Precalculus Trigonometry Test 1: Step-by-Step Study Guidance

Q1. Rewrite the equation in standard form and graph it: $x^2 - 8x + y^2 + 6y = 11$

Background

Topic: Conic Sections (Circles)

This question tests your ability to rewrite a quadratic equation in standard form by completing the square, and to identify and graph the conic section it represents.

Key Terms and Formulas

• Standard form of a circle: $(x - h)^2 + (y - k)^2 = r^2$

• Completing the square: A method to rewrite quadratic expressions in the form $(x - h)^2$ or $(y - k)^2$

Step-by-Step Guidance

1. Group the $x$ and $y$ terms: $x^2 - 8x + y^2 + 6y = 11$

2. Complete the square for the $x$ terms: $x^2 - 8x$

3. Complete the square for the $y$ terms: $y^2 + 6y$

4. Rewrite the equation using the completed squares, and adjust the right side accordingly.

Try solving on your own before revealing the answer!

Final Answer:

Standard form: $(x - 4)^2 + (y + 3)^2 = 36$

This is a circle centered at $(4, -3)$ with radius $6$.

Q2. Rewrite the equation in standard form and graph it: $x^2 - 8x - y^2 + 6y = -8$

Background

Topic: Conic Sections (Hyperbolas)

This question tests your ability to rewrite a general quadratic equation in standard form for a hyperbola by completing the square, and to graph the resulting curve.

Key Terms and Formulas

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

• Completing the square for both $x$ and $y$ terms

Step-by-Step Guidance

1. Group $x$ and $y$ terms: $x^2 - 8x - (y^2 - 6y) = -8$

2. Complete the square for $x^2 - 8x$ and $y^2 - 6y$ (note the negative sign in front of the $y$ terms).

3. Rewrite the equation using the completed squares, and move constants to the right side.

4. Divide both sides by the constant on the right to get the equation in standard hyperbola form.

Try solving on your own before revealing the answer!

Final Answer:

Standard form: $\frac{(x - 4)^2}{9} - \frac{(y - 3)^2}{16} = 1$

This is a hyperbola centered at $(4, 3)$, opening left and right, with $a = 3$, $b = 4$.

Q3. Find the coordinates of the foci for the curve in question #2

Background

Topic: Hyperbolas – Foci

This question tests your understanding of how to find the foci of a hyperbola given its standard form.

Key Terms and Formulas

• For $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$, the foci are at $(h \pm c, k)$, where $c = \sqrt{a^2 + b^2}$

Step-by-Step Guidance

1. Identify $h$, $k$, $a$, and $b$ from the standard form found in Q2.

2. Calculate $c$ using $c = \sqrt{a^2 + b^2}$.

3. Write the coordinates of the foci as $(h + c, k)$ and $(h - c, k)$.

Try solving on your own before revealing the answer!

Final Answer:

The foci are at $(4 + 5, 3) = (9, 3)$ and $(4 - 5, 3) = (-1, 3)$.

Here, $a = 3$, $b = 4$, so $c = 5$.

Q4. Rewrite the equation in standard form and graph it: $x^2 - 2x + 9y^2 + 54y + 46 = 0$

Background

Topic: Conic Sections (Ellipses)

This question tests your ability to rewrite a quadratic equation in standard form for an ellipse by completing the square, and to graph the ellipse.

Key Terms and Formulas

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

• Completing the square for $x$ and $y$ terms

Step-by-Step Guidance

1. Group $x$ and $y$ terms: $x^2 - 2x + 9(y^2 + 6y) = -46$

2. Complete the square for $x^2 - 2x$ and $y^2 + 6y$ (remember to factor out the $9$from the$y$ terms before completing the square).

3. Rewrite the equation using the completed squares, and move constants to the right side.

4. Divide both sides by the constant on the right to get the equation in standard ellipse form.

Try solving on your own before revealing the answer!

Final Answer:

Standard form: $\frac{(x - 1)^2}{16} + \frac{(y + 3)^2}{16} = 1$

This is a circle (special case of ellipse) centered at $(1, -3)$ with radius $4$.

Q5. Rewrite the equation in standard form and graph it: $4x - y^2 + 4y + 16 = 0$

Background

Topic: Conic Sections (Parabolas)

This question tests your ability to rewrite a quadratic equation in standard form for a parabola by completing the square, and to graph the parabola.

