Step-by-Step Guidance for Plane Trigonometry Practice Test (Ch 4–6.1)

Q1. Graph the function $y = 2 \cos\left(x - \frac{\pi}{10}\right)$ and answer the following:

• (a) Amplitude

• (b) Any x-axis reflection

• (c) Period

• (d) Any phase shift

• (e) Any vertical translation

• (f) 5 key points (in a table or as ordered pairs)

Background

Topic: Graphs of Trigonometric Functions (Cosine)

This question tests your understanding of how to analyze and graph cosine functions, including identifying amplitude, period, phase shift, vertical translation, and key points.

Key Terms and Formulas

• General form: $y = a \cos(bx - c) + d$

• Amplitude: $|a|$

• Period: $\frac{2\pi}{|b|}$

• Phase shift: $\frac{c}{b}$

• Vertical translation: $d$

Step-by-Step Guidance

1. Identify the values of $a$, $b$, $c$, and $d$ in the given function $y = 2 \cos\left(x - \frac{\pi}{10}\right)$ by comparing it to the general form.

2. Determine the amplitude by taking the absolute value of $a$.

3. Find the period using $\frac{2\pi}{|b|}$.

4. Calculate the phase shift by evaluating $\frac{c}{b}$.

5. Check for any vertical translation by identifying $d$.

6. List 5 key points by selecting $x$-values within two periods, substituting into the function, and finding corresponding $y$-values.

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Q2. Graph the function $y = \sec\left(x - \frac{\pi}{4}\right)$ over one period and answer:

• a) The reciprocal function

• b) The reciprocal function amplitude and any vertical stretch/shrink (state which)

• c) The period

• d) Any x-axis reflection

• e) Any phase shift

• f) Any vertical translation

• g) 3 key points and 2 vertical asymptote equations for the first period OR 2 key points and 3 vertical asymptote equations for the first period

Background

Topic: Graphs of Reciprocal Trigonometric Functions (Secant)

This question tests your ability to analyze and graph the secant function, including identifying its reciprocal, amplitude, period, phase shift, vertical translation, and asymptotes.

Key Terms and Formulas

• Secant is the reciprocal of cosine: $\sec x = \frac{1}{\cos x}$

• Period of $\sec x$ is the same as $\cos x$: $\frac{2\pi}{|b|}$

• Vertical asymptotes occur where $\cos(x - \frac{\pi}{4}) = 0$

Step-by-Step Guidance

1. Identify the reciprocal function (cosine) and write its equation.

2. Determine the amplitude and whether there is a vertical stretch or shrink by examining the coefficient of the reciprocal function.

3. Calculate the period using $\frac{2\pi}{|b|}$.

4. Find the phase shift by evaluating $\frac{c}{b}$.

5. Locate the vertical asymptotes by solving $\cos(x - \frac{\pi}{4}) = 0$ for $x$ within one period.

6. Choose 2 or 3 key points (where the function is defined) and list their coordinates.

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Q3. An object is attached to a coiled spring and pulled down a distance of $s_0$ units, then released. The time for one complete oscillation is $\frac{3}{7}$ seconds.

• a) Give an equation that models the position $s(t)$ at time $t$.

• b) Determine the position at time $t = 1.46$ sec.

• c) Find the frequency (oscillations per second).

Background

Topic: Simple Harmonic Motion (Trigonometric Modeling)

This question tests your ability to model periodic motion using trigonometric functions, specifically sine or cosine, and to interpret period and frequency.

Key Terms and Formulas

• General model: $s(t) = a \cos(\omega t)$ or $s(t) = a \sin(\omega t)$

• Period: $T = \frac{2\pi}{\omega}$

• Frequency: $f = \frac{1}{T}$

Step-by-Step Guidance

1. Identify the amplitude $a$ (the initial displacement from equilibrium).

2. Use the given period $T = \frac{3}{7}$ to solve for $\omega$ using $T = \frac{2\pi}{\omega}$.

3. Write the equation for $s(t)$ using the values of $a$ and $\omega$.

4. To find the position at $t = 1.46$ sec, substitute $t$ into your equation for $s(t)$.

5. Calculate the frequency $f = \frac{1}{T}$.

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Q4. Write the expression $4 \sin 2x \sin 4x$ as a sum or difference of trigonometric functions.

Background

Topic: Product-to-Sum Formulas

This question tests your ability to use product-to-sum identities to rewrite products of sines as sums or differences.

Key Terms and Formulas

• Product-to-sum: $\sin A \sin B = \frac{1}{2}[\cos(A - B) - \cos(A + B)]$

Step-by-Step Guidance

1. Identify $A = 2x$ and $B = 4x$ in the formula.

2. Apply the product-to-sum identity to $\sin 2x \sin 4x$.

3. Multiply the result by 4 to match the original expression.

4. Simplify the resulting expression as much as possible.

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Q5. Verify the identity: $-\sin 2\theta = \frac{-2 \cot \theta}{1 + \cot^2 \theta}$

Background

Topic: Trigonometric Identities (Double Angle, Cotangent)

This question tests your ability to verify trigonometric identities, especially those involving double angles and cotangent.

