compute the final values yet.

4. Prepare to convert the angle to degrees and minutes if needed.

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Q11. Convert the following points:

(a) Convert $(4, -8)$ from rectangular to polar form. Keep $r$ as a simplified radical and $\theta$ rounded to the hundredths place.

Background

Topic: Rectangular and Polar Coordinates

This question tests your ability to convert between rectangular and polar coordinates.

Key Terms and Formulas

• $r = \sqrt{x^2 + y^2}$

• $\theta = \arctan\left(\frac{y}{x}\right)$ (adjust for quadrant)

Step-by-Step Guidance

1. Calculate $r = \sqrt{4^2 + (-8)^2}$ and simplify the radical.

2. Find $\theta = \arctan\left(\frac{-8}{4}\right)$ and determine the correct quadrant for the angle.

3. Set up the polar form $(r, \theta)$, but do not compute the final values yet.

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(b) Convert $(5, 120^\circ)$ from polar to rectangular form. Give exact values using the unit circle and simplify.

Background

Topic: Polar to Rectangular Coordinates

This question tests your ability to convert from polar to rectangular coordinates using trigonometric values.

Key Terms and Formulas

• $x = r \cos \theta$

• $y = r \sin \theta$

Step-by-Step Guidance

1. Write $x = 5 \cos 120^\circ$ and $y = 5 \sin 120^\circ$.

2. Recall the exact values for $\cos 120^\circ$ and $\sin 120^\circ$ from the unit circle.

3. Set up the rectangular coordinates $(x, y)$ using these values, but do not compute the final values yet.

Try solving on your own before revealing the answer!