Key Terms and Formulas

• Standard form of a parabola (vertical): $(y - k)^2 = 4p(x - h)$

• Completing the square for $y$ terms

Step-by-Step Guidance

1. Rewrite the equation to group $y$ terms: $-y^2 + 4y = -4x - 16$

2. Factor out $-1$ from the $y$ terms: $-(y^2 - 4y) = -4x - 16$

3. Complete the square for $y^2 - 4y$.

4. Rewrite the equation in the form $(y - k)^2 = 4p(x - h)$.

Try solving on your own before revealing the answer!

Final Answer:

Standard form: $(y - 2)^2 = -4(x + 3)$

This is a parabola opening to the left, vertex at $(-3, 2)$.

Q6. Find the coordinates of the focus and an equation of the directrix for the curve in question #5

Background

Topic: Parabolas – Focus and Directrix

This question tests your understanding of how to find the focus and directrix of a parabola given its standard form.

Key Terms and Formulas

• For $(y - k)^2 = 4p(x - h)$, the vertex is at $(h, k)$

• Focus: $(h + p, k)$ for a parabola opening left/right

• Directrix: $x = h - p$

Step-by-Step Guidance

1. Identify $h$, $k$, and $p$ from the standard form found in Q5.

2. Calculate the focus using $(h + p, k)$.

3. Write the equation of the directrix: $x = h - p$.

Try solving on your own before revealing the answer!

Final Answer:

Focus: $(-4, 2)$

Directrix: $x = -2$

Here, $p = -1$, $h = -3$, $k = 2$.

Q7. For the given parametric equations $x = -t$, $y = t^2$, find its rectangular form, graph it, and indicate the orientation of the line.

Background

Topic: Parametric Equations

This question tests your ability to eliminate the parameter to find the rectangular (Cartesian) form of a curve, and to analyze its graph and orientation.

Key Terms and Formulas

• Parametric equations: $x = f(t)$, $y = g(t)$

• Rectangular form: An equation relating $x$ and $y$ directly

Step-by-Step Guidance

1. Solve $x = -t$ for $t$ in terms of $x$.

2. Substitute this expression for $t$ into $y = t^2$ to get $y$ in terms of $x$.

3. Analyze the resulting equation to determine the type of curve and its orientation as $t$ increases.

Try solving on your own before revealing the answer!

Final Answer:

Rectangular form: $y = x^2$

This is a parabola opening upward. As $t$ increases, $x$ decreases, so the orientation is from right to left along the parabola.

Q8. Find two different angles, one positive and one negative, coterminal with $\theta = -400^\circ$

Background

Topic: Coterminal Angles

This question tests your understanding of how to find angles that share the same terminal side as a given angle by adding or subtracting multiples of $360^\circ$.

Key Terms and Formulas

• Coterminal angles: $\theta + 360^\circ k$, where $k$ is any integer

Step-by-Step Guidance

1. Add $360^\circ$ to $-400^\circ$ until you get a positive angle.

2. Subtract $360^\circ$ from $-400^\circ$ to get a more negative coterminal angle.

Try solving on your own before revealing the answer!

Final Answer:

Positive coterminal angle: $320^\circ$

Negative coterminal angle: $-40^\circ$

Q9. Find two different angles, one positive and one negative, coterminal with $s = 3\pi$

Background

Topic: Coterminal Angles (Radians)

This question tests your ability to find coterminal angles in radians by adding or subtracting multiples of $2\pi$.

Key Terms and Formulas

• Coterminal angles: $\theta + 2\pi k$, where $k$ is any integer

Step-by-Step Guidance

1. Subtract $2\pi$ from $3\pi$ to get a positive coterminal angle less than $2\pi$.

2. Subtract $2\pi$ again to get a negative coterminal angle.

Try solving on your own before revealing the answer!

Final Answer:

Positive coterminal angle: $\pi$

Negative coterminal angle: $-\pi$

Q10. Convert degree measure to radians: $\theta = 120^\circ$

Background

Topic: Angle Conversion (Degrees to Radians)

This question tests your ability to convert an angle from degrees to radians using the conversion factor $\frac{\pi}{180^\circ}$.