Key Terms and Formulas

• Double angle: $\sin 2\theta = 2 \sin \theta \cos \theta$

• $\cot \theta = \frac{\cos \theta}{\sin \theta}$

• $1 + \cot^2 \theta = \csc^2 \theta$

Step-by-Step Guidance

1. Start with the right side: $\frac{-2 \cot \theta}{1 + \cot^2 \theta}$.

2. Express $\cot \theta$ and $1 + \cot^2 \theta$ in terms of sine and cosine.

3. Simplify the denominator using the Pythagorean identity.

4. Simplify the entire expression to see if it matches $-\sin 2\theta$.

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Q6. Given $\cos s = -\frac{4}{5}$ and $\sin t = \frac{5}{13}$, with $s$ and $t$ in quadrant II, find:

• (a) $\sin(s + t)$

• (b) $\tan(s + t)$

• (c) The quadrant of $s + t$

Background

Topic: Sum and Difference Formulas

This question tests your ability to use sum formulas for sine and tangent, and to determine the quadrant of an angle.

Key Terms and Formulas

• $\sin(s + t) = \sin s \cos t + \cos s \sin t$

• $\tan(s + t) = \frac{\tan s + \tan t}{1 - \tan s \tan t}$

Step-by-Step Guidance

1. Find $\sin s$ and $\cos t$ using the Pythagorean identity and the given quadrant information.

2. Plug the values into the sum formula for $\sin(s + t)$.

3. Find $\tan s$ and $\tan t$ using $\sin$ and $\cos$ values.

4. Plug into the tangent sum formula to find $\tan(s + t)$.

5. Determine the sign of $\sin(s + t)$ and $\cos(s + t)$ to identify the quadrant of $s + t$.

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Q7. Given $\cos \theta = -\frac{20}{29}$ and $\sin \theta < 0$, determine $\sin 2\theta$ and $\cos 2\theta$.

Background

Topic: Double Angle Formulas

This question tests your ability to use double angle identities for sine and cosine, and to determine the correct sign based on quadrant information.

Key Terms and Formulas

• $\sin 2\theta = 2 \sin \theta \cos \theta$

• $\cos 2\theta = 2 \cos^2 \theta - 1$

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

Step-by-Step Guidance

1. Use the Pythagorean identity to solve for $\sin \theta$ (choose the correct sign based on $\sin \theta < 0$).

2. Calculate $\sin 2\theta$ using $2 \sin \theta \cos \theta$.

3. Calculate $\cos 2\theta$ using $2 \cos^2 \theta - 1$.

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Q8. Find the exact value of $y = \arccos\left(\frac{3}{2}\right)$ if it exists.

Background

Topic: Inverse Trigonometric Functions

This question tests your understanding of the domain and range of the arccosine function.

Key Terms and Formulas

• Domain of $\arccos x$: $-1 \leq x \leq 1$

• Range of $\arccos x$: $0 \leq y \leq \pi$

Step-by-Step Guidance

1. Check if $\frac{3}{2}$ is within the domain of the arccosine function.

2. If it is not, state that the value does not exist.

3. If it is, find the angle whose cosine is $\frac{3}{2}$.

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Q9. Give the exact value of $\sin\left(2 \cos^{-1}\left(\frac{1}{13}\right)\right)$.

Background

Topic: Double Angle and Inverse Trigonometric Functions

This question tests your ability to combine inverse trigonometric functions with double angle identities.

Key Terms and Formulas

• Let $\theta = \cos^{-1}\left(\frac{1}{13}\right)$, so $\cos \theta = \frac{1}{13}$

• $\sin 2\theta = 2 \sin \theta \cos \theta$

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

Step-by-Step Guidance

1. Let $\theta = \cos^{-1}\left(\frac{1}{13}\right)$, so $\cos \theta = \frac{1}{13}$.

2. Find $\sin \theta$ using the Pythagorean identity (choose the positive root if $\theta$ is in $[0, \pi]$).

3. Use the double angle formula $\sin 2\theta = 2 \sin \theta \cos \theta$.

4. Substitute the values for $\sin \theta$ and $\cos \theta$ into the formula.

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Q10. Write $\sec\left(\arccot\left(\frac{4 - u^2}{u}\right)\right)$ as an algebraic expression in $u$, for $u > 0$.

Background

Topic: Trigonometric Expressions and Inverses

This question tests your ability to rewrite trigonometric expressions involving inverse cotangent in terms of algebraic expressions.

Key Terms and Formulas

• If $\theta = \arccot(x)$, then $\cot \theta = x$

• $\sec \theta = \frac{1}{\cos \theta}$

• Right triangle relationships: $\cot \theta = \frac{\text{adjacent}}{\text{opposite}}$

Step-by-Step Guidance

1. Let $\theta = \arccot\left(\frac{4 - u^2}{u}\right)$, so $\cot \theta = \frac{4 - u^2}{u}$.

2. Draw a right triangle with adjacent side $4 - u^2$ and opposite side $u$.

3. Find the hypotenuse using the Pythagorean theorem.

4. Express $\cos \theta$ in terms of $u$ and use $\sec \theta = \frac{1}{\cos \theta}$.

Try solving on your own before revealing the answer!