Key Terms and Formulas

• Conversion: $\text{radians} = \text{degrees} \times \frac{\pi}{180^\circ}$

Step-by-Step Guidance

1. Multiply $120^\circ$ by $\frac{\pi}{180^\circ}$.

2. Simplify the resulting fraction.

Try solving on your own before revealing the answer!

Final Answer:

$120^\circ = \frac{2\pi}{3}$ radians

Q11. Convert radian measure to degrees: $s = -\frac{\pi}{10}$

Background

Topic: Angle Conversion (Radians to Degrees)

This question tests your ability to convert an angle from radians to degrees using the conversion factor $\frac{180^\circ}{\pi}$.

Key Terms and Formulas

• Conversion: $\text{degrees} = \text{radians} \times \frac{180^\circ}{\pi}$

Step-by-Step Guidance

1. Multiply $-\frac{\pi}{10}$ by $\frac{180^\circ}{\pi}$.

2. Simplify the resulting expression.

Try solving on your own before revealing the answer!

Final Answer:

$-\frac{\pi}{10}$ radians $= -18^\circ$

Q12. Find the complement and supplement of angle $\theta = 51^\circ$

Background

Topic: Complementary and Supplementary Angles

This question tests your understanding of how to find the complement and supplement of a given angle in degrees.

Key Terms and Formulas

• Complement: $90^\circ - \theta$

• Supplement: $180^\circ - \theta$

Step-by-Step Guidance

1. Subtract $51^\circ$ from $90^\circ$ to find the complement.

2. Subtract $51^\circ$ from $180^\circ$ to find the supplement.

Try solving on your own before revealing the answer!

Final Answer:

Complement: $39^\circ$

Supplement: $129^\circ$

Q13. Find the complement and supplement of angle $s = \frac{\pi}{5}$

Background

Topic: Complementary and Supplementary Angles (Radians)

This question tests your ability to find the complement and supplement of a given angle in radians.

Key Terms and Formulas

• Complement: $\frac{\pi}{2} - s$

• Supplement: $\pi - s$

Step-by-Step Guidance

1. Subtract $\frac{\pi}{5}$ from $\frac{\pi}{2}$ to find the complement.

2. Subtract $\frac{\pi}{5}$ from $\pi$ to find the supplement.

3. Simplify the resulting fractions.

Try solving on your own before revealing the answer!

Final Answer:

Complement: $\frac{3\pi}{10}$

Supplement: $\frac{4\pi}{5}$

Q14. Subtract $13.123^\circ - 6^\circ 51' 39''$, express your answer in degrees/minutes/seconds

Background

Topic: Angle Subtraction (DMS Format)

This question tests your ability to subtract angles given in decimal degrees and degrees/minutes/seconds (DMS), and to express the result in DMS format.

Key Terms and Formulas

• 1 degree = 60 minutes, 1 minute = 60 seconds

• Convert decimal degrees to DMS if needed

Step-by-Step Guidance

1. Convert $13.123^\circ$ to DMS format.

2. Subtract $6^\circ 51' 39''$ from the DMS value, borrowing as needed.

3. Express the result in degrees, minutes, and seconds.

Try solving on your own before revealing the answer!

Final Answer:

$13.123^\circ = 13^\circ 7' 22.8''$

Difference: $6^\circ 15' 43.8''$

Q15. The terminal point corresponding to angle $s$ has coordinates $\left(\frac{12}{13}, -\frac{5}{13}\right)$. Find the values of all six circular functions of angle $s$

Background

Topic: Circular Functions (Trigonometric Ratios)

This question tests your ability to find the sine, cosine, tangent, cosecant, secant, and cotangent of an angle given a point on the unit circle.

Key Terms and Formulas

• $\sin s = y/r$

• $\cos s = x/r$

• $\tan s = y/x$

• $\csc s = r/y$

• $\sec s = r/x$

• $\cot s = x/y$

• Here, $r = \sqrt{x^2 + y^2}$

Step-by-Step Guidance

1. Calculate $r$ using $r = \sqrt{x^2 + y^2}$.

2. Find $\sin s$, $\cos s$, and $\tan s$ using the definitions above.

3. Find $\csc s$, $\sec s$, and $\cot s$ as reciprocals of the primary functions.

Try solving on your own before revealing the answer!

Final Answer:

$r = 1$

$\sin s = -\frac{5}{13}$, $\cos s = \frac{12}{13}$, $\tan s = -\frac{5}{12}$

$\csc s = -\frac{13}{5}$, $\sec s = \frac{13}{12}$, $\cot s = -\frac{12}{5}$

Q16. Identify the amplitude, frequency, period, and the shifts of the equation $y = 5 \cos\left(2x - \frac{\pi}{2}\right) - 3$

Background

Topic: Graphing Trigonometric Functions

This question tests your ability to identify key characteristics of a cosine function, including amplitude, period, phase shift, and vertical shift.

Key Terms and Formulas

• General form: $y = A \cos(Bx - C) + D$

• Amplitude: $|A|$

• Period: $\frac{2\pi}{|B|}$

• Frequency: $|B|$

• Phase shift: $\frac{C}{B}$

• Vertical shift: $D$

Step-by-Step Guidance

1. Identify $A$, $B$, $C$, and $D$ from the equation.

2. Calculate the amplitude as $|A|$.

3. Calculate the period as $\frac{2\pi}{|B|}$.

4. Find the phase shift as $\frac{C}{B}$.

5. State the vertical shift $D$.

Try solving on your own before revealing the answer!

Final Answer:

Amplitude: $5$

Frequency: $2$

Period: $\pi$

Phase shift: $\frac{\pi}{4}$ to the right

Vertical shift: $-3$

Q17. Graph the basic period of $y = 5 \cos\left(2x - \frac{\pi}{2}\right) - 3$

Background

Topic: Graphing Trigonometric Functions

This question tests your ability to graph a cosine function with amplitude, period, phase shift, and vertical shift.

Key Terms and Formulas

• Use the information from Q16: amplitude, period, phase shift, vertical shift

• Plot key points: maximum, minimum, and midline crossings

Step-by-Step Guidance

1. Mark the phase shift and vertical shift on your axes.

2. Determine the interval for one period using the period found in Q16.

3. Plot the maximum, minimum, and midline points for one period of the cosine function.

4. Sketch the curve connecting these points smoothly.

Try solving on your own before revealing the answer!

Final Answer:

The graph is a cosine wave with amplitude $5$, period$\pi$, phase shift$\frac{\pi}{4}$to the right, and vertical shift$-3$. The maximum is at$y = 2$, minimum at$y = -8$.

Q18. Identify the amplitude, frequency, period, and the shifts of the equation $y = -2 \sin(x + \pi) + 1$

Background

Topic: Graphing Trigonometric Functions

This question tests your ability to identify the amplitude, period, phase shift, and vertical shift of a sine function.

Key Terms and Formulas

• General form: $y = A \sin(Bx + C) + D$

• Amplitude: $|A|$

• Period: $\frac{2\pi}{|B|}$

• Frequency: $|B|$

• Phase shift: $-\frac{C}{B}$

• Vertical shift: $D$

Step-by-Step Guidance

1. Identify $A$, $B$, $C$, and $D$ from the equation.

2. Calculate the amplitude as $|A|$.

3. Calculate the period as $\frac{2\pi}{|B|}$.

4. Find the phase shift as $-\frac{C}{B}$.

5. State the vertical shift $D$.

Try solving on your own before revealing the answer!

Final Answer:

Amplitude: $2$

Frequency: $1$

Period: $2\pi$

Phase shift: $-\pi$ (to the left)

Vertical shift: $1$

Q19. Graph the basic period of $y = -2 \sin(x + \pi) + 1$

Background

Topic: Graphing Trigonometric Functions

This question tests your ability to graph a sine function with amplitude, period, phase shift, and vertical shift.

Key Terms and Formulas

• Use the information from Q18: amplitude, period, phase shift, vertical shift

• Plot key points: maximum, minimum, and midline crossings

Step-by-Step Guidance

1. Mark the phase shift and vertical shift on your axes.

2. Determine the interval for one period using the period found in Q18.

3. Plot the maximum, minimum, and midline points for one period of the sine function.

4. Sketch the curve connecting these points smoothly.

Try solving on your own before revealing the answer!

Final Answer:

The graph is a sine wave with amplitude $2$, period $2\pi$, phase shift $-\pi$ (to the left), and vertical shift $1$. The maximum is at$y = 3$, minimum at$y = -1